nature physics insight. foundations of quantum mechanics. april 2014 volume 10. no 4

NATURE PHYSICS | VOL 10 | APRIL 2014 | www.nature.com/naturephysics 253

INSIGHT | CONTENTS

CON

TEN

TS

NPG LONDONThe Macmillan Building, 4 Crinan Street, London N1 9XWT: +44 207 833 4000 F: +44 207 843 [email protected]

EDITOR ALISON WRIGHT

INSIGHT EDITOR IULIA GEORGESCU

PRODUCTION/ART EDITOR ALLEN BEATTIE

COPY EDITOR KEVIN SHERIDAN

SENIOR COPY EDITOR JANE MORRIS

EDITORIAL ASSISTANTEMMA BURCH

MARKETINGSARAH-JANE BALDOCK

PUBLISHERRUTH WILSON

EDITOR-IN-CHIEF, NATURE PUBLICATIONSPHILIP CAMPBELL

INSIGHT | CONTENTS

COVER IMAGE From science to technology and back again: state-of-the-art tools developed for quantum technologies, in theory and experiment, are allowing researchers to revisit the foundations of quantum theory and to explore the terra incognita that may lie beyond.

IMAGE: CARTA MARINA, OPUS OLAI MAGNI GOTTI LINCOPENSIS, EX TYPIS ANTONII LAFRERI SEQUANI, ROM, 1572 COLOURED ENGRAVING, NATIONAL LIBRARY OF SWEDEN, MAP COLLECTION, KOB, KARTOR, 1 AB

Foundations of quantum mechanics

The fields of quantum information theory and quantum technology exploded in the late 1990s the very decade that marked the rise of the internet. Labelled the second quantum revolution, this new wave of multidisciplinary research was fuelled by the quest for faster computers and secure communication. But exploiting purely quantum mechanical features for information processing requires a deeper understanding of their origin and role in different physical systems, as well as exquisite experimentalcontrol.

More than two decades of research have resulted in remarkable theoretical progress and experimental capabilities that now enable us to revisit the very foundations of quantum theory. To make a cartographic analogy, our present understanding of quantum mechanics is like an island containing still uncharted regions and with indistinct coastlines; even less is known of what may lie beyond the surrounding seas. This Nature Physics Insight covers some of the exploratory attempts to improve our map of the quantum world.

Experimental advances in the creation of macroscopic superposition states are pushing the limits of quantum theory to establish whether (or where) the quantum description eventually breaks down and the classical one takes over. Such studies might even betray gravitational corrections

to quantum mechanics and could therefore be useful in quantum gravity research. In parallel, photonic experiments are providing new insight into nonlocality and complementarity recent work seems to suggest that these too could be exploited to test models of quantum gravity, taking that quest from astrophysical observations to Earth-based experiments.

On the theoretical side, intriguing concepts are emerging such as possible nonlocal correlations that are stronger than those predicted by quantum mechanics, or the existence of an indefinite causal structure. These concepts could be exploited in new quantum information processing tasks, and they illustrate the two-way relationship that exists between quantum information theory and the foundations of quantum mechanics. And, as we celebrate fifty years of Bells theorem this year, it seems timely to consider entanglement and its previously unsuspected connections to other areas of physics, such as thermodynamics and many-body theory.

It would be impossible to cover all of the exciting research directions in this very active field, hence the aim of this Insight on the foundations of quantum mechanics is to provide merely a taste and to encourage a deeper exploration of thesubject.

Iulia Georgescu, Associate Editor

COMMENTARYGravity in quantum mechanicsGiovanni Amelino-Camelia 254Quantum entanglementVlatko Vedral 256

PROGRESS ARTICLEQuantum causalityaslav Brukner 259

REVIEW ARTICLESNonlocality beyond quantum mechanicsSandu Popescu 264Testing the limits of quantum mechanical superpositionsMarkus Arndt and Klaus Hornberger 271Testing foundations of quantum mechanics with photonsPeter Shadbolt, Jonathan C. F. Mathews, Anthony Laing and Jeremy L. OBrien 278

2014 Macmillan Publishers Limited. All rights reserved

254 NATURE PHYSICS | VOL 10 | APRIL 2014 | www.nature.com/naturephysics

COMMENTARY | INSIGHT

Gravity in quantum mechanicsGiovanni Amelino-Camelia

Gravity and quantum mechanics tend to stay out of each others way, but this might change as we devise new experiments to test the applicability of quantum theory to macroscopic systems and larger lengthscales.

The remarkable accomplishments of twentieth-century physics revolve around the success of two theoretical paradigms. On the one hand, we have phenomena described by quantum mechanics and involving interactions that are governed by the standard model of particle physics. On the other hand, we have (general) relativity and its description of gravitational phenomena. These two very different theories manage to share quarters by keeping clear of each other1. Gravity is negligible in the typical applications of quantum mechanics, which involve microscopic particles and relatively short distances. Analogously, quantum mechanical effects are usually inconsequential when we studymacroscopic bodies and largedistance scales, where gravity is incharge.

Still, we do not expect gravity to be truly absent at microscopic scales or that quantum mechanics should somehow switch off at macroscopic distances: it is just that the effects they produce in those regimes are very small and we have not yet managed to develop the technologies and devise the experiments capable of seeing such small effects. But this frustratingly leaves us without a clue about how these very different theories manage to cooperate when they both must be taken intoaccount.

The early Universe is the prototypical example of where we expect both theories to produce large effects1,2. However, with no direct experimental access to the conditions in the early Universe we have to look elsewhere, and our best chances are in regimes where one of the two theories dominates the description of the dynamics and yet the smaller effects of the other theory might come within the reach of some high-sensitivity experiments. More simply put, we should then be looking either for (i) modifications of gravity by quantum mechanics or, conversely, for (ii) modifications of quantum mechanics bygravity.

Among the numerous approaches1 used to define the interface between relativity and quantum mechanics, I find it easiest to focus on the one centred on modified uncertainty relations and/or modified commutator relations. A well-studied scenario3,4 assumes that the uncertainty relations for measuring the position coordinates xj and momenta pk are produced by non-commutativity of the relevant observables of the form:

[xj, pk]=ijkh(1 + 2p2) (1)

where jk is 1 only when j = k (0 otherwise) and is a length-scale characteristic of the modification to be determined experimentally. The standard Heisenberg commutator is recovered in the limiting case where the effects of can be neglected. In addition, there may also be new uncertainty principles and non-trivial commutators involving pairs of position coordinates of the form5,6:

[xj, xk]=ijk+imjkxm (2)

where the matrices jk and mjk would have to be characterized by small length scales, small enough to explain why quantum mechanics was so far successful ignoringthem.

For measurements involving large distance scales, the term mjk in equation(2) could be important in two very different ways. There will be situations that we usually describe only in terms of relativity and gravitational effects and in these cases the analysis of the spacetime properties will be affected by the new properties of spacetime coordinates governed by mjk. And there will be situations that we usually describe using quantum theory alone here the analysis of the quantum uncertainties might receive small corrections of quantum-gravitational origin governed by mjk.

The first of these two possibilities has already been studied intensely, particularly over the past decade2. These efforts focused

on phenomena of a mainly relativistic and gravitational nature that are studied with experimental sensitivities for which one might expect tiny effects originating from the interface between gravity and quantum mechanics. Some of the most interesting opportunities for such tests concern the description of the propagation of particles over astrophysical distances. Relativity makes firm predictions for these laws of propagation assuming, however, that spacetime coordinates are unaffected by uncertainty principles. But new uncertainty principles, such as the oneencoded through equation (2), would affect the structure of the signal in photons and neutrinos seen by telescopes monitoring distant explosions in astrophysical bodies.

