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10QBism, the Perimeter of Quantum Bayesianism

Christopher A. FuchsPerimeter Institute for Theoretical Physics

Waterloo, Ontario N2L 2Y5, Canada

cfuchs@perimeterinstitute.ca

This article summarizes the Quantum Bayesian [17] pointof view of quantum mechanics, with special emphasis on theviews outer edgesdubbed QBism.1 QBism has its rootsin personalist Bayesian probability theory, is crucially depen-dent upon the tools of quantum information theory, and mostrecently, has set out to investigate whether the physical worldmight be of a type sketched by some false-started philosophiesof 100 years ago (pragmatism, pluralism, nonreductionism,and meliorism). Beyond conceptual issues, work at PerimeterInstitute is focussed on the hard technical problem of findinga good representation of quantum mechanics purely in termsof probabilities, without amplitudes or Hilbert-space opera-tors. The best candidate representation involves a mysteriousentity called a symmetric informationally complete quantummeasurement. Contemplation of it gives a way of thinkingof the Born Rule as an addition to the rules of probabil-ity theory, applicable when an agent considers gambling onthe consequences of his interactions with a newly recognizeduniversal capacity: dimension (formerly Hilbert-space dimen-sion). (The word capacity should conjure up an image ofsomething like gravitational massa bodys mass measuresits capacity to attract other bodies. With hindsight one cansay that the founders of quantum mechanics discovered an-other universal capacity, dimension.) The article ends byshowing that the egocentric elements in QBism represent noimpediment to pursuing quantum cosmology and outliningsome directions for future work.

I. A FEARED DISEASE

The start of the new decade has just passed and so hasthe media frenzy over the H1N1 flu pandemic. Both arewelcome events. Yet, as misplaced as the latter turnedout to be, it did serve to remind us of a basic truth: Thata healthy body can be stricken with a fatal disease which

1Quantum Bayesianism, as it is called in the literature, usu-ally refers to a point of view on quantum states originallydeveloped by C. M. Caves, C. A. Fuchs, and R. Schack. Thepresent work, however, goes far beyond those statements inthe metaphysical conclusions it drawsso much so that theauthor cannot comfortably attribute the thoughts herein tothe triumvirate as a whole. Thus, the term QBism to marksome distinction from the known common ground of QuantumBayesianism. Needless to say, the author takes sole responsi-bility for any inanities herein.

to outward appearances is nearly identical to a commonyearly annoyance. There are lessons here for quantummechanics. In the history of physics, there has neverbeen a healthier body than quantum theory; no theoryhas ever been more all-encompassing or more powerful.Its calculations are relevant at every scale of physical ex-perience, from subnuclear particles, to table-top lasers, tothe cores of neutron stars and even the first three min-utes of the universe. Yet since its founding days, manyphysicists have feared that quantum theorys commonannoyancethe continuing feeling that something at thebottom of it does not make sensemay one day turn outto be the symptom of something fatal.There is something about quantum theory that is dif-

ferent in character from any physical theory posed before.To put a finger on it, the issue is this: The basic state-ment of the theorythe one we have all learned from ourtextbooksseems to rely on terms our intuitions balk atas having any place in a fundamental description of re-ality. The notions of observer and measurement aretaken as primitive, the very starting point of the theory.This is an unsettling situation! Shouldnt physics be talk-ing about what is before it starts talking about what willbe seen and who will see it? Perhaps no one has put thepoint more forcefully than John Stewart Bell [8]:

What exactly qualifies some physical systems toplay the role of measurer? Was the wavefunc-tion of the world waiting to jump for thousands ofmillions of years until a single-celled living crea-ture appeared? Or did it have to wait a littlelonger, for some better qualified system . . . witha PhD?

One sometimes gets the feelingand this is what unifiesmany a diverse quantum foundations researcherthatuntil this issue is settled, fundamental physical theoryhas no right to move on. Worse yet, that to the extent itdoes move on, it does so only as the carrier of somethinginsidious, something that will eventually cause the wholeorganism to stop in its tracks. Dark matter and darkenergy? Might these be the first symptoms of somethingsystemic? Might the problem be much deeper than get-ting our quantum fields wrong? This is the kind offear at work here.So the field of quantum foundations is not unfounded;

it is absolutely vital to physics as a whole. But what con-stitutes progress in quantum foundations? How wouldone know progress if one saw it? Through the years, itseems the most popular strategy has taken its cue (even

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if only subliminally) from the tenor of John Bells quote:The idea has been to remove the observer from the theoryjust as quickly as possible, and with surgical precision. Inpractice this has generally meant to keep the mathemat-ical structure of quantum theory as it stands (complexHilbert spaces, operators, tensor products, etc.), but, byhook or crook, find a way to tell a story about the math-ematical symbols that involves no observers at all.In short, the strategy has been to reify or objectify

all the mathematical symbols of the theory and thenexplore whatever comes of the move. Three examplessuffice to give a feel: In the de Broglie Bohm pilotwave version of quantum theory, there are no funda-mental measurements, only particles flying around ina 3N -dimensional configuration space, pushed around bya wave function regarded as a real physical field in thatspace. In spontaneous collapse versions, systems areendowed with quantum states that generally evolve uni-tarily, but from time-to-time collapse without any needfor measurement. In Everettian or many-worlds quan-tum mechanics, it is only the world as a wholetheycall it a multiversethat is really endowed with an in-trinsic quantum state, and that quantum state evolvesdeterministically, with only an illusion from the inside ofprobabilistic branching.The trouble with all these interpretations as quick fixes

for Bells hard-edged remark is that they look to be justthat, really quick fixes. They look to be interpretivestrategies hardly compelled by the particular details ofthe quantum formalism, giving only more or less arbi-trary appendages to it. This already explains in partwhy we have been able to exhibit three such differentstrategies, but it is worse: Each of these strategies givesrise to its own set of incredibilitiesones which, if onewere endowed with Bells gift for the pen, one could makelook just as silly. Pilot-wave theories, for instance, giveinstantaneous action at a distance, but not actions thatcan be harnessed to send detectable signals. If so, thenwhat a delicately balanced high-wire act nature presentsus with. Or take the Everettians. Their world purportsto have no observers, but then it has no probabilities ei-ther. What are we then to do with the Born Rule forcalculating quantum probabilities? Throw it away andsay it never mattered? It is true that quite an efforthas been made by the Everettians to rederive the rulefrom decision theory. Of those who take the point seri-ously, some think it works [9], some dont [10]. But out-side the sprachspiel who could ever believe? No amountof sophistry can make decision anything other than ahollow concept in a predetermined world.

II. QUANTUM STATES DO NOT EXIST

There is another lesson from the H1N1 virus. It isthat sometimes immunities can be found in unexpectedpopulations. To some perplexity, it seems that people

over 65a population usually more susceptible to fatali-ties with seasonal flufare better than younger folk withH1N1. No one knows exactly why, but the leading the-ory is that the older population, in its years of otherexposures, has developed various latent antibodies. Theantibodies are not perfect, but they are a start. And soit may be for quantum foundations.Here, the latent antibody is the concept of information,

and the perfected vaccine, we believe, will arise in partfrom the theory of single-case, personal probabilitiesthe branch of probability theory called Bayesianism.Symbolically, the older population corresponds to someof the very founders of quantum theory (Heisenberg,Pauli, Einstein)2 and some of the younger disciples ofthe Copenhagen school (Rudolf Peierls, John ArchibaldWheeler, Asher Peres), who, though they disagreed onmany details of the visionWhose information? Infor-mation about what?were unified on one point: Thatquantum states are not something out there, in the exter-nal world, but instead are expressions of information. Be-fore there were people using quantum theory as a branchof physics, before they were calculating neutron-capturecross-sections for uranium and working on all the otherpractical problems the theory suggests, there were noquantum states. The world may be full of stuff and thingsof all kinds, but among all the stuff and all the things,there is no unique, observer-independent, quantum-statekind of stuff.The immediate payoff of this strategy is that it elimi-

nates the conundrums arising in the various objectified-state interpretations. A paraphrase of a quote by JamesHartle makes the point decisively [11]:

A quantum-mechanical state being a sum-mary of the observers information about an indi-vidual physical system changes both by dynam-ical laws, and whenever the observer acquiresnew information about the system through theprocess of measurement. The existence of twolaws for the evolution of the state vector becomesproblematical only if it is believed that the statevector is an objective property of the system. If,however, the state of a system is defined as alist of [experimental] propositions together withtheir [probabilities of occurrence], it is not sur-prising that after a measurement the state mustbe changed to be in accord with [any] new infor-mation. The reduction of the wave packet doestake place in the consciousness of the observer,not because of any unique physical process whichtakes place there, but only because the state isa construct of the observer and not an objectiveproperty of the physical system.

2I feel guilty not mentioning Bohr here, but he so rarelytalked directly about quantum states that I fear anything Isay would be misrepresentative.