Such imprints are now being sought with the Fermi (see Fig. 1), HESS and MAGIC telescopes for photons, and with IceCube for neutrinos. There is a determined effort to find evidence of spacetime fuzziness effects. The data analysis would be very simple if we could assume that the astrophysical source emits a burst of high-energy photons and neutrinos all in exact simultaneity: if such a short-duration burst propagates in a classical spacetime, then all particles in the burst must reach our telescope (nearly) simultaneously. One of the possible implications of the modified uncertainty relations is that the signal would propagate with some fuzziness and the particles would not reach our telescope simultaneously the arrival times might therefore exhibit some sort of statistical spread.

However, we know that the duration of particle bursts from astrophysical sources is not ideally small: in the best cases the bursts last a few seconds. This decreases the sensitivity of the studies, but we are learning how to compensate for these aspects of the emission mechanisms. The results so far have been negative, but the expected pace of improvement in sensitivities for the next decade or so



INSIGHT | COMMENTARY

provides hope that a discovery might be just around the next corner.

Much less has been done for the second possibility, the case of phenomena primarily governed by standard quantum mechanics, but affected by small corrections originating from the interface between gravity and quantum mechanics. Nevertheless, over the past couple of years there have been some studies that I believe might set the stage for quick progress in this line of research.

In this respect I feel it is very significant that techniques are being developed for testing some of the most striking features of quantum mechanics such as entanglement in experiments involving the exchange of particles over truly macroscopic distances, including the possibility of exchanging particles between a ground laboratory and a satellite7 (see also the Review by Shadboltetal. in this issue8). On the theory side, we will have to catch up with these experimental opportunities and we have just recently started to make progress9,10 in understanding how the non-commutativity of coordinates (equation(2)) could affect entanglement and other striking aspects of quantum theory. For instance, we expect that the presence of further contributions to the uncertainties that grow with distance would produce a loss of coherence in the quantum states that becomes increasingly significant over large distances. This coherence loss would lead to a gradual loss of entanglement. So with an entangled state shared between Earth and a distant satellite, we could find an otherwise unexpected loss of coherence possibly signalling a quantum-gravityeffect.

It would be very interesting to look for such an effect, even though the theoretical efforts aimed at modelling such phenomena cannot give us much guidance yet. We understand qualitatively the mechanism that should produce the loss of coherence, but we are still looking for a satisfactory phenomenological description for guiding these long-distance quantum-theory tests.

Another promising direction is the development of experimental techniques for testing the applicability of quantum theory to macroscopic systems. In particular, it is now possible to observe the quantum behaviour of the observables of truly macroscopic mechanical oscillators11,12 (see also the Review by Arndt and Hornberger in this issue13). If the gravitational corrections to the quantum mechanics of macroscopic bodies are very different from those for microscopic particles, this could be

exploited in the search of manifestations of the interface between gravity and quantum mechanics.

It is useful to contemplate an idealized description12,14,15 of a macroscopic body composed of N identical particles in terms of its centre-of-mass coordinates Xj =

1N-Nn=1xjn and Pj =

1N-Nn=1pjn where

xnj and pnj denote the coordinates and momentum of the nth particle. In the current formulation of quantum mechanics, the validity of standard commutators for the constituent particles [xmj, xnk]=0 and [xmj, pnk]=inmjkh implies that also [Xj,Xk]=0 and [Xj, Pk]=ijkh, meaning that the same commutation relations apply to the centre-of-mass degree of freedom of the macroscopic body. This striking property of standard quantum mechanics turns out to be fragile, and as soon as any of the parameters , jk, mjk in equations (1) and (2) become non-negligible, the correspondence between the quantum properties of the microscopic particles and those of the macroscopic body is lost12,14,15.

This should encourage accurate tests of quantum mechanics with macroscopic bodies, especially if we can manage to take differential measurements that compare the quantum properties of a macroscopic body to those of one of its constituents. But here too theory needs to advance to a level where it can feed back to experiments: the sort of descriptions of macroscopic bodies for which the relevant quantum-gravity scenarios have been so far analysed do not go much further than the idealized description of a macroscopic body that

I used here. More realistic theoretical descriptions of macroscopic bodies could provide guidance for these experiments.

I, for one, am not at all frustrated by the fact that theory might have to catch up with experiments. For a long time a time that might be eventually viewed as the dark ages of quantum-gravity research it seemed that the study of the interface between gravity and quantum mechanics should be a unique case of pure-theory science. It was not expected that experiments would ever reach the level of the theory. But things are now changing, and it would be extremely exciting if experiments took the lead in some areas of quantum gravityresearch.

Giovanni Amelino-Camelia is in the Physics Department, Sapienza University of Rome, Rome00185, Italy. e-mail: [email protected]

References1. Carlip, S. Rep. Prog. Phys. 64, 885942 (2001).2. Amelino-Camelia, G. Living Rev. Rel. 16, 5 (2013).3. Kempf, A. etal. Phys. Rev. D 52, 11081118 (1995).4. Ali, A.F. etal. Phys. Lett. B 678, 497499 (2009). 5. Doplicher, S. etal. Phys. Lett. B 331, 3944 (1994).6. Majid, S. & Ruegg, H. Phys. Lett. B 334, 348354 (1994). 7. Rideout, D. etal. Class. Quant. Grav. 29, 224011 (2012).8. Shadbolt, P., Mathews, J.C.F., Laing, A. & OBrien, J.L.

Nature Phys. 10, 278286 (2014).9. Adhikari, S. etal. Phys. Rev. A 79, 042109 (2009).10. Ghorashi, S.A.A. & Bagheri Harouni, M. Phys. Lett. A

377,952956(2013). 11. Chen, Y.J. Phys. B 46, 104001 (2013).12. Pikovski, I. etal. Nature Phys. 8, 393397 (2012). 13. Arndt, M. & Hornberger, K. Nature Phys. 10, 271277 (2014).14. Quesne, C. & Tkachuk, V.M. Phys. Rev. A

81, 012106 (2010).15. Amelino-Camelia, G. Phys. Rev. Lett. 111, 101301 (2013).

NA

SA/D

OE/

FERM

I LAT

CO

LLA

BORA

TIO

N

Ursa Major

Leo

GRB 130427A

Figure 1 | Searching the skies for the tiny effects that originate from the interface between gravity and quantum mechanics. Gamma-ray bursts are short-lived and very bright. The highest-energy light ever detected from such an event (GRB130427A) was observed in 2013. The image in the left panel taken by NASAs Fermi Gamma-ray telescope shows how the northern galactic hemisphere of the gamma-ray sky looked just before the GRB130427A burst depicted in the right panel.




Quantum entanglementVlatko Vedral

Recent advances in quantum information theory reveal the deep connections between entanglement and thermodynamics, many-body theory, quantum computing and its link to macroscopicity.

Quantum physics started with MaxPlancks act of desperation, in which he assumed that energy is quantized in order to explain the intensity profile of the black-body radiation. Some twenty-five years later, WernerHeisenberg, MaxBorn, PascualJordan, ErwinSchrdinger and PaulDirac wrote down the complete laws of quantum theory. A pertinent question then immediately came up and was subsequently hotly debated by the founding fathers of quantum physics: what features of quantum theory make it different from classical mechanics? Is it Plancks quantization, Bohrs complementarity, Heisenbergs uncertainty principle or the superposition principle1?

Schrdinger felt that the answer was none of the above. In some sense, each of these features can also be either present or mimicked within classical physics: energy can be coarse-grained classically by brute force if nothing else; waves can be superposed; and complementarity and uncertainty can be found in the trade-off between the knowledge of the wavelength and position of the wave. But the one effect Schrdinger thought had no classical counterpart whatsoever the characteristic trait of quantum physics is entanglement2.

The reason entanglement is so counterintuitive and presents a radical departure from classical physics can be nicely explained in terms of modern quantum information theory mixed with some of Schrdingers jargon. The states of quantum systems are described by what Schrdinger called catalogues of information (psi-wavefunctions). These catalogues contain the probabilities for all possible outcomes of the measurements we can make on the system. Schrdinger thought it odd that when we have two entangled physical systems, their joint catalogue of information can be better specified than the catalogue of each individual system. In other words, the whole can be less uncertain than either of itsparts!

This is impossible, classically speaking. Imagine that someone asks you to predict the toss of a single (fair) coin. Most likely you would not bet too much on it because

the outcome is completely uncertain. But consider that tossing two coins becomes less uncertain. Indeed, quantum mechanically, the state of two coins could be completely known, whereas the state of each of the coins is still maximallyuncertain.

In quantum information theory, this leads to negative conditional entropies. When it comes to quantum coins, as we know the outcome, two predictable tosses have zero entropy. However, if we only toss one coin, the outcome is completely uncertain and therefore has one unit of entropy. If we were to quantify the entropy of the second toss, given that the first has been conducted, we would come up with one negative bit that is, the entropy of two tosses minus the entropy of one toss:01=1bit.

It is precisely because of such peculiarities that the pioneers of quantum physics considered entanglement weird and counterintuitive. However, after around twenty years of intense research in this area, we are now accustomed to entanglement and, moreover, as we learn more about it we discover that entanglement emerges in unexpectedplaces.

Negative entropies have a physical meaning in thermodynamics. My colleagues and I have shown3 that negative entropy refers to the situation where we can erase the state of the system, but at the same time obtain some useful work from it. In classical physics we need to invest work in order to erase information a process known as Landauers erasure4, but quantum mechanically we can have it both ways. This is possible because the system erasing the information could be entangled with the system that is having its information erased. In that case, the total state could have zero entropy, so it can be reset without doing work. Moreover, the eraser now also results in a zero-entropy state and so it can be used to obtain one unit ofwork.

Furthermore, we realized that entanglement can exist in many-body systems (with arbitrarily large numbers of particles) as well as at finite temperature5. Entanglement can be witnessed using macroscopic observables, such as the heat capacity (see ref. 6 for recent experiments).

In fact, entanglement also serves as an order parameter characterizing quantum phase transitions7, and there is growing evidence that quantum topological phase transitions can only be understood in terms of entanglement8. A quantum phase transition is a macroscopic change driven by a variation in the ground state of a many-body system at zero temperature9. But, in contrast to an ordinary phase, no local order parameter can distinguish between the ordered and the disordered topological phases10. For instance, because the change from non-magnetic to magnetic behaviour constitutes an ordinary phase transition, we can check whether an ordinary phase is magnetic by measuring the state of just one spin. However, a topological phase transition cannot be characterized by a local parameter it requires an understanding of the global entanglement of the wholestate.

This is good news for stable encoding of quantum information. The idea is to use topological phases as quantum memories11. This is precisely because topological states are gapped (that is, the energy gap between the ground and excited states is finite) and no local noise can kick the topological state out of the protected subspace. The ground states are also degenerate, meaning that there are different states with the same level of robustness that can be used to encodeinformation.

Quantum information theory has also expanded our understanding of entanglement in other areas. Exciting recent work focuses on ways of quantifying entanglement. The most fruitful general idea is to quantify entanglement by measuring how different quantum states are from their best possible classical approximations. But, there are many non-equivalent ways of capturing this difference, which leads to a great deal of ongoing research12. For instance, nonlocality13, which, strictly speaking, means that no local realistic model can be found to explain the outcomes of measurements performed on entangled systems, is not the same as inseparability. This is because separable states are still quantum states, whereas local hidden variables can be drawn from more general probabilistic theories. Moreover, quantum



INSIGHT | COMMENTARY

nonlocality is just one possible way of violating Bells inequalities and one can always imagine more nonlocal theories (see also the Review by Popescu in this issue14). In addition, there is the notion of non-contextuality the fact that different quantum measurements do not necessarily commute15 and on top of that there are many different types of entanglement (bipartite, multipartite and global) and they can all be quantified in differentways16,17.

Why is the question of quantifying entanglement important? First, if we can estimate how close classical states are to quantum ones, we can tell how easy it would be to simulate quantum states of many-body systems. This is the logic behind a very powerful numerical method called matrix

product states that has revolutionized some aspects of solid-state physics17. The idea is simple: given as few as twenty half-spins (or qubits), we would need 220 bits to store their quantum state this is already practically intractable for todays classical computers. However, if we know that no two groups of qubits are entangled by more than one unit of entanglement, the size of the approximation can be drastically reduced. Take ten qubits versus the other ten qubits in principle we need 210 states to describe the entanglement between the two subsystems, but given that we know that they contain only one unit of entanglement, only two states for each subsystem willsuffice.

Second, if we think of quantum entanglement as a resource in quantum

cryptography and protocols such as quantum teleportation and super-dense coding, then being able to quantify entanglement is crucial for characterizing the efficiency of such protocols. Entanglement was initially thought necessary for facilitating the speed-up in quantum computation. More precisely, if our quantum computer never contains more than a certain finite number of entangled qubits, then it can never be universal. This is true for computers with registers that are always pure. It is simple to understand why: a universal computer should be capable of preparing any physical state, but if entanglement must always be bounded, then those states with more entanglement cannot be reached18. However, when it comes to

The set of many-body (in this case, many-qubit) states is broadly divided into entangled and separable (or disentangled) states. Among separable states there is a tiny, nowhere-dense subset of zero discord, which is called a classically correlated state. Classical states can be written as mixtures of states |a1a2aN, |b1b2 bN, where {|a1,|b1} form an orthogonal basis for qubit1, {|a2,|b2} form an orthogonal basis for qubit 2 and so on for all qubits up to N. The rest of the separable states contain non-zero discord. The geometry of this set is not well understood (other than perhaps for the special case of two qubits). The set of all states is convex and so is the set of separable states.

The global entanglement of a given state can be measured, for instance, by the relative entropy of entanglement, which is defined as the relative entropy S( || )=tr(loglog), between the state () and the closest separable state to it, (ref.5). Separable states are those obtained by mixing product states of the form 12N.

Some important classes of states are shown in the figure. The (many-qubit) GHZ states, |000+|111, are close to separable states because they always have one unit of entanglement independent of the number of qubits. In terms of global entanglement they are close to, for instance, the product states |000, which are disentangled. Another example of a product state is the state |++++, where every qubit is in a superposition of the basis states|+=|0+|1.

The W state is a symmetric superposition of states containing a fixed ratio of zeroes and ones, such as

|001+|010+|100. With respect to global entanglement, W states are more entangled than the GHZ states, and entanglement scales with the logarithm of the number ofqubits.

Cluster states are even more entangled, where entanglement scales as N/2, with N being the number of qubits. The most entangled states are typical states, which are simply random states of N qubits. Their entanglement scales as N. The maximally mixed state (in which all possible qubit states are mixed with equal probability) is completely disentangled and contains no correlations of any type.

The dephased GHZ states are mixtures of the states |000 and |111 and are classically correlated, but have no quantum discord (since the states |0 and

|1 are orthogonal). Mixtures of the states |0000 and |++++, on the other hand, are not only classically correlated but also contain quantum discord, although they are not entangled. States like this are thought to still be useful for some quantum computations, but their exact power is not fully understood at present.