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It says that the real substance of Bells fear is just that,the fear itself. To succumb to it is to block the way tounderstanding the theory on its own terms. Moreover,the shriller notes of Bells rhetoric are the least of theworries: The universe didnt have to wait billions of yearsto collapse its first wave functionwave functions are notpart of the observer-independent world.But this much of the solution is an elderly and some-

what ineffective antibody. Its presence is mostly a callfor more clinical research. Luckily the days for this areripe, and it has much to do with the development of thefield of quantum information theory in the last 15 yearsthat is, the multidisciplinary field that has brought aboutquantum cryptography, quantum teleportation, and willone day bring about full-blown quantum computation.Terminology can say it all: A practitioner in this field,whether she has ever thought an ounce about quantumfoundations, is just as likely to say quantum informa-tion as quantum state when talking of any |. Whatdoes the quantum teleportation protocol do? A nowcompletely standard answer would be: It transfers quan-tum information from Alices site to Bobs. What wehave here is a change of mindset [6].What the facts and figures, protocols and theorems

of quantum information pound home is the idea thatquantum states look, act, and feel like information in thetechnical sense of the wordthe sense provided by prob-ability theory and Shannons information theory. Thereis no more beautiful demonstration of this than RobertSpekkenss toy model for mimicking various features ofquantum mechanics [12]. In that model, the toys areeach equipped with four possible mechanical configura-tions; but the players, the manipulators of the toys, areconsistently impededfor whatever reason!from hav-ing more than one bit of information about each toysactual configuration. (Or a total of two bits for each twotoys, three bits for each three toys, and so on.) The onlythings the players can know are their states of uncer-tainty about the configurations. The wonderful thing isthat these states of uncertainty exhibit many of the char-acteristics of quantum information: from the no-cloningtheorem to analogues of quantum teleportation, quantumkey distribution, entanglement monogamy, and even in-terference in a Mach-Zehnder interferometer. More thantwo dozen quantum phenomena are reproduced qualita-tively, and all the while one can always pinpoint the un-derlying cause of the occurrence: The phenomena arise inthe uncertainties, never in the mechanical configurations.It is the states of uncertainty that mimic the formal ap-paratus of quantum theory, not the toys so-called onticstates (states of reality).What considerations like this tell the -ontologists3

3Not to be confused with Scientologists. This neologism wascoined by Chris Granade, a Perimeter Scholars International

i.e., those who to attempt to remove the observertoo quickly from quantum mechanics by giving quan-tum states an unfounded ontic statuswas well put bySpekkens:

[A] proponent of the ontic view might argue thatthe phenomena in question are not mysterious ifone abandons certain preconceived notions aboutphysical reality. The challenge we offer to such aperson is to present a few simple physical prin-ciples by the light of which all of these phe-nomena become conceptually intuitive (and notmerely mathematical consequences of the formal-ism) within a framework wherein the quantumstate is an ontic state. Our impression is that thischallenge cannot be met. By contrast, a singleinformation-theoretic principle, which imposes aconstraint on the amount of knowledge one canhave about any system, is sufficient to derive allof these phenomena in the context of a simple toytheory . . .

The point is, far from being an appendage cheaply tackedon to the theory, the idea of quantum states as informa-tion has a simple unifying power that goes some waytoward explaining why the theory has the very mathe-matical structure it does.4 By contrast, who could takethe many-worlds idea and derive any of the structure ofquantum theory out of it? This would be a bit like try-ing to regrow a lizard from the tip of its chopped-off tail:The Everettian conception never purported to be morethan a reaction to the formalism in the first place.There are, however, aspects of Bells challenge (or at

least the mindset behind it), that remain a worry. Andupon these, all could still topple. There are the oldquestions of Whose information? and Information aboutwhat?these certainly must be addressed before any vac-cination can be declared a success. It must also be settledwhether quantum theory is obligated to give a criterionfor what counts as an observer. Finally, because no onewants to give up on physics, we must tackle head-on themost crucial question of all: If quantum states are notpart of the stuff of the world, then what is? What sort

student at Perimeter Institute, and brought to the authorsattention by R. W. Spekkens, who pounced on it for its beau-tiful subtlety.4We say goes some way toward because, though the toymodel makes about as compelling a case as we have ever seenthat quantum states are states of information (an extremelyvaluable step forward), it gravely departs from quantum the-ory in other aspects. For instance, by its nature, it can give noBell inequality violations or analogues of the Kochen-Speckernoncolorability theorems. Later sections of this paper will in-dicate that the cause of the deficit is that the toy model differscrucially from quantum theory in its answer to the questionInformation about what?

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of stuff does quantum mechanics say the world is madeof?Good immunology does not come easily. But this much

is sure: The glaringly obvious (that a large part of quan-tum theory, the central part in fact, is about information)should not be abandoned rashly: To do so is to lose gripof the theory as it is applied in practice, with no bet-ter grasp of reality in return. If on the other hand, oneholds fast to the central point about information, initiallyfrightening though it may be, one may still be able to re-construct a picture of reality from the unfocused edge ofvision. Often the best stories come from there anyway.

III. QUANTUM BAYESIANISM

Every area of human endeavor has its bold extremes.Ones that say, If this is going to be done right, we mustgo this far. Nothing less will do. In probability theory,the bold extreme is the personalist Bayesian account ofit [13]. It says that probability theory is of the characterof formal logica set of criteria for testing consistency.In the case of formal logic, the consistency is betweentruth values of propositions. However logic itself does nothave the power to set the truth values it manipulates. Itcan only say if various truth values are consistent or in-consistent; the actual values come from another source.Whenever logic reveals a set of truth values to be incon-sistent, one must dip back into the source to find a wayto alleviate the discord. But precisely in which way toalleviate it, logic gives no guidance. Is the truth valuefor this one isolated proposition correct? Logic itself ispowerless to say.The key idea of personalist Bayesian probability the-

ory is that it too is a calculus of consistency (or coher-ence as the practitioners call it), but this time for onesdecision-making degrees of belief. Probability theory canonly say if various degrees of belief are consistent or in-consistent with each other. The actual beliefs come fromanother source, and there is nowhere to pin their respon-sibility but on the agent who holds them. Dennis Lindleyput it nicely in his book Understanding Uncertainty [14]:

The Bayesian, subjectivist, or coherent, para-digm is egocentric. It is a tale of one personcontemplating the world and not wishing to bestupid (technically, incoherent). He realizes thatto do this his statements of uncertainty must beprobabilistic.

A probability assignment is a tool an agent uses to makegambles and decisionsit is a tool he uses for navigat-ing life and responding to his environment. Probabilitytheory as a whole, on the other hand, is not about asingle isolated belief, but about a whole mesh of them.When a belief in the mesh is found to be incoherent withthe others, the theory flags the inconsistency. However,it gives no guidance for how to mend any incoherencesit finds. To alleviate the discord, one can only dip back

into the source of the assignmentsspecifically, the agentwho attempted to sum up all his history, experience, andexpectations with those assignments in the first place.This is the reason for the terminology that a probabilityis a degree of belief rather than a degree of truth ordegree of facticity.Where personalist Bayesianism breaks away the most

from other developments of probability theory is that itsays there are no external criteria for declaring an iso-lated probability assignment right or wrong. The onlybasis for a judgment of adequacy comes from the inside,from the greater mesh of beliefs the agent may have thetime or energy to access when appraising coherence.It was not an arbitrary choice of words to title the

previous section QUANTUM STATES DO NOT EXIST,but a hint of the direction we must take to develop a per-fected vaccine. This is because the phrase has a precursorin a slogan Bruno de Finetti, the founder of personalistBayesianism, used to vaccinate probability theory itself.In the preface to his seminal book [15], de Finetti writes,centered in the page and in all capital letters,

PROBABILITY DOES NOT EXIST.

It is a powerful statement, constructed to put a finger onthe single most-significant cause of conceptual problemsin pre-Bayesian probability theory. A probability is nota solid object, like a rock or a tree that the agent mightbump into, but a feeling, an estimate inside himself.Previous to Bayesianism, probability was often

thought to be a physical property5something objectiveand having nothing to do with decision-making or agentsat all. But when thought so, it could be thought onlyinconsistently so. And hell hath no fury like an inconsis-tency scorned. The trouble is always the same in all itsvaried and complicated forms: If probability is to be aphysical property, it had better be a rather ghostly oneone that can be told of in campfire stories, but never quiteprodded out of the shadows. Heres a sample dialogue:

Pre-Bayesian: Ridiculous, probabilities arewithout doubt objective. They can be seenin the relative frequencies they cause.

Bayesian: So if p = 0.75 for some event, after1000 trials well see exactly 750 such events?

Pre-Bayesian: You might, but most likely youwont see that exactly. Youre just likely tosee something close to it.

Bayesian: Likely? Close? How do you define orquantify these things without making refer-ence to your degrees of belief for what willhappen?

5Witness Richard von Mises, who even went so far as towrite, Probability calculus is part of theoretical physics inthe same way as classical mechanics or optics, it is an entirelyself-contained theory of certain phenomena . . . [16].

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Pre-Bayesian: Well, in any case, in the infinitelimit the correct frequency will definitelyoccur.

Bayesian: How would I know? Are you sayingthat in one billion trials I could not pos-sibly see an incorrect frequency? In onetrillion?

Pre-Bayesian: OK, you can in principle seean incorrect frequency, but itd be ever lesslikely!

Bayesian: Tell me once again, what does likelymean?

This is a cartoon of course, but it captures the essenceand the futility of every such debate. It is better to admitat the outset that probability is a degree of belief, anddeal with the world on its own terms as it coughs up itsobjects and events. What do we gain for our theoreticalconceptions by saying that along with each actual eventthere is a ghostly spirit (its objective probability, itspropensity, its objective chance) gently nudging itto happen just as it did? Objects and events are enoughby themselves.Similarly for quantum mechanics. Here too, if ghostly

spirits are imagined behind the actual events producedin quantum measurements, one is left with conceptualtroubles to no end. The defining feature of QuantumBayesianism [16] is that it says along the lines of deFinetti, If this is going to be done right, we must gothis far. Specifically, there can be no such thing as aright and true quantum state, if such is thought of as de-fined by criteria external to the agent making the assign-ment: Quantum states must instead be like personalist,Bayesian probabilities.The direct connection between the two foundational

issues is this. Quantum states, through the Born Rule,can be used to calculate probabilities. Conversely, if oneassigns probabilities for the outcomes of a well-selectedset of measurements, then this is mathematically equiva-lent to making the quantum-state assignment itself. Thetwo kinds of assignments determine each other uniquely.Just think of a spin- 12 system. If one has elicited onesdegrees of belief for the outcomes of a x measurement,and similarly ones degrees of belief for the outcomes ofy and z measurements, then this is the same as speci-fying a quantum state itself: For if one knows the quan-tum states projections onto three independent axes, thenthat uniquely determines a Bloch vector, and hence aquantum state. Something similar is true of all quan-tum systems of all sizes and dimensionality. There isno mathematical fact embedded in a quantum state that is not embedded in an appropriately chosen set ofprobabilities.6 Thus generally, if probabilities are per-

6See Section IV where this statement is made precise in alldimensions.

sonal in the Bayesian sense, then so too must be quantumstates.What this buys interpretatively, beside airtight consis-

tency with the best understanding of probability theory,is that it gives each quantum state a home. Indeed, ahome localized in space and timenamely, the physicalsite of the agent who assigns it! By this method, oneexpels once and for all the fear that quantum mechan-ics leads to spooky action at a distance, and expels aswell any hint of a problem with Wigners friend [17]. Itdoes this because it removes the very last trace of confu-sion over whether quantum states might still be objective,agent-independent, physical properties.The innovation here is that, for most of the history

of efforts to take an informational point of view aboutquantum states, the supporters of the idea have tried tohave it both ways: that on the one hand quantum statesare not real physical properties, yet on the other there isa right quantum state independent of the agent after all.For instance, one hears things like, The right quantumstate is the one the agent should adopt if he had all theinformation. The tension in these two desires leavestheir holders open to attack on both flanks and generalconfusion all around.Take first instantaneous action at a distancethe hor-

ror of this idea is often one of the strongest motivationsfor those seeking to take an informational stance on quan-tum states. But, now an opponent can say:

If there is a right quantum state, then whynot be done with all this squabbling and call thestate a physical fact to begin with? It is surelyexternal to the agent if the agent can be wrongabout it. But, once you admit that (and youshould admit it), youre sunk: For, now what re-course do you have to declare no action at a dis-tance when a delocalized quantum state changesinstantaneously?

Here I am with a physical system right infront of me, and though my probabilities for theoutcomes of measurements I can do on it mighthave been adequate a moment ago, there is anobjectively better way to gamble now becauseof something that happened far in the distance?(Far in the distance and just now.) How couldthat not be the signature of action at a dis-tance? You can try to defend yourself by say-ing quantum mechanics is all about relations7

7A typical example is of a woman traveling far from homewhen her husband divorces her. Instantaneously she becomesunmarriedmarriage is a relational property, not somethinglocalized at each partner. It seems to be popular to give thisexample and say, Quantum mechanics might be like that.The conversation usually stops without elaboration, but letscarry it a little further: Suppose the woman is right in frontof me. Would the far-off divorce mean that there is instanta-

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or some other feel-good phrase, but Im talkingabout measurements right here, in front of me,with outcomes I can see right now. Ones enter-ing my awarenessnot outcomes in the mind ofGod who can see everything and all relations. Itis that which I am gambling upon with the helpof the quantum formalism. An objectively betterquantum state would mean that my gambles andactions, though they would have been adequatea moment ago, are now simply wrong in the eyesof the worldthey could have been better. Howcould the quantum system in front of me generateoutcomes instantiating that declaration withoutbeing privy to what the eyes of the world alreadysee? Thats action at a distance, I say, or at leasta holism that amounts to the same thingtheresnothing else it could be.

Without the protection of truly personal quantum-state assignments, action at a distance is there asdoggedly as it ever was. And things only get worse withWigners friend if one insists there be a right quantumstate. As it turns out, the method of mending this co-nundrum displays one of the most crucial ingredients ofQBism. Let us put it in plain sight.Wigners friend is the story of two agents, Wigner

and his friend, and one quantum systemthe only de-viation we make from a more common presentation8 isthat we put the story in informational terms. It starts offwith the friend and Wigner having a conversation: Sup-pose they both agree that some quantum state | cap-tures their mutual beliefs about the quantum system.9

Furthermore suppose they agree that at a specified timethe friend will make a measurement on the system ofsome observable (outcomes i = 1, . . . , d). Finally, theyboth note that if the friend gets outcome i, he will (andshould) update his beliefs about the system to some newquantum state |i. There the conversation ends and theaction begins: Wigner walks away and turns his back tohis friend and the supposed measurement. Time passesto some point beyond when the measurement should havetaken place.

neously a different set of probabilities I could use for weighingthe consequences of trying to seduce her? Not at all. I wouldhave no account to change my probabilities (not for this rea-son anyway) until I became aware of her changed relation,however long it might take that news to get to me.8For instance, [18] is about as common as they get.9Being Bayesians, of course, they dont have to agree at thisstagefor recall | is not a physical fact for them, only acatalogue of beliefs. But suppose they do agree.

FIG. 1. In contemplating a quantum measurement, onemakes a conceptual split in the world: one part is treatedas an agent, and the other as a kind of reagent or catalyst(one that brings about change in the agent itself). The latteris a quantum system of some finite dimension d. A quantummeasurement consists first in the agent taking an action onthe quantum system. The action is represented formally bya set of operators {Ei}a positive-operator-valued measure.The action generally leads to an incompletely predictable con-sequence Ei for the agent. The quantum state | makes noappearance but in the agents head; for it captures his degreesof belief concerning the consequences of his actions, and, incontrast to the quantum system itself, has no existence in theexternal world. Measurement devices are depicted as pros-thetic hands to make it clear that they should be consideredan integral part of the agent. The sparks between the mea-surement-device hand and the quantum system represent theidea that the consequence of each quantum measurement is aunique creation within the previously existing universe. Twopoints are decisive in distinguishing this picture of quantummeasurement from a kind of solipsism: 1) The conceptual splitof agent and external quantum system: If it were not needed,it would not have been made. 2) Once the agent chooses anaction {Ei} to take, the particular consequence Ek of it is be-yond his controlthat is, the actual outcome is not a productof his whim and fancy.

What now is the correct quantum state each agentshould have assigned to the quantum system? We havealready concurred that the friend will and should assignsome |i. But what of Wigner? If he were to consistentlydip into his mesh of beliefs, he would very likely treat hisfriend as a quantum system like any other: one with someinitial quantum state capturing his (Wigners) beliefsof it (the friend), along with a linear evolution operator10

10We suppose for the sake of introducing less technicality

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U to adjust those beliefs with the flow of time.11 Sup-pose this quantum state includes Wigners beliefs abouteverything he assesses to be interacting with his friendin old parlance, suppose Wigner treats his friend as anisolated system.From this perspective, before any furtherinteraction between himself and the friend or the othersystem, the quantum state Wigner would assign for thetwo together would be U

( ||

)U most gener-

ally an entangled quantum state. The state of the systemitself for Wigner would be gotten from this larger stateby a partial trace operation; in any case, it will not bean |i.Does this make Wigners new state assignment incor-

rect? After all, if he had all the information (i.e., allthe facts of the world) wouldnt that include knowing thefriends measurement outcome? Since the friend shouldassign some |i, shouldnt Wigner himself (if he had allthe information)? Or is it the friend who is incorrect?For if the friend had all the information, wouldnt hesay that he is neglecting that Wigner could put the sys-tem and himself into the quantum computational equiv-alent of an iron lung and forcefully reverse the so-calledmeasurement? I.e., Wigner, if he were sufficiently sophis-ticated, should be able to force

U( ||

)U || . (1)

And so the back and forth goes. Who has the right stateof information? The conundrums simply get too heavy ifone tries to hold to an agent-independent notion of cor-rectness for otherwise personalistic quantum states. TheQuantum Bayesian dispels these and similar difficulties ofthe aha, caught you! variety by being conscientiouslyforthright. Whose information? Mine! Informationabout what? The consequences (for me) of my actionsupon the physical system! Its all I-I-me-me mine, asthe Beatles sang.The answer to the first question surely comes as no sur-

prise by now, but why on earth the answer for the second?Its like watching a Quantum Bayesian shoot himself inthe foot, a friend once said. Why something so ego-centric, anthropocentric, psychology-laden, myopic, andpositivistic (weve heard any number of expletives) asthe consequences (for me) of my actions upon the sys-tem? Why not simply say something neutral like theoutcomes of measurements? Or, fall in line with Wolf-gang Pauli and say [21]:

that U is a unitary operation, rather than the more generalcompletely positive trace-preserving linear maps of quantuminformation theory [19]. This, however, is not essential to theargument.11For an explanation of the status of unitary operations fromthe QBist perspective, as personal judgments directly anal-ogous to quantum states themselves, see Footnote 22 andRefs. [2,5,20].