The scaling of global entanglement with the number of qubits does not necessarily reflect other measures of entanglement, such as quantum macroscopicity. This notion is designed to capture the state in Schrdinger cat experiments, namely a superposition of two, or more, macroscopically distinct quantum states (as in the GHZ state, which has a high degree ofmacroscopicity).

|+++... +

21 (|000... 000...| + |111... 111...|)

|00...0 |GHZ |W

21 (|00...0 00...0|+|++...+ ++...+|)

BAC

KGRO

UN

D IM

AG

E:

ALE

XEY

PAV

LUTS

/ A

LAM

Y

Box 1 | Charting the set of many-body physical states.




mixed states (Box1), there are examples of computations that, despite requiring a small amount of entanglement (never more than a single entangled bit), can still achieve an exponential speed-up relative to their classical counterparts. It has been suggested that these computers exploit a more general type of quantum correlation, known as quantum discord (see Box1 and ref.12 for a review). Unfortunately, even the amount of discord is bounded during these computations, so it is hard to see how it could make adifference.

The third point is perhaps the most intriguing as it touches on the issue of macroscopicity (see also the Review by Arndt and Hornberger in this issue19). Namely, is there a limit to how big a system can be and still exhibit sizable quantum mechanical features? Here it seems appropriate to invoke Schrdinger again. But instead of his deadalive cat thought experiment, what about superposing 1018 atoms in two places separated by, say, a millimetre? I deliberately chose these numbers since we can just about see (assuming 20/20 vision) this collection of atoms (comparable to the size of an amoeba) and resolve its location with the naked eye. Now, curiously, this is a particular quantum mechanical state called a GreenbergerHorneZeilinger (GHZ) state, which is written as |000+|111, where the state |0 signifies the atom in one location and |1 the atom in the other location and there are 1018 zeroes and ones. Based on the proximity of classical states, it is not very entangled. In fact, it is just about as close to a classical state as an

entangled state can be (Box1). The global entanglement (measured, for instance, by the relative entropy between this state and the closest separable state) is always 1 for GHZ states, no matter how many particles areinvolved.

GHZ states are examples of states that are difficult to prepare in practice20, but are very easy to simulate classically. States that are difficult to simulate are, in general, the ones where entanglement scales with the number of particles in the system. This is the case for typical many-body interacting systems, as well as for cluster states used in measurement-based quantum computation. On the other hand, cluster states do not usually exhibit quantum macroscopicity.

Although it seems to be a problem, the dichotomy between macroscopicity and the amount of entanglement could in fact be fortuitous. It is usually said that being able to build a large-scale universal quantum computer is tantamount to testing the macroscopic limits (if any) of quantum theory. But, it could be that for whatever unknown reason large GHZ states cannot be made, yet, at the same time, quantum computers can be designed that far outperform the existing classical ones. That would be a curious state of affairsindeed!

These research directions have practical and fundamental implications. Technologically, it is still not fully understood how far quantum computers can be scaled up, nor can the full range of their applications be easily predicted. On the fundamental side, the problem is how to bridge the gap between the micro and the macro domains. Can thermodynamics

be fully reconciled with quantum entanglement21, and how far into the macro domain do quantum effects really need to be taken into account? This brings up a whole new set of exciting questions ranging from whether living organisms could also exploit entanglement22 to whether quantum effects can ever have an impact in the gravitationaldomain.

Vlatko Vedral is at the Clarendon Laboratory, University of Oxford, Oxford OX1 3PU, UK and the Centre for Quantum Technologies, National University of Singapore, Singapore117543,Singapore. e-mail: [email protected]

References1. Kumar, M. Quantum: Einstein, Bohr and the Great Debate about

the Nature of Reality (Icon Books, 2008).2. Schrdinger, E. Proc. Camb. Phil. Soc. 31, 553563 (1935).3. del Rio, L. etal. Nature 474, 6163 (2011). 4. Maruyama, K., Nori, F. & Vedral, V. Rev. Mod. Phys.

81, 123 (2009).5. Amico, L. etal. Rev. Mod. Phys. 80, 517576 (2008).6. Sigh, H. et. al. New J.Phys. 15, 113001 (2013).7. Osterloh, A. etal. Nature 416, 608610 (2002).8. Kitaev, A. & Preskill, J. Phys. Rev. Lett. 96, 110404 (2006).9. Sachdev, S. Quantum Phase Transitions (Cambridge Univ.

Press,2011). 10. Wen, X.G. Phys. Rev. B 65, 165113 (2002).11. Kitaev, A.Y. Ann. Phys. 303, 230 (2003).12. Modi, K. etal. Rev. Mod. Phys. 84, 16551707 (2012).13. Bell, J.S. Speakable and Unspeakable in Quantum Mechanics

(Cambridge Univ. Press, 1987).14. Popescu, S. Nature Phys. 10, 264270(2014).15. Grudka, A. et. al. Preprint at http://arxiv.org/

abs/1209.3745(2012). 16. Horodecki, R. etal. Rev. Mod. Phys. 81, 865942 (2009).17. Eisert, J. etal. Rev. Mod. Phys. 82, 277306 (2010).18. Jozsa, R. & Linden, N. Proc. R.Soc. A 459, 20112032 (2003).19. Arndt, M. & Hornberger, K. Nature Phys. 10, 271277 (2014).20. Korsbakken, J.I., Wilhelm, F.K. & Whaley, K.B. Europhys. Lett.

89, 30003 (2010). 21. Dorner, R. etal. Phys. Rev. Lett. 109, 160601 (2012).22. Sarovar, M. etal. Nature Phys. 6, 462467 (2010).



The concept of causality has been the subject of heated debates in literature about the metaphysics and philosophy of science for centuries. I have no intention of entering into these discus-sions here, but instead adopt a rudimentary, pragmatic approach. Causal thinking spontaneously arises in a child at about the time when she or he realizes that by exerting forces on nearby objects, the child can make these objects move according to their will. Causal relations are revealed by observing what would happen in the world (for instance, with the child's object) if a given parameter (the child's will) were separated out from the rest of the world and could be chosen freely.

The distinction between statistical and causal relations is echoed in the famous slogan correlation does not imply causation. Whereas the former are definable in terms of joint probabilities for observed variables, the latter require specification of conditional probabilities to provide an analysis of how the probability distribution ought to change under external interventions1. Here, I will say that an A has a causal influence on B if conditional probability P(B|A) for B observ-ably changes under free variation of A. But how can we be sure that such a variation is really free? We cannot, but this does not prevent us from considering a variation to be free whenever we have every reason to believe that it is. For all practical purposes, it is sufficient to toss a coin or use a quantum random generator to produce a free variable. And even if that free variable were produced in a deter-ministic way for example by taking the current temperature in Celsius, multiplying it by the number of my next-door neighbours children plus one, I would still regard it as being free.

In quantum physics, it is assumed that the background time or definite causal structure pre-exists such that for every pair of events A and B at distinct spacetime regions one has either A is in the past of B, or B is in the past of A, or the two are space-like separated (see Fig.1a,b). But thanks to the theorems developed by Kochen and Specker2, and by Bell3, we know that quantum mechan-ics is incompatible with the view that physical observables possess pre-existing values independent of the measurement context. (This incompatibility still holds if one assigns probabilities to the possible values of observables independently of the measurement context, rather than determining which particular result will be obtained in a single run of the experiment.) Do the theorems extend to causal structures as well?

If one assumes that quantum mechanical laws can be applied to causal relations, one might have situations in which the causal order of events is not always fixed, but is subject to quantum uncertainty, just like position or momentum. Indefinite causal structures could correspond to superpositions of situations where, roughly speaking, A is in the past of B and B is in the past of A jointly. One may spec-ulate that such situations could arise when both general relativity

Quantum causalityaslav Brukner

Traditionally, quantum theory assumes the existence of a fixed background causal structure. But if the laws of quantum mechanics are applied to the causal relations, then one could imagine situations in which the causal order of events is not always fixed, but is subject to quantum uncertainty. Such indefinite causal structures could make new quantum information processing tasks possible and provide methodological tools in quantum theories of gravity. Here, I review recent theoretical progress in this emerging area.

and quantum physics become relevant. A simple example could involve a single massive object in a superposition of two or more distinct spatial positions. Because the object is in a spatial super-position, the gravitational field it produces will also be in a super-position of states, and so will the spacetime geometry itself. This may lead to situations in which it is not fixed in advance whether a particular separation between two events is time-like orspace-like.

The consequences of having indefinite causal order would be enormous, as this would imply that spacetime and causal order might not be the truly basic ingredients of nature. It has been repeat-edly pointed out that the notion of time might be at the origin of the persistent difficulties in formulating a quantum theory of gravity49. But how do we formulate quantum theory without the assumption of an underlying causal structure and background time? What new phenomenology would be implied by the idea of indefinite causal structures? If such structures cannot be excluded on a logical basis, do they exist in nature? And if they do, why have they not yet been observed in quantum experiments?

In 2005, Lucien Hardy proposed to address these penetrating questions by developing frameworks in which causal structures may be considered to be dynamic, as in general relativity, and indefinite, similar to quantum observables10,11. He introduced one such frame-work based on a new mathematical object, the causaloid, which contains information about the causal relations between different spacetime regions. Since then, researchers, particularly in Pavia12, Vienna13 and the Perimeter Institute14, have applied the powerful tools and concepts of quantum information to shed new light on the relation between the nature of time, causality and the formalism of quantum theory a subject that has been traditionally studied within the general relativity and quantum gravity communities. In a similar vein, recent rigorous theorems in quantum information have been developed, which relate the probabilistic structure of quantum theory to the three-dimensionality of space15,16. By making plausi-ble assumptions on how (microscopic) systems are manipulated by (macroscopic) laboratory devices, it was shown that the structure of the underlying probabilistic theory cannot be modified (for exam-ple by replacing quantum theory with a more general probabilistic theory) without changing the dimensionality of space.

In conventional (causal) formulation of quantum theory, cor-relations between results obtained in causally related and acausally related experiments are mathematically described in very different ways. For example, correlations between results obtained on a pair of space-like separated systems are described by a joint state on the tensor product of two Hilbert spaces, whereas those obtained from measuring a single system at different times are described by an ini-tial state and a map on a single Hilbert space. (The causal structure of quantum theory is unrelated apriori to the causal structure of

Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria and Institute for Quantum Optics and Quantum Information (IQOQI), Boltzmanngasse 3, 1090 Vienna, Austria. e-mail: [email protected]

INSIGHT | PROGRESS ARTICLEPUBLISHED ONLINE: 1 APRIL 2014|DOI: 10.1038/NPHYS2930



relativity, as, for example, the measurements on two systems may be time-like separated in the relativistic sense and still be described by a tensor product of Hilbert spaces. Here, for simplicity, I will use causally related and time-like, as well as acausally related and space-like, interchangeably.) Very much at the focus of recent research on causality in quantum theory is the objective of finding a unified way of representing correlations between space-like and time-like regions. Once such a representation is found, we may be able to use it for the description of general quantum correlations, for which the causal ordering of events and whether they take place between space-like or time-like regions is not fixed.

Various results in the past have indicated that a unified quan-tum description may be possible for experiments involving distinct systems at one time and those involving a single system at distinct times. For example, it has been shown that there is an isomor-phism between spatial and temporal quantum correlations1720. This has conceptual and practical implications for the correspondence between quantum bounds on violation of the Bell inequality21, and its temporal analogue, the LeggettGarg inequality2225. The cor-respondence between the communication costs in the classical simulation of spatial correlations and the memory costs in the simu-lation of temporal correlations is an example of this17,26. Eventually all these developments led to several approaches towards a caus-ally neutral formulation of quantum theory: the causaloid5 already mentioned above, and further developments in terms of duoten-sors27, the quantum combs28, quantum processes13 and quantum conditionalstates14.

Notwithstanding the differences among the approaches, they all make use of the ChoiJamiolkowski (CJ) isomorphism29,30 to pro-vide a unified framework for representing a composition of opera-tions as well as tensor products of system states. If an operation is

performed on a quantum state described by a density matrix , M() describes the updated state after the operation (up to normaliza-tion), where M is a completely positive (CP), trace-non-increasing map (because we want our maps to lead to positive probabilities not larger than one) from the space of matrices over the input Hilbert space A to the one over the output Hilbert space B, which we write as M:L(A)L(B) (the two Hilbert spaces can have differ-ent dimensions, as the operation may involve additional quantum systems). The CJ isomorphism enables us to represent the opera-tions by operators rather than maps. It associates a bipartite state MAB L (AB) to a CP map, as given by MAB=JM(|++|), where indicates tensor product, |+=dAj=1|jjAB is a (non-normalized) maximally entangled state, the set of states {|jdAj=1} is an orthonormal basis of A with dimension dA, and J is the iden titymap.

In the comb28 and duotensor27 framework, one associates CJ operators with arbitrary regions of spacetime in which an observer might possibly perform a quantum operation. These operators can be combined to obtain the operator for a bigger, composite, region, using methods that are motivated in part by the graphical repre-sentation of categorical quantum mechanics31. In the framework of quantum conditional states, MattLeifer and RobSpekkens14 have developed a causally neutral formulation of quantum theory using a quantum generalization of Bayesian conditioning32. They intro-duce a conditional state B|A, playing an analogous role to condi-tional probability P(B|A) in classical probability theory. If A and B are space-like separated regions, their joint state AB is inferred from the conventional formalism, and the conditional state is derived from the joint, whereas if they are time-like separated it is the con-ditional state B|A that is inferred from the conventional formalism (for example through a map B|A = M(A)), and the joint state is derived. In either case the relation between the conditional and joint state is given by AB=B|A*A where the *-product is a particular product (defined by B|A*A

-AB|A

-A, where I have dropped the

identity operators and tensor products), which is analogous to the Bayes relation in classical probability theory, P(AB)=P(B|A)P(A). But the approach has limitations, for example in treating multiple temporal correlations, mostly owing to the fact that the *-product is non-commutative and non-associative. (The approaches1214,27,28 dif-fer among themselves in the insertion of partial transposes in the definition of CJ operators.)

With the notable exception of ref. 14, which has an epistemo-logical flavour, all other approaches are typically formulated opera-tionally; instead of using the notions of traditional physics such as position, momentum or energy, the focus lies on instrument set-tings and the outcomes of measurements. The operational idea of a causal influence is best illustrated by considering two scientists, Alice and Bob, who work in two separate laboratories. At every run of the experiment, each of them receives a physical system and performs an operation on it, after which they send their respective system out of the laboratory. During the operations of each experi-menter, the laboratory is shielded from the rest of the worldit is only opened for the system to come in and to go out, but except for these two events, it is kept closed and a signal can neither enter into nor leak out of the laboratory. Each laboratory features a device with an input and an output connector. If input a is chosen on Alices side (or, respectively, b on Bobs side), she will perform an operation on the system and send it out of the laboratory. The device will output measurement result x (respectively y) according to a certain prob-ability distribution p(x,y|a,b). The operations a and b, for example, could be the flip of a classical bit in the classical world or the unitary transformation, or in general a CP trace-non-increasing map in the quantum world.