The objectivity of physics is . . . fully ensured inquantum mechanics in the following sense. Al-though in principle, according to the theory, it isin general only the statistics of series of experi-ments that is determined by laws, the observeris unable, even in the unpredictable single case,to influence the result of his observationas forexample the response of a counter at a particu-lar instant of time. Further, personal qualitiesof the observer do not come into the theory inany waythe observation can be made by objec-tive registering apparatus, the results of whichare objectively available for anyones inspection.

To the uninitiated, our answer for Information aboutwhat? surely appears to be a cowardly, unnecessary re-treat from realism. But it is the opposite. The answer wegive is the very injunction that keeps the potentially con-flicting statements of Wigner and his friend in check,12 atthe same time as giving each agent a hook to the externalworld in spite of QBisms egocentric quantum states.You see, for the QBist, the real world, the one both

agents are embedded inwith its objects and eventsis taken for granted. What is not taken for grantedis each agents access to the parts of it he has nottouched. Wigner holds two thoughts in his head: 1)that his friend interacted with a quantum system, elicit-ing some consequence of the interaction for himself, and2) after the specified time, for any of Wigners own fur-ther interactions with his friend or system or both, heought to gamble upon their consequences according toU( ||

)U . One statement refers to the friends

potential experiences, and one refers to Wigners own.So long as it is kept clear that U

( ||

)U refers to

the latterhow Wigner should gamble upon the thingsthat might happen to himmaking no statement what-soever about the former, there is no conflict. The worldis filled with all the same things it was before quantumtheory came along, like each of our experiences, that rockand that tree, and all the other things under the sun; itis just that quantum theory provides a calculus for gam-bling on each agents own experiencesit doesnt giveanything else than that. It certainly doesnt give oneagent the ability to conceptually pierce the other agentspersonal experience. It is true that with enough effortWigner could enact Eq. (1), causing him to predict thathis friend will have amnesia to any future questions on hisold measurement results. But we always knew Wignercould do thata mallet to the head would have beengood enough.

12Paulis statement certainly wouldnt have done that. Re-sults objectively available for anyones inspection? This isthe whole issue with Wigners friend in the first place. Ifboth agents could just look at the counter simultaneouslywith negligible effect in principle, we would not be having thisdiscussion.

7

The key point is that quantum theory, from this light,takes nothing away from the usual world of common ex-perience we already know. It only adds.13 At the veryleast it gives each agent an extra tool with which to nav-igate the world. More than that, the tool is here fora reason. QBism says when an agent reaches out andtouches a quantum systemwhen he performs a quantummeasurementthat process gives rise to birth in a nearlyliteral sense. With the action of the agent upon the sys-tem, the no-go theorems of Bell and Kochen-Specker as-sert that something new comes into the world that wasntthere previously: It is the outcome, the unpredictableconsequence for the very agent who took the action. JohnArchibald Wheeler said it this way, and we follow suit,Each elementary quantum phenomenon is an elemen-tary act of fact creation. [23]With this much, QBism has a story to tell on both

quantum states and quantum measurements, but what ofquantum theory as a whole? The answer is found in tak-ing it as a universal single-user theory in much the sameway that Bayesian probability theory is. It is a usersmanual that any agent can pick up and use to help makewiser decisions in this world of inherent uncertainty.14

To say it in a more poignant way: In my case, it is aworld in which I am forced to be uncertain about theconsequences of most of my actions; and in your case, itis a world in which you are forced to be uncertain aboutthe consequences of most of your actions. And what ofGods case? What is it for him? Trying to give him aquantum state was what caused this trouble in the firstplace! In a quantum mechanics with the understandingthat each instance of its use is strictly single-userMymeasurement outcomes happen right here, to me, and Iam talking about my uncertainty of them.there is noroom for most of the standard, year-after-year quantum

13This point will be much elaborated on in the Section VI.14 Most of the time one sees Bayesian probabilities character-ized (even by very prominent Bayesians like Edwin T. Jaynes[22]) as measures of ignorance or imperfect knowledge. Butthat description carries with it a metaphysical commitmentthat is not at all necessary for the personalist Bayesian, whereprobability theory is an extension of logic. Imperfect knowl-edge? It sounds like something that, at least in imagination,could be perfected, making all probabilities zero or oneoneuses probabilities only because one does not know the true,pre-existing state of affairs. Language like this, the readerwill notice, is never used in this paper. All that matters for apersonalist Bayesian is that there is uncertainty for whateverreason. There might be uncertainty because there is igno-rance of a true state of affairs, but there might be uncertaintybecause the world itself does not yet know what it will givei.e., there is an objective indeterminism. As will be argued inlater sections, QBism finds its happiest spot in an unflinch-ing combination of subjective probability with objectiveindeterminism.

mysteries.

FIG. 2. The Born Rule is not like the other classic lawsof physics. Its normative nature means, if anything, it ismore like the Biblical Ten Commandments. The classic lawson the left give no choice in their statement: If a field isgoing to be an electromagnetic field at all, it must satisfyMaxwells equations; it has no choice. Similarly for the otherclassic laws. Their statements are intended to be statementsconcerning nature just exactly as it is. But think of the TenCommandments. Thou shalt not steal. People steal all thetime. The role of the Commandment is to say, You have thepower to steal if you think you can get away with it, but itsprobably not in your best interest to do so. Something bad islikely to happen as a result. Similarly for Thou shalt notkill, and all the rest. It is the worshippers choice to obeyeach or not, but if he does not, he ought to count on somethingpotentially bad in return. The Born Rule guides, Gamble insuch a way that all your probabilities mesh together throughme. The agent is free to ignore the advice, but if he does so,he does so at his own peril.

The only substantive conceptual issue left before syn-thesizing a final vaccine15 is whether quantum mechanicsis obligated to derive the notion of agent for whose aidthe theory was built in the first place? The answer comesfrom turning the tables: Thinking of probability theoryin the personalist Bayesian way, as an extension of for-mal logic, would one ever imagine that the notion of anagent, the user of the theory, could be derived out ofits conceptual apparatus? Clearly not. How could youpossibly get flesh and bones out of a calculus for makingwise decisions? The logician and the logic he uses aretwo different substancesthey live in conceptual cate-gories worlds apart. One is in the stuff of the physicalworld, and one is somewhere nearer to Platos heaven ofideal forms. Look as one might in a probability textbookfor the ingredients to reconstruct the reader himself, one

15Not to worry, there are still plenty of technical ones, aswell as plenty more conceptual ones waiting for after thevaccination.

8

will never find them. So too, the Quantum Bayesian saysof quantum theory.With this we finally pin down the precise way in which

quantum theory is different in character from any phys-ical theory posed before. For the Quantum Bayesian,quantum theory is not something outside probabilitytheoryit is not a picture of the world as it is, as sayEinsteins program of a unified field theory hoped to bebut rather it is an addition to probability theory itself.As probability theory is a normative theory, not sayingwhat one must believe, but offering rules of consistencyan agent should strive to satisfy within his overall meshof beliefs, so it is the case with quantum theory.To take this substance into ones mindset is all the

vaccination one needs against the threat that quantumtheory carries something viral for theoretical physics as awhole. A healthy body is made healthier still. For withthis protection, we are for the first time in a positionto ask, with eyes wide open to what the answer couldnot be, just what after all is the world made of? Farfrom being the last word on quantum theory, QBism, webelieve, is the start of a great adventure. An adventurefull of mystery and danger, with hopes of triumph . . .and all the marks of life.

IV. SEEKING SICS THE BORN RULE AS

FUNDAMENTAL

You know how men have always hankered after unlaw-ful magic, and you know what a great part in magicwords have always played. If you have his name, . . .you can control the spirit, genie, afrite, or whateverthe power may be. Solomon knew the names of all thespirits, and having their names, he held them subjectto his will. So the universe has always appeared to thenatural mind as a kind of enigma, of which the keymust be sought in the shape of some illuminating orpower-bringing word or name. That word names theuniverses principle, and to possess it is after a fashionto possess the universe itself.

But if you follow the pragmatic method, you cannotlook on any such word as closing your quest. You mustbring out of each word its practical cash-value, set itat work within the stream of your experience. It ap-pears less as a solution, then, than as a program formore work, and more particularly as an indication ofthe ways in which existing realities may be changed.

Theories thus become instruments, not answers to

enigmas, in which we can rest. We dont lie back uponthem, we move forward, and, on occasion, make natureover again by their aid.

William James

If quantum theory is a users manual, one cannot forgetthat the world is its author. And from its writing style,one may still be able to tell something of the author her-self. The question is how to tease out the psychology ofthe style, frame it, and identify the underlying motif.Something that cannot be said of the Quantum

Bayesian program is that it has not had to earn its keep

in the larger world of quantum interpretations. Sincethe beginning, the promoters of the view have been onthe run proving technical theorems whenever required toclose a gap in its logic or negate an awkwardness inducedby its new way of speaking. It was never enough to lieback upon the pronouncements: They had to be shownto have substance, something that would drive physicsitself forward. A case in point is the quantum de Finettitheorem [3,24].This is a theorem that arose from contemplating the

meaning of one of the most common phrases of quantuminformation theorythe unknown quantum state. Theterm is ubiquitous: Unknown quantum states are tele-ported, protected with quantum error correcting codes,used to check for quantum eavesdropping, and arisein innumerable other applications. From a Quantum-Bayesian point of view, however, the phrase can only bean oxymoron, something that contradicts itself: If quan-tum states are compendia of beliefs, and not states ofnature, then the state is known to someone, at the veryleast the agent who holds it. But if so, then what are theexperimentalists doing when they say they are perform-ing quantum-state tomography in the laboratory? Thevery goal of the procedure is to characterize the unknownquantum state a piece of laboratory equipment is repeti-tively preparing. There is certainly no little agent sittingon the inside of the device devilishly sending out quan-tum systems representative of his beliefs, and smiling asan experimenter on the outside slowly homes in on thoseprivate thoughts through his experiments. What gives?The quantum de Finetti theorem is a technical result

that allows the story of quantum-state tomography to betold purely in terms of a single agentnamely, the exper-imentalist in the laboratory. In a nutshell, the theoremis this. Suppose the experimentalist walks into the labo-ratory with the very minimal belief that, of the systemshis device is spitting out (no matter how many), he couldinterchange any two of them and it would not change thestatistics he expects for any measurements he might per-form. Then the theorem says coherence alone requireshim to make a quantum-state assignment (n) (for any nof those systems) that can be represented in the form:

(n) =

P () nd , (2)

where P () d is some probability measure on the spaceof single-system density operators and n = represents an n-fold tensor product of identical quantumstates. To put it in words, this theorem licenses the ex-perimenter to act as if each individual system has somestate unknown to him, with a probability density P ()representing his ignorance of which state is the true one.But it is only as ifthe only active quantum state in thepicture is the one the experimenter (the agent) actuallypossesses, namely (n). The right-hand side of Eq. (2),though necessary among the possibilities, is just one ofmany representations for (n). When the experimenter

9

performs tomography, all he is doing is gathering datasystem-by-system and updating, via Bayes rule [25], thestate (n) to some new state (k) on a smaller number ofremaining systems. Particularly, one can prove that thisform of quantum-state assignment leads the agent to ex-pect that with more data, he will approach ever moreclosely a posterior state of the form (k) = k. This iswhy one gets into the habit of speaking of tomographyas revealing the unknown quantum state.This example is just one of several [3,26,27,20], and

what they all show is that the point of view has sometechnical crunch16it is not just stale, lifeless philoso-phy. It stands a chance to make nature over again byits aid. What better way to master a writers intentionsthan to edit her draft and see if she tolerates the changes,admitting in the end that the story flows more easily?In this regard, no question of QBism tests natures tol-

erance more probingly than this. If quantum theory isso closely allied with probability theory, if it can even beseen as an addition to it, then why is it not written in alanguage that starts with probability, rather than a lan-guage that ends with it? Why does quantum theory in-voke the mathematical apparatus of complex amplitudes,Hilbert spaces, and linear operators? This brings us topresent-day research at Perimeter Institute.For, actually there are ways to pose quantum theory

purely in terms of probabilitiesindeed, there are manyways, each with a somewhat different look and feel [29].The work of W. K. Wootters is an example, and as heemphasized long ago [30],

It is obviously possible to devise a formula-tion of quantum mechanics without probabilityamplitudes. One is never forced to use any quan-tities in ones theory other than the raw resultsof measurements. However, there is no reason toexpect such a formulation to be anything otherthan extremely ugly. After all, probability am-plitudes were invented for a reason. They are notas directly observable as probabilities, but theymake the theory simple. I hope to demonstratehere that one can construct a reasonably prettyformulation using only probabilities. It may notbe quite as simple as the usual formulation, butit is not much more complicated.

What has happened in the intervening years is that themathematical structures of quantum information theoryhave grown significantly richer than the ones he hadbased his considerations onso much so that we maynow be able to optimally re-express the theory. What

16In fact, the quantum de Finetti theorem has long left itsfoundational roots behind and found far more widespreadrecognition with its applications to quantum cryptography[28].

was once not much more complicated, now has thepromise of being downright insightful.The key ingredient is a hypothetical structure called a

symmetric informationally complete positive-operator-valued measure, or SIC (pronounced seek) for short.This is a set of d2 rank-one projection operators i =|ii| on a finite d-dimensional Hilbert space such that

i|j2 = 1d+ 1

whenever i 6= j . (3)

Because of their extreme symmetry, it turns out thatsuch sets of operators, when they exist, have three veryfine-tuned properties: 1) the operators must be linearlyindependent and span the space of Hermitian operators,2) there is a sense in which they come as close to an or-thonormal basis for operator space as they can (underthe constraint that all the elements in a basis be pos-itive semi-definite), and 3) after rescaling, they form aresolution of the identity operator, I =

i1di.

The symmetry, positive semi-definiteness, and prop-erties 1 and 2 are significant because they imply thatan arbitrary quantum state pure or mixedcan beexpressed as a linear combination of the i. Further-more, the expansion is likely to have some significantfeatures not found in other, more arbitrary expansions.The most significant of these becomes apparent when onetakes property 3 into account. Because the operatorsHi =

1di are positive semi-definite and form a resolu-

tion of the identity, they can be interpreted as labelingthe outcomes of a quantum measurement devicenot astandard-textbook, von Neumann measurement devicewhose outcomes correspond to the eigenvalues of someHermitian operator, but to a measurement device of themost general variety allowed by quantum theory, theso-called positive-operator-valued measures (POVMs)[19,31]. Particularly noteworthy is the smooth relationbetween the probabilities P (Hi) = tr

(Hi

)given by the

Born Rule for the outcomes of such a measurement17 andthe expansion coefficients for in terms of the i:

=

d2i=1

((d+ 1)P (Hi)

1

d

)i . (4)

There are no other operator bases that give rise to sucha simple formula connecting probabilities with densityoperators, and it suggests that this is just the place theQuantum Bayesian should seek his motif.

17There is a slight ambiguity in notation here, asHi is duallyused to denote an operator and an outcome of a measurement.For the sake of simplicity, we hope the reader will forgive thisand similar abuses.

10

| = 3

2

f+5f8f+3f8f+5f

+ei+

3

2

8f7f+

f7f+8f9f

++e

i

3

2

8f9f+8f7f+

f7f

where

= eipi/12

f =

3

3

g =

6

21 18

=

721

14

=

7 +

21

14

21 4228

ei =1

2

(46 6

21 6g i

18 + 6

21 6g

)13

FIG. 3. D. M. Applebys pencil-and-paper SIC indimension 6. This is an example of one vector | amongthe 36 that go together to form the simplest known SIC ind = 6. One of the many problems facing a proof of generalSIC existence is that no one has yet latched onto a universalpattern in the existing analytic solutionsevery dimensionappears to be of a distinct character.

Before getting to that, however, we should reveal whatis so consternating about the SICs: It is the questionof whether they exist at all. Despite 10 years of grow-ing effort since the definition was first introduced [3234](there are now nearly 50 papers on the subject), no onehas been able to show that they exist in completely gen-eral dimension. All that is known firmly is that they existin dimensions 2 through 67 [35]. Dimensions 2 15, 19,24, 35, and 48 are known through direct or computer-automated analytic proof; the remaining solutions areknown through numerical simulation, satisfying Eq. (3)to within a precision of 1038. How much evidence isthis that SICs exist generally? The reader must answerthis one for himself (certainly there can be no reader-independent answer to something so subjective!), but forthe remainder of the article we will proceed as if theydo always exist for finite d. At least this is the conceitof our story. We note in passing, however, that the SICexistence problem is not without wider context: if theydo exist, they solve at least three other (more practical,non-foundational) optimality problems in quantum infor-mation theory [3639]it would be a nasty trick if SICsdidnt always exist!

FIG. 4. Any quantum measurement can be conceptualizedin two ways. Suppose an arbitrary von Neumann measure-ment on the ground, with outcomes Dj = 1, . . . , d. Itsprobabilities P (Dj) can be derived by cascading it with afixed fiducial SIC measurement in the sky (of outcomesHi = 1, . . . , d

2). Let P (Hi) and P (Dj |Hi) represent anagents probabilities, assuming the measurement in the skyis actually performed. The probability Q(Dj) represents in-stead the agents probabilities under the assumption that themeasurement in the sky is not performed. The Born Rule,in this language, says that P (Dj), P (Hi), and P (Dj |Hi) arerelated by the Bayesian-style Eq. (8).