The correlations are non-signalling if no observable change can take place in Alices laboratory as a consequence of any-thing that may be done in Bobs laboratory and vice versa. More

Future

or

Past

a b c d e

10

BA

AB

AB

BA

1

0

0

0

Figure 1 | Different causal relations between events in Alice's and Bobs laboratories. In a definite causal structure, a global background time determines whether a, Alice is before Bob, b, Bob is before Alice, or c,the two are causally neutral. Whereas in a and b signalling is always one-way, from the past to the future, there is no signalling in c because the two laboratories are space-like separated. The latter is a typical situation in tests of Bells inequalities on entangled states. d, In a closed time-like causal structure, the signalling is two-way, which gives rise to the grandfather paradox. To illustrate this paradox, consider the following example. Alice performs an identity operation on her input bit of value 0. The unchanged bit leaves her laboratory and is sent to Bob, who performs a bit flip and outputs a bit of value 1. The bit travels backwards in time to enter Alices laboratory as her input. Hence, the logical contradiction 10 for the value of Alices input arises. This can be seen as an instance of the grandfather paradox if the bit values 1 and 0 are taken to represent killing Alices grandfather and not killing Alices grandfather, respectively. e,In an indefinite causal structure, Bob can, by choosing his measurement basis, end up before or after Alice with a certain probability. The vector on the circle next to Bobs laboratory represents a resource a process which gives rise to quantum correlations with indefinite causal order. If he performs a measurement in the red (green) basis, he projects the process such that his actions occur after (before) Alices operations.

NATURE PHYSICS DOI: 10.1038/NPHYS2930PROGRESS ARTICLE | INSIGHT



specifically, this implies that Alices marginal distributionwhich is obtained by summing up the joint probability distribution over Bobs resultsis independent of Bobs input choice and vice versa: yp(x,y|a,b)=p(x,a) and xp(x,y|a,b)=p(y,b) for all a, b, x and y. The correlations are one-way signalling if one of the two conditions does not hold, and two-way signalling if neither condition holds.

It can easily be seen that a fixed causal order will impose restric-tions on the ways in which Alice and Bob can communicate. Imagine that Alice exists in Bobs past. She can act on her system and encode her input a into it before sending it to Bob. That way, his device can output y=a and the signalling is perfect. Because each party receives each system only once, Bob cannot signal to Alice. Consequently, two-way signalling is impossible. I will denote by pA B(x,y|a,b) (by pB A(x,y|a,b)) the general probability distribution in which signalling from Alice to Bob (from Bob to Alice) is possible. In a definite causal structure, it may still be the case that the causal rela-tion between events is not known with certainty. A situation where Alice exists before Bob with a probability of 01 and Bob exists before Alice with a probability of 1 is represented by a probabil-ity of the form of p(x,y|a,b)=pA B(x,y|a,b)+(1)pB A(x,y|a,b).

Are more general causal structures possible? Can Alice and Bob have two-way communication even though the exchanged system enters each of the two laboratories only once? At first sight this seems impossible, except in a world with closed causal loops, where a signal may go back and forth from Alice to Bob. Such closed time-like curves (CTCs) were first proposed by Kurt Gdel in 1949. Gdel was an Austrian logician who discovered, surprisingly, that general relativity equations allow CTC solutions33. But the existence of CTCs seems to imply logical paradoxes, most notably the grandfather paradox in which an agent goes back in time to kill his grandfather (see Fig.1d). Possible solutions have been proposed in which quantum mechanics and CTCs can coexist and such paradoxes are avoided, but not with-out modifying quantum theory into a nonlinear one (refs 3437, and unpublished results by C.H.Bennett and B.Schumacher). Nonlinear theories themselves are problematic38. The question remains: is it possible to keep the (linear) framework of quantum theory, have no paradoxes and still go beyond definite causal structures?

One such framework was proposed recently by OgnyanOreshkov, Fabio Costa and I13. There, we assumed that operations in local laboratories are described by quantum mechanics (that is, they are CP maps). Using the CJ isomorphism, the probability for a pair M x,a A1A2 and M y,b B1B2 of local CP maps performed by Alice between the local input A1 and output A2 Hilbert spaces, and by Bob between the Hilbert spaces B1 and B2, are represented as a bilinear function of the corresponding CJ operators as follows: p(x,y|a,b) = Tr[WA1B1A2B2(M x,a A1A2 M y,b B1B2)]. Here WA1B1A2B2 belongs to the space of matrices over the tensor product of the input A1,B1 and the output A2,B2 Hilbert spaces of two parties. It is the cen-tral object of the formalism and represents a new resource called process a generalization of the notion of state. The matrix W is a positive matrix, and it returns unit probability for CP trace-preserving maps. Just like in the aforementioned approaches, it provides a unified way to represent correlations in casually related and acausally related experiments. Although the notion of causal structure is built within the local laboratories insofar as the output of an operation is causally influenced by the input, no reference is made to any global causal relations between the operations in two laboratories. Most interestingly, we have found situations where two operations are neither causally ordered nor in a probabilistic mix-ture of definite causal orders: that is, one cannot say that one opera-tion is either before or after the other. In these cases the process is not a probabilistic mixture of the processes with definite causal order: WABWA B+(1+)WB A, where 01, and WA B is the process in which Alice can signal to Bob, and WB Ais that in which Bob can signal to Alice.

In terms of probability distributions, this can be written as p(a,b|x,y)pA B(a,b|x,y)+(1)pB A(a,b|x,y). Because the correla-tions are incompatible with any underlying definite causal structure, we call them quantum correlations with indefinite causal order.

The existence of the new correlations can be demonstrated in a theory-independent way on the basis of recorded data in an experimental test. These new correlations violate a causal inequal-ity which is satisfied if events take place in a causal sequence. This stands in direct analogy to the famous violation of Bells inequality in quantum mechanics, which is satisfied if the measured quan-tities have predefined local values3. The causal inequality is best explained in terms of a game involving two players, Alice and Bob again, each of whom receives a random input bit value, 0 or 1.The point of the game is that each player tries to guess the input of the other player. One of the players, say Bob, receives an additional random bit, which specifies who will need to guess whose bit in a given run of the game. It can easily be seen that in every causal scenario the success probability of the game is bounded by 3/4. Without loss of generality, consider that Alice is in Bobs past, as illustrated in Fig. 1a. Then she can always send her input bit to him and they will accomplish their task perfectly if he is required to guess her bit, whereas if she is asked to guess his bit, she cannot do better than giving a random answer. This gives an overall suc-cess probability of 3/4. But if Alice and Bob share quantum cor-relations with indefinite causal order, they can achieve a success probability as high as13 (2 +

-2)/4. Whereas causal correlations allow signalling in no more than one fixed direction, correlations with indefinite causal order allow Bob, depending on his choice of measured observable, to effectively end up before or after Alice with a probability of 1/

-2 (see Fig.1e). All causal loops and para-doxes are avoided: in every single run, only one-way signalling is realized, but the signalling direction, from Alice to Bob or from Bob to Alice, is in Bobs control and may vary from run to run. It is intriguing that both the classical bound and the quantum vio-lation of the causal inequality match the corresponding numbers in the ClauserHorneShimonyHolt version39 of Bells inequality. Most recently, a process for three parties has been found in which perfect signalling correlations among three parties are possible, whereas the same is impossible in any causal scenario40 (this is analogous to the all versus nothing type of argument against local hiddenvariables41).