So suppose they do. Thinking of a quantum state as lit-erally an agents probability assignment for the outcomesof a potential SIC measurement leads to a new way to ex-press the Born Rule for the probabilities associated withany other quantum measurement. Consider the diagramin Figure 4. It depicts a SIC measurement in the sky,with outcomes Hi, and any standard von Neumann mea-surement on the ground.18 For the sake of specificity,let us say the latter has outcomesDj = |jj|, the vectors|j representing some orthonormal basis. We conceive oftwo possibilities (or two paths) for a given quantumsystem to get to the measurement on the ground: Path1 is that it proceeds directly to the measurement onthe ground. Path 2 is that it proceeds first to themeasurement in the sky and only subsequently to the

18Do not, however, let the designation SIC sitting in thesky make the device seem too exalted and unapproachable.Actual implementations have already been built for bothqubits [40] and qutrits [42].

11

measurement on the groundthe two measurements arecascaded.Suppose now, we are given the agents personal prob-

abilities P (Hi) for the outcomes in the sky and his con-ditional probabilities P (Dj |Hi) for the outcomes on theground subsequent to the sky. I.e., we are given the prob-abilities the agent would assign on the supposition thatthe quantum system follows Path 2. Then coherencealone (in the Bayesian sense) is enough to tell whatprobabilities P (Dj) the agent should assign for the out-comes of the measurement on the groundit is given bythe Law of Total Probability applied to these numbers:

P (Dj) =i

P (Hi)P (Dj |Hi) . (5)

That takes care of Path 2, but what of Path 1? Is thisenough information to recover the probability assignmentQ(Dj) the agent would assign for the outcomes on Path1 via a normal application of the Born Rule? That is,that

Q(Dj) = tr(Dj) (6)

for some quantum state ? Maybe, but the answer willclearly not be P (Dj). One has

Q(Dj) 6= P (Dj) (7)

simply because Path 2 is not a coherent process (in thequantum sense!) with respect to Path 1there is a mea-surement that takes place in Path 2 that does not takeplace in Path 1.What is remarkable about the SIC representation is

that it implies that, even though Q(Dj) is not equal toP (Dj), it is still a function of it. Particularly,

Q(Dj) = (d+ 1)P (Dj) 1

= (d+ 1)

d2i=1

P (Hi)P (Dj |Hi) 1 . (8)

The Born Rule is nothing but a kind of QuantumLaw of Total Probability! No complex amplitudes, nooperatorsonly probabilities in, and probabilities out.Indeed, it is seemingly just a rescaling of the old law,Eq. (5). And in a way it is.

But beware: One should not interpret Eq. (8) as inval-idating probability theory itself in any way: For the oldLaw of Total Probability has no jurisdiction in the settingof our diagram, which compares a factual experiment(Path 1) to a counterfactual one (Path 2).19 Indeed as

19Indeed, as we have emphasized, there is a trace of a veryold antibody in QBism. While writing this essay, it came tolight in the nice historical study of Ref. [41] that Born and

any Bayesian would emphasize, if there is a distinguish-ing mark in ones considerationssay, the fact of twodistinct experiments, not onethen one ought to takethat into account in ones probability assignments (atleast initially so). Thus there is a hidden, or at least sup-pressed, condition in our notation: Really we should havebeen writing the more cumbersome, but honest, expres-sions P (Hi|E2), P (Dj|Hi, E2), P (Dj|E2), and Q(Dj |E1)all along. With this explicit, it is no surprise that,

Q(Dj |E1) 6=i

P (Hi|E2)P (Dj |Hi, E2) . (9)

The message is that quantum theory supplies some-thinga new form of Bayesian coherence, though em-pirically based (as quantum theory itself is)that rawprobability theory does not. The Born Rule in theselights is an addition to Bayesian probability, not in thesense of a supplier of some kind of more-objective prob-abilities, but in the sense of giving extra normative rulesto guide the agents behavior when he interacts with thephysical world.It is a normative rule for reasoning about the conse-

quences of ones proposed actions in terms of the poten-tial consequences of an explicitly counterfactual action.It is like nothing else physical theory has contemplatedbefore. Seemingly at the heart of quantum mechanicsfrom the QBist view is a statement about the impact ofcounterfactuality. The impact parameter is metered by asingle, significant number associated with each physicalsystemits Hilbert-space dimension d. The larger the dassociated with a system, the more Q(Dj) must deviatefrom P (Dj). Of course this point must have been implicitin the usual form of the Born Rule, Eq. (6). What is im-portant from the QBist perspective, however, is how thenew form puts the significant parameter front and center,displaying it in a way that one ought to nearly trip over.Understanding this as the goal helps pinpoint the role

of SICs in our considerations. The issue is not that quan-tum mechanics must be rewritten in terms of SICs, butthat it can be.20 Certainly no one is going to drop theusual operator formalism and all the standard methodslearned in graduate school to do their workaday calcu-lations in SIC language exclusively. It is only that theSICs form an ideal coordinate system for a particularproblem (an important one to be sure, but nonetheless

Heisenberg, already at the 1927 Solvay conference, refer to the

calculation |cn(t)|2 =

mSmn(t)cm(0)

2 and say, it shouldbe noted that this interference does not represent a contra-diction with the rules of the probability calculus, that is, withthe assumption that the |Snk|

2 are quite usual probabilities.Their reasons for saying this may have been different fromour own, but at least they had come this far.20If everything goes right, that is, and the damned thingsactually exist in all dimensions!

12

a particular one)the problem of interpreting quantummechanics. The point of all the various representationsof quantum mechanics (like the various quasi-probabilityrepresentations of [29], the Heisenberg and Schrodingerpictures, and even the path-integral formulation) is thatthey give a means for isolating one or another aspect ofthe theory that might be called for by a problem at hand.Sometimes it is really important to do so, even for deepconceptual issues and even if all the representations arelogically equivalent.21 In our case, we want to bring intoplain view the idea that quantum mechanics is an ad-dition to Bayesian probability theorynot a generaliza-tion of it [43], not something orthogonal to it altogether[44], but an addition. With this goal in mind, the SICrepresentation is a particularly powerful tool. Throughit, one sees the Born Rule as a functional of a usage ofthe Law of Total Probability that one would have madein another (counterfactual) context.22 The SICs empha-

21Just think of the story of Eddington-Finkelstein coordi-nates in general relativity. Once upon a time it was notknown whether a Schwarzschild black hole might have, be-side its central singularity, a singularity in the gravitationalfield at the event horizon. Apparently it was a heated debate,yes or no. The issue was put to rest, however, with the de-velopment of the coordinate system. It allowed one to writedown a solution to the Einstein equations in a neighborhoodof the horizon and check that everything was alright after all.22 Furthermore it is similarly so of unitary time evolution ina SIC picture. To explain what this means, let us change con-siderations slightly and make the measurement on the grounda unitarily rotated version of the SIC in the sky. This con-trasts with the von Neumann measurement we have previ-ously restricted the ground measurement to be. In this set-ting, Dj =

1

dUjU

, which in turn implies a slight modifica-tion to Eq. (8),

Q(Dj) = (d+ 1)

d2i=1

P (Hi)P (Dj |Hi)1

d, (10)

for the probabilities on the ground. Note what this is saying!As the Born Rule is a functional of the Law of Total Prob-ability, unitary time evolution is a functional of it as well.For, if we thought in terms of the Schodinger picture, P (Hi)and Q(Dj) would be the SIC representations for the initialand final quantum states under an evolution given by U.The similarity is no accident. This is because in both casesthe conditional probabilities P (Dj |Hi) completely encode theidentity of a measurement on the ground.Moreover, it makes abundantly clear another point of QBismthat has not been addressed so much in the present paper.Since a personalist Bayesian cannot turn his back on the clar-ification that all probabilities are personal judgments, place-holders in a calculus of consistency, he certainly cannot turnhis back on the greater lesson Eqs. (8) and (10) are tryingto scream out. Just as quantum states are personal judg-ments P (Hi), quantum measurement operators Dj and uni-

size and make this point clear. At the end of the dayhowever, after all the foundational worries of quantumtheory are finally overcome, the SICs might in principlebe thrown away, just as the scaffolding surrounding anyfinished construction would be.Much of the most intense research of Perimeter Insti-

tutes QBism group is currently devoted to seeing howmuch of the essence of quantum theory is captured byEq. (8). For instance, one way to approach this is totake Eq. (8) as a fundamental axiom and ask what furtherassumptions are required to recover all of quantum the-ory? To give some hint of how a reconstruction of quan-tum theory might proceed along these lines, note Eq. (4)again. What it expresses is that any quantum state can be reconstructed from the probabilities P (Hi) thestate gives rise to. This, however, does not imply thatplugging just any probability distribution P (Hi) into theequation will give rise to a valid quantum state. A gen-eral probability distribution P (Hi) in the formula willlead to a Hermitian operator of trace one, but it may notlead to an operator with nonnegative eigenvalues. In-deed it takes further restrictions on the P (Hi) to makethis true. That being the case, the Quantum Bayesianstarts to wonder if these restrictions might arise from therequirement that Eq. (8) simply always make sense. Fornote, if P (Dj) is too small, Q(Dj) will go negative; andif P (Dj) is too large, Q(Dj) will become larger than 1.So, P (Dj) must be restricted. But that in turn forces theset of valid P (Hi) to be restricted as well. And so theargument goes. For sure, some amount of quantum the-ory (and maybe all of it) is reconstructed in this fashion[5,4547].Another exciting development comes from loosening

the form of Eq. (8) to something more generic:

Q(Dj) =

ni=1

P (Hi)P (Dj |Hi) , (11)

where there is initially no assumed relation between ,, and n as there is in Eq. (8). Then, under a few furtherconditions with only the faintest hint of quantum theory

tary time evolutions U are personal judgments tooin thiscase P (Dj |Hi). The only distinction is the technical one,that one expression is an unconditioned probability, while theother is a collection of conditionals. Most importantly, it set-tles the age-old issue of why there should be two kinds of stateevolution at all. When Hartle wrote, A quantum-mechanicalstate being a summary of the observers information about anindividual physical system changes both by dynamical laws,and whenever the observer acquires new information aboutthe system through the process of measurement, what is hisdynamical law making reference to? There are not two thingsthat a quantum state can do, only one: Strive to be consis-tent with all the agents other probabilistic judgments on theconsequences of his actions, factual and counterfactual.