The possibility of indefinite causal orders has also been discussed in the context of quantum computation42. The idea of (causal) quan-tum computation, or quantum circuits, may be illustrated as a set of gates physically connected by wires through which quantum systems propagate. As the systems pass those gates, they change their states. This is repeated in succession until the computation is

A

|0B

A

B|1

Figure 2 | Superposition of quantum circuits. The causal succession in which the physical boxes A and B are applied to the computers state depends on the state of the control qubit. By projecting the control qubit in a particular basis (-2 -

1 (|0+|1); see text for more details), the network is in a quantum superposition of being used in a circuit with causal structure A B and of being used in a circuit with causal structure B A.

INSIGHT | PROGRESS ARTICLENATURE PHYSICS DOI: 10.1038/NPHYS2930



finished. Once two gates have been inserted in a circuit in a given order, there is no way to invert their causal relation.

Recently, Chiribella et al. have proposed that the geometry of the wires between the gates could be controlled by the quantum state of a controlled qubit, a quantum switch12. In this way it would be possible to build a superposition of circuits in which the gates of a given set act in a different order, depending on the state of the switch (Fig.2). This more general model of computation can out-perform causal quantum computers in specific tasks, such as dis-criminating between two non-signalling channels43. In a simpler version, the task is to distinguish whether a pair of boxeswhich represent two unitaries A and Bcommute or anticommute, that is, whether AB=+ BA. Realizing such a task within the standard circuit model would unavoidably require at least one of the uni-taries to be applied twice12. But there is a simple algorithm that exploits superpositions of causal circuits and makes it possible to use each box only once. It coherently applies the two unitaries on the initial state | of the computer (for example an internal degree of freedom of the flying qubit) in two possible orders, depend-ing on the state of a control qubit. If the control qubit is prepared in the superposition -2 -

1 (|0 + |1), the output of the algorithm is:

-2 -1 (AB||0+BA||1)=-2 -1 AB|(|0+|1) with the phase of the

control qubit state +1 or 1 depending on whether the two unitaries commute or anticommute, respectively. A simple measurement of the control qubit in the bases -2 -

1(|0+|1) finally solves the task. In this elementary example, the number of oracle queries is reduced from 2 to 1 with respect to causal computation, but superpositions of quantum circuits can be exploited to solve a related computa-tional problem of size n by using O(n) black-box queries, whereas the best-known quantum algorithm with fixed order between the gates requires O(n2) queries44. In general, any superposition of quan-tum circuits can be simulated (with, at most, polynomial overhead) by a standard causal quantum circuit.

The causal succession of gate operations in a quantum circuit is the circuit analogue of the events separated by a time-like interval. This analogy can be pushed further. The networks in which geom-etry of the wires between the gates are entangled with the state of a control qubit can be thought of as a toy model for the quantum gravity situation introduced in the text above. If a massive object is put in a spatial superposition, then the metric produced, and hence spatio-temporal distances between events in the gravitational field of the object, get entangled with the object's position.

We have only begun to scratch the surface of this new field of quantum causality. The deeper we dig, the more questions arise. Where should we search for phenomenological evidence of indefinite causal orders? At the Planck length? In superpositions of large masses and spacetime backgrounds? Although it is evi-dent how to implement superpositions of wires in a circuit, they are apparently not sufficient for realizing indefinite causal processes that violate the causal inequality. Are these processes just mathematical artefacts of the theory? What is the quantum bound on violating the causal inequality, and which physical principles might constrain it? What is the zoo of complexity classes for quantum computers exploiting indefinite causal structures? If spacetime and causality are not the most fundamental ingredients of nature, what, then, are the basic building blocks? And might the former emerge in some sort of decoherence from the latter?

In the same way that quantum entanglement and coherence as working concepts have given rise to quantum-enhanced informa-tion processing45, for instance with Shors factoring algorithms or secure quantum key distribution, the power of quantum computa-tion on indefinite causal structures may lead to new protocols and procedures, maybe even changing the character of quantum infor-mation science itself. But the present research programme will not reach fulfilment if it does not provide new insights into the challenge

of finding a theory of quantum gravity. The difficulties that arise when attempting to merge quantum theory and general relativity are so complex, and have lasted for so long, that some have come to suspect that they are not merely technical and mathematical in nature, but rather conceptual and fundamental. Research on quan-tum causality may lead to answers.

Received 3 December 2013; accepted 25 February 2014; published online 1 April 2014.

References1. Pearl, J. Causality: Models, Reasoning, and Inference

(CambridgeUniv.Press,2000).2. Kochen, S. & Specker, E.P. The problem of hidden variables in quantum

mechanics. J.Math. Mech. 17, 59 (1967).3. Bell, J.S. On the EinsteinPodolskyRosen paradox. Physics 1, 195200 (1964). 4. DeWitt, B.S. Quantum theory of gravity I. The canonical theory. Phys. Rev.

160, 11131148 (1967).5. Peres, A. Measurement of time by quantum clocks. Am. J.Phys.

48, 552557(1980).6. Wooters, W.K. Time replaced by quantum correlations. Int. J.Theor. Phys.

23,701711 (1984).7. Rovelli, C. Quantum mechanics without time: a model. Phys. Rev. D

42,26382646 (1990).8. Isham, C. in Integrable Systems, Quantum Groups and Quantum Field Theory

(ed. Ibort, R.) 157287 (Kluwer, 1993). 9. Gambini, R., Porto, R.A. & Pullin, J. A relational solution to the problem of

time in quantum mechanics and quantum gravity: a fundamental mechanism for quantum decoherence. New J.Phys. 6, 45 (2004).

10. Hardy, L. Probability theories with dynamic causal structure: a new framework for quantum gravity. Preprint at http://arxiv.org/abs/gr-qc/0509120 (2005).

11. Hardy, L. Towards quantum gravity: a framework for probabilistic theories with non-fixed causal structure. J.Phys. A 40, 3081 (2007).

12. Chiribella, G., DAriano, G.M., Perinotti, P. & Valiron, B. Quantum computations without definite causal structure. Phys. Rev. A 88, 022318 (2013).

13. Oreshkov, O., Costa, F. & Brukner, C. Quantum correlations with no causal order. Nature Commun. 3, 1092 (2012).

14. Leifer, M.S. & Spekkens, R.W. Towards a formulation of quantum theory as a causally neutral theory of Bayesian inference. Phys. Rev. A 88, 052130 (2013).

15. Mller, M.P. & Masanes, L. Three-dimensionality of space and the quantum bit: an information-theoretic approach. New J.Phys. 15, 053040 (2013).

16. Dakic, B. & Brukner, C. in Quantum Theory: Informational Foundations and Foils (eds Chiribella, G. & Spekkens, R.) (Springer, in the press); Preprint at http://arxiv.org/abs/1307.3984 (2013).

17. Brukner, C., Taylor, S., Cheung, S. & Vedral, V. Quantum entanglement in time. Preprint at http://arxiv.org/abs/quant-ph/0402127v1 (2004).

18. Leifer, M.S. Quantum dynamics as an analog of conditional probability. Phys. Rev. A 74, 042310 (2006).

19. Marcovitch, S. & Reznik, B. Structural unification of space and time correlations in quantum theory. Preprint at http://arxiv.org/abs/1103.2557(2011).

20. Fitzsimons, J., Jones, J. & Vedral, V. Quantum correlations which imply causation. Preprint at http://arxiv.org/abs/1302.2731 (2013).

21. Cirelson, B.C. Quantum generalizations of Bells inequality. Lett. Math. Phys. 4, 93 (1980).

22. Leggett, A.J. & Garg, A. Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks? Phys. Rev. Lett. 54, 857 (1985).

23. Fritz, T. Quantum correlations in the temporal ClauserHorneShimonyHolt (CHSH) scenario. New J.Phys. 12, 083055 (2010).