13

in themfor instance, that there should exist measure-ments on the ground for which, under appropriate con-ditions, one can have certainty for their outcomesoneimmediately gets a significantly more restricted form forthis relation:

Q(Dj) =

(1

2qd+ 1

) ni=1

P (Hi)P (Dj |Hi)1

2q , (12)

where very interestingly the parameters q and d canonly take on integer values, q = 0, 1, 2, . . . , and d =2, 3, 4, . . . ,, and n = 12qd(d 1) + d.The q = 2 case can be identified with the quantum

mechanical one we have seen before. On the other hand,the q = 0 case can be identified with the usual vision ofthe classical world: A world where counterfactuals sim-ply do not matter, for the world just is. In this case, anagent is well advised to take Q(Dj) = P (Dj), meaningthat there is no operational distinction between exper-iments E1 and E2 for him. It should not be forgottenhowever, that this rule, trivial though it looks, is still anaddition to raw probability theory. It is just one thatmeshes well with what had come to be expected by mostclassical physicists. To put it yet another way, in theq = 0 case, the agent says to himself that the fine de-tails of his actions do not matter. This to some extentauthorizes the view that observation is a passive pro-cess in principleagain the classical worldview. Finally,the cases q = 1 and q = 4, though not classical, trackstill other structures that have been explored previously:They correspond to what the Born Rule would look likeif alternate versions of quantum mechanics, those overreal [48] and quaternionic [49] vector spaces, were ex-pressed in the equivalent of SIC terms.23

Formula (12) from the general setting indicates morestrongly than ever that it is the role of dimension thatis key to distilling the motif of our users manual. Quan-tum theory, seen as a normative addition to probabilitytheory, is just one theory (the second rung above classi-cal) along an infinite hierarchy. What distinguishes thelevels of this hierarchy is the strength q with which di-mension couples the two paths in our diagram of Fig-ure 2. It is the strength with which we are compelledto deviate from the Law of Total Probability when wetransform our thoughts from the consequences of coun-terfactual actions upon a ds worth of the worlds stuffto the consequences of our factual ones. Settling upon

23The equivalent of SICs (i.e., informationally complete setsof equiangular projection operators) certainly do not existin general dimensions for the real-vector-space caseinsteadthese structures only exist in a sparse set of dimensions,d = 2, 3, 7, 23, . . . . With respect to the quaternionic theory, itappears from numerical work that they do not generally existin that setting either [50]. Complex quantum mechanics, likebaby bears possessions, appears to be just right.

q = 2 (i.e., settling upon quantum theory itself) setsthe strength of the coupling, but the d variable remains.Different systems, different d, different deviations from anaive application of the Law of Total Probability.In some way yet to be fully fleshed out, each quan-

tum system seems to be a seat of active counterfactualityand possibility, whose outward effect is as an agent ofchange for the parts of the world that come into con-tact with it. Observer and system, agent and reagent,might be a way to put it. Perhaps no metaphor is morepregnant for QBisms next move than this: If a quan-tum system is comparable to a chemical reagent, then dis comparable to a valence. But valence for what moreexactly?

V. THE ESSENCE OF BELLS THEOREM,

QBISM STYLE

It is easy enough to say that a quantum system (andhence each piece of the world) is a seat of possibility. Ina spotty way, certain philosophers have been saying sim-ilar things for 150 years. What is unique about quantumtheory in the history of thought is the way in which itsmathematical structure has pushed this upon us to ourvery surprise. It wasnt that all these grand statementson the philosophical structure of the world were builtinto the formalism, but that the formalism reached outand shook its users until they opened their eyes. Bellstheorem and all its descendants are examples of that.So when the users opened their eyes, what did they

see? From the look of several recent prominent exposi-tions on the subject [5153], it was nonlocality everlast-ing! That the world really is full of spooky action at adistancelive with it and love it. But conclusions drawnfrom even the most rigorous of theorems can only be ad-ditions to ones prior understanding and beliefs when thetheorems do not contradict those beliefs flat out. Suchwas the case with Bells theorem. It has just enoughroom in it to not contradict a misshapen notion of prob-ability, and that is the hook and crook that the lovers ofStar Trek have thrived on. The Quantum Bayesian, how-ever, with a different understanding of probability anda commitment to the idea that quantum measurementoutcomes are personal, draws quite a different conclu-sion from the theorem. In fact it is a conclusion from thefar opposite end of the spectrum: It tells of a world un-known to most monist and rationalist philosophies: Theuniverse, far from being one big nonlocal block, should bethought of as a thriving community of marriageable, butotherwise autonomous entities. That the world shouldviolate Bells theorem remains, even for QBism, the deep-est statement ever drawn from quantum theory. It saysthat quantum measurements are moments of creation.This language has already been integral to our presen-

tation, but seeing it come about in a formalism-drivenway like Bells makes the issue particularly vivid. Here

14

we devote some effort to showing that the language ofcreation is a consequence of three things: 1) the quan-tum formalism, 2) a personalist Bayesian interpretationof probability, and 3) the elementary notion of what itmeans to be two objects rather than one. We do not doit however with Bells theorem precisely, but with an ar-gument that more directly implicates the EPR criterionof reality as the source of trouble with quantum theory.The thrust of it is that it is the EPR criterion that shouldbe jettisoned, not locality.Our starting point is like our previous setupan agent

and a systembut this time we make it two systems:One of them, the left-hand one, is ready. The other,the right-hand one, is waiting. The agent will eventuallymeasure each in turn.24 Simple enough to say, but thingsget hung at the start with the issue of what is meant bytwo systems? A passage from a 1948 paper of Einstein[54] captures the essential issue well:

If one asks what is characteristic of the realmof physical ideas independently of the quantum-theory, then above all the following attracts ourattention: the concepts of physics refer to a realexternal world, i.e., ideas are posited of thingsthat claim a real existence independent of theperceiving subject (bodies, fields, etc.), and theseideas are, on the one hand, brought into as securea relationship as possible with sense impressions.Moreover, it is characteristic of these physicalthings that they are conceived of as being ar-ranged in a space-time continuum. Further, itappears to be essential for this arrangement ofthe things introduced in physics that, at a specifictime, these things claim an existence independentof one another, insofar as these things lie in dif-ferent parts of space. Without such an assump-tion of the mutually independent existence (thebeing-thus) of spatially distant things, an as-sumption which originates in everyday thought,physical thought in the sense familiar to us wouldnot be possible. Nor does one see how physicallaws could be formulated and tested without sucha clean separation. . . .

For the relative independence of spatially dis-tant things (A and B), this idea is characteristic:an external influence on A has no immediate ef-fect on B; this is known as the principle of localaction, . . . . The complete suspension of this ba-sic principle would make impossible the idea of(quasi-) closed systems and, thereby, the estab-lishment of empirically testable laws in the sensefamiliar to us.

We hope it is clear to the reader by now that QBism

24It should be noted how we depart from the usual presenta-tion here: There is only the single agent and his two systems.There is no Alice and Bob accompanying the two systems.

concurs with every bit of this. Quantum states may notbe the stuff of the world, but QBists never shudder frompositing quantum systems as real existences externalto the agent. And just as the agent has learned fromlong, hard experience that he cannot reach out and touchanything but his immediate surroundings, so he imaginesof every quantum system, one to the other. What is itthat A and B are spatially distant things but that theyare causally independent?This notion, in Einsteins hands,25 led to one of the

nicest, most direct arguments that quantum states can-not be states of reality, but must be something more likestates of information, knowledge, expectation, or belief[56]. The argument is importantlet us repeat the wholething from Einsteins most thorough version of it [57]. Itmore than anything sets the stage for a QBist develop-ment of a Bell-style contradiction.

Physics is an attempt conceptually to graspreality as it is thought independently of its beingobserved. In this sense on speaks of physicalreality. In pre-quantum physics there was nodoubt as to how this was to be understood. InNewtons theory reality was determined by a ma-terial point in space and time; in Maxwells the-ory, by the field in space and time. In quantummechanics it is not so easily seen. If one asks:does a -function of the quantum theory repre-sent a real factual situation in the same sensein which this is the case of a material system ofpoints or of an electromagnetic field, one hesi-tates to reply with a simple yes or no; why?What the -function (at a definite) time asserts,is this: What is the probability for finding a def-inite physical magnitude q (or p) in a definitelygiven interval, if I measure it at time t? The prob-ability is here to be viewed as an empirically de-terminable, therefore certainly as a real quan-tity which I may determine if I create the same -function very often and perform a q-measurementeach time. But what about the single measuredvalue of q? Did the respective individual systemhave this q-value even before this measurement?To this question there is no definite answer withinthe framework of the theory, since the measure-ment is a process which implies a finite distur-bance of the system from the outside; it wouldtherefore be thinkable that the system obtains adefinite numerical value for q (or p) the measurednumerical value, only through the measurementitself. For the further discussion I shall assumetwo physicists A and B, who represent a different

25Beware! This is not to say in the hands of EPREinstein,Podolsky, and Rosen. The present argument is not their ar-gument. For a discussion of Einsteins dissatisfaction with theone appearing in the EPR paper itself, see [55].

15

conception with reference to the real situation asdescribed by the -function.

A. The individual system (before the measure-ment) has a definite value of q (or p) for allvariables of the system, and more specifi-cally, that value which is determined by ameasurement of this variable. Proceedingfrom this conception, he will state: The -function is no exhaustive description of thereal situation of the system but an incom-plete description; it expresses only what weknow on the basis of former measurementsconcerning the system.

B. The individual system (before the measure-ment) has no definite value of q (or p).The value of the measurement only arisesin cooperation with the unique probabilitywhich is given to it in view of the -functiononly through the act of measurement it-self. Proceeding from this conception, hewill (or, at least, he may) state: The -function is an exhaustive description of thereal situation of the system.

We now present to these two physicists the fol-lowing instance: There is to be a system whichat the time t of our observation consists of twopartial systems S1 and S2, which at this timeare spatially separated and (in the sense of clas-sical physics) are without significant reciprocity.The total system is to be completely describedthrough a known -function 12 in the sense ofquantum mechanics. All quantum theoreticiansnow agree upon the following: If I make a com-plete measurement of S1, I get from the results ofthe measurement and from 12 an entirely defi-nite -function 2 of the system S2. The charac-ter of 2 then depends upon what kind of mea-surement I undertake on S1.

Now it appears to me that one may speak ofthe real factual situation of the partial systemS2. Of this real factual situation, we know tobegin with, before the measurement of S1, evenless than we know of a system described by the-function. But on one supposition we should,in my opinion, absolutely hold fast: The real fac-tual situation of the system S2 is independent ofwhat is done with the system S1, which is spa-tially separated from the former. According tothe type of measurement which I make of S1, Iget, however, a very different 2 for the secondpartial system. Now, however, the real situationof S2 must be independent of what happens toS1. For the same real situation of S2 it is pos-sible therefore to find, according to ones choice,different types of -function. . . .

If now the physicists, A and B, accept thisconsideration as valid, then B will have to giveup his position that the -function constitutes acomplete description of a real factual situation.

For in this case it would be impossible that twodifferent types of -functions could be coordi-nated with the identical factual situation of S2.

Aside from asserting a frequentistic conception of prob-ability, the argument is nearly perfect.26 It tells us oneimportant reason why we should not be thinking of quan-tum states as the -ontologists do. Particularly, it is onewe should continue to bear in mind as we move to a Bell-type setting: Even there, there is no reason to waiveron its validity. It may be true that Einstein implicitlyequated incomplete description with there must exista hidden-variable account (though we do not think hedid), but the argument as stated neither stands nor fallson this issue.There is, however, one thing that Einstein does miss

in his argument, and this is where the structure of Bellsthinking steps in. Einstein says, to this question there isno definite answer within the framework of the theorywhen speaking of whether quantum measurements aregenerative or simply revealing of their outcomes. Ifwe accept everything he has already said, then with alittle clever combinatorics and geometry one can indeedsettle the question.Let us suppose that the two spatially separated sys-

tems in front of the agent are two ququarts (i.e., each sys-tem is associated with a four-dimensional Hilbert spaceH4), and that the agent ascribes a maximally entangledstate to the pair, i.e., a state | in H4H4 of the form,

| =1

2

4i=1

|i|i . (13)

Then we know that there exist pairs of measurements,one for each of the separate systems, such that if the out-come of one is known (whatever the outcome), one willthereafter make a probability-one statement concerningthe outcome of the other. For instance, if a nondegener-ate Hermitian operator H is measured on the left-handsystem, then one will thereafter ascribe a probability-oneassignment for the appropriate outcome of the transposedoperatorHT on the right-hand system. What this meansfor a Bayesian agent is that after performing the firstmeasurement he will bet his life on the outcome of thesecond.But how could that be if he has already recognized two

systems with no instantaneous causal influence betweeneach other? Mustnt it be that the outcome on the right-hand side is already there simply awaiting confirmationor registration? It would seem Einsteins physicist B isalready living in a state of contradiction.

26You see, there really was a reason for including Einsteinwith Heisenberg, Pauli, Peierls, Wheeler, and Peres at thebeginning of the article. Still, please reread Footnote 14.

16

Indeed it must be this kind of thinking that led Ein-steins collaborators Podolsky and Rosen to their famoussufficient criterion for an element of [preexistent] real-ity [55]:

If, without in any way disturbing a system, wecan predict with certainty (i.e., with probabil-ity equal to unity) the value of a physical quan-tity, then there exists an element of reality cor-responding to that quantity.

Without doubt, no personalist Bayesian would ever ut-ter such a notion: Just because he believes somethingwith all his heart and soul and would gamble his life onit, it would not make it necessarily so by the powers ofnatureeven a probability-one assignment is a state ofbelief for the personalist Bayesian. But he might stillentertain something not unrelated to the EPR criterionof reality. Namely, that believing a particular outcomewill be found with certainty on a causally disconnectedsystem entails that one also believes the outcome to bealready there simply awaiting confirmation.But it is not so, and the Quantum Bayesian has al-

ready built this into his story of measurement. Let usshow this presently27 by combining all the above witha beautifully simple Kochen-Specker style constructiondiscovered by Cabello, Estebaranz, and Garca-Alcaine(CEGA) [61]. Imagine some measurement H on the left-hand system; we will denote its potential outcomes as acolumn of letters, like this

a

bc

d

(14)

Further, since there is a fixed transformation taking anyH on the left-hand system to a corresponding HT on theright-hand one, there is no harm in identifying the no-tation for the outcomes of both measurements. That isto say, if the agent gets outcome b (to the exclusion ofa, c, and d) for H on the left-hand side, he will make aprobability-one prediction for b on the right-hand side,even though that measurement strictly speaking is a dif-ferent one, namely HT. If the agent further subscribes to(our Bayesian variant of) the EPR criterion of reality, hewill say that he believes b to be TRUE of the right-handsystem as an element of reality.Now let us consider two possible measurements, H1

and H2 for the left-hand side, with potential outcomes

ab

cd

and

ef

gh

(15)

27Overall this particular technique has its roots in Stairs [58],and seems to bear some resemblance to the gist of Conwayand Kochens Free Will Theorem [59,60].

respectively. Both measurements cannot be performedat once, but it might be the case that if the agent getsa specific outcome for H1, say c particularly, then notonly will he make a probability-one assignment for c ina measurement of HT1 on the right-hand side, but alsofor e in a measurement of HT2 on it. Similarly, if H2were measured on the left, getting an outcome e; then hewill make a probability-one prediction for c in a measure-ment of HT1 on the right. This would come about if H1and H2 (and consequently H

T1 and H

T2 ) share a common

eigenvector. Supposing so and that c was actually theoutcome for H1 on the left, what conclusion would theEPR criterion of reality draw? It is that both c and e areelements of reality on the right, and none of a, b, d, f ,g, or h are. Particularly, since the right-hand side couldnot have known whether H1 or H2 was measured on theleft, whatever c and e stands for, it must be the samething, the same property. In such a case, we discard theextraneous distinction between c and e in our notationand write

a

bc

d

and

c

fg

h

(16)

for the two potential outcome sets for a measurement onthe right.We now have all the notational apparatus we need to

have some fun. The genius of CEGA was that they wereable to find a set of nine interlocking Hermitian oper-ators H1, H2, . . . , H9 for the left, whose set of potentialoutcomes for the corresponding operators on the rightwould look like this:

ab

cd

ae

fg

hi

cj

hk

gl

be

mn

ik

no

pq

dj

pr

fl

qr

mo

(17)

Take the second column as an example. It means that ifH2 were measured on the left-hand system, only one of a,e, f , or g would occurthe agent cannot predict whichbut if a occurred, he would be absolutely certain of it alsooccurring in a measurement of HT1 on the right. And ife were to occur on the left, then he would be certain ofgetting e as well in a measurement of HT5 on the right.And similarly with f and g, wit