24. Budroni, C., Moroder, T., Kleinmann, M. & Ghne, O. Bounding temporal quantum correlations. Phys. Rev. Lett. 111, 020403 (2013).

25. Markiewicz, M., Przysiezna, A., Brierley, S. & Paterek, T. Genuinely multi-point temporal quantum correlations. Preprint at http://arxiv.org/abs/1309.7650(2013).

26. Kleinmann, M., Ghne, O., Portillo, J.R., Larsson, J-. & Cabello, A. Memory cost of quantum contextuality. New J.Phys. 13, 113011 (2011).

27. Hardy, L. Reformulating and reconstructing quantum theory. Preprint at http://arxiv.org/abs/1104.2066 (2011).

28. Chiribella, G., DAriano, G.M. & Perinotti, P. Theoretical framework for quantum networks. Phys. Rev. A 80, 022339 (2009).

29. Jamiokowski, A. Linear transformations which preserve trace and positive semidefiniteness of operators. Rep. Math. Phys. 3, 4, 275278 (1972).

30. Choi, M-D. Completely positive linear maps on complex matrices. Lin. Alg. Applic. 10, 285290 (1975).

NATURE PHYSICS DOI: 10.1038/NPHYS2930PROGRESS ARTICLE | INSIGHT



31. Coecke, B. Kindergarten quantum mechanics. Preprint at http://arxiv.org/abs/quant-ph/0510032 (2005).

32. Caves, C.M., Fuchs, C.A. & Schack, R. Quantum probabilities as Bayesian probabilities. Phys. Rev. A 65, 022305 (2002).

33. Gdel, K. An example of a new type of cosmological solution of Einsteins field equations of gravitation. Rev. Mod. Phys. 21, 447450 (1949).

34. Deutsch, D. Quantum mechanics near closed timelike lines. Phys. Rev. D 44,31973217 (1991).

35. Greenberger, D.M. & Svozil, K. in Quo Vadis Quantum Mechanics? (edsElitzurA. Dolev, S. & Kolenda, N.) 6372 (Springer, 2005).

36. Svetlichny, G. Effective quantum time travel. Int. J.Theor. Phys. 50, 3903 (2011).

37. Lloyd S. et al. Closed timelike curves via post-selection: theory and experimental demonstration. Phys. Rev. Lett. 106, 040403 (2011).

38. Gisin, N. Weinbergs non-linear quantum mechanics and supraluminal communications, Phys. Lett. A 143, 12 (1990).

39. Clauser, J.F., Horne, M.A., Shimony, A. & Holt, R.A. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. 23, 880 (1969).

40. Baumeler, . & Wolf, S. Perfect signaling among three parties violating predefined causal order. Preprint at http://arxiv.org/ abs/1312.5916 (2013).

41. Greenberger, D., Horne, M., Shimony, A. & Zeilinger, A. Bells theorem without inequalities. Am. J.Phys. 58, 1131 (1990).

42. Hardy, L. in Proc. Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle: Int. Conf. in Honour of Abner Shimony. Preprint at http://arxiv.org/abs/quant-ph/0701019(2007).

43. Chiribella, G. Perfect discrimination of no-signalling channels via quantum superposition of causal structures. Phys. Rev. A 86, 040301(R) (2012).

44. Arajo, M., Costa, F. & Brukner, C. Decrease in query complexity for quantum computers with superposition of circuits. Preprint at http://arxiv.org/abs/1401.8127(2014).

45. Nielsen, M.A. & Chuang, I.L. Quantum Computation and Quantum Information (Cambridge Univ. Press, 2000).

AcknowledgementsI thank F. Costa, O. Oreshkov and J. Pienaar for discussions. This work was supported by the Austrian Science Fund (FWF) through FoQuS and individual project 24621, the European Commission Project RAQUEL, FQXi, and the John TempletonFoundation.

Additional informationReprints and permissions information is available online at www.nature.com/reprints. Correspondence and requests for materials should be addressed to C.B.

Competing financial interestsThe authors declare no competing financial interests.

INSIGHT | PROGRESS ARTICLENATURE PHYSICS DOI: 10.1038/NPHYS2930



Quantum mechanics is, without any doubt, our best theory of nature. Apart from gravity, quantum mechanics explains vir-tually all known phenomena, from the structure of atoms, the rules of chemistry and properties of condensed matter to nuclear structure and the physics of elementary particles. And it does all this to an unprecedented level of accuracy. Yet, almost 80 years since its discovery, there is a general consensus that we still lack a deep understanding of quantum mechanics. Indeed, novel, puzzling and even paradoxical situations are frequently discovered. And Im not talking about the well-known interpretational puzzles related to the measurement problem, but about a variety of quan-tum effects, from the AharonovBohm effect1, which was hidden in plain sight, to Bell inequality violations2, to the multitude of strange effects related to entanglement and quantum information. They are all puzzling and paradoxical only because we do not yet have the intuition and understanding that would allow us to predict and expect them.

Surprisingly however, with very few notable exceptions, for many years research on the fundamental aspects of quantum mechanics was put on the back burner; there seemed to always be more impor-tant, pressing issues. During the past couple of years, however, there has been a strong renewed interest in the subject and there seems to be hope that we will finally reach a much deeper understanding of the nature of quantum mechanics. In what follows, I will describe a small part of thisresearch.

As was noted long ago, the axioms of quantum mechanics are far less natural, intuitive and physical than those of other theo-ries, such as special relativity. Special relativity can be completely deduced from two axioms: (1) all inertial frames of reference are equivalent and (2) there is a finite maximum speed for propaga-tions of signals. Contrast these with the very mathematical and physically obscure axioms of quantum mechanics: every state is a vector in a complex Hilbert space, every observable corresponds to a Hermitian operator acting on that Hilbert space, and so on. Furthermore, when trying to make physical statements about nature, they are all sort of negative: nature is uncertain, we cannot predict the result of a measurement, if we measure this we disturb that, and so on. Clearly there is no way to reconstruct the whole theory from such physical statements. Yet, there is a glimmer of hope. As both Aharonov (ref. 3 and personal communication) and Shimony4 independently noticed, the fundamental non-determin-ism of quantum mechanics, one of the most unpleasant aspects of the theory and the very subject of Einsteins famous complaint God doesnt play dice, actually plays a positive role: it opens the window to a new phenomenon nonlocality. And Aharonov even went a step further (ref.3 and personal communication). He remarked that it is possible, in principle, to have a theory that is non-deter-ministic without being nonlocal. On the other hand, it is impossible to have a nonlocal theory that respects relativistic causality but is

Nonlocality beyond quantum mechanicsSandu Popescu

Nonlocality is the most characteristic feature of quantum mechanics, but recent research seems to suggest the possible exist-ence of nonlocal correlations stronger than those predicted by theory. This raises the question of whether nature is in fact more nonlocal than expected from quantum theory or, alternatively, whether there could be an as yet undiscovered principle limiting the strength of nonlocal correlations. Here, I review some of the recent directions in the intensive theoretical effort to answer this question.

deterministic. Indeed, very roughly speaking, if by moving some-thing here, something else instantaneously wiggles there, the only way in which this doesnt lead to instantaneous communication is if that wiggling thing is uncertain and the wiggling can be only spotted a posteriori. The bottom line, therefore, is that if we take nonlocality to be the starting point, then fundamental non-deter-minism the most characteristic property of quantum mechan-ics immediately follows as a consequence. Hence, we should consider nonlocality and not non-determinism as a basic axiom of quantummechanics.

In the years following Aharonov and Shimonys suggestion and due to the advent of quantum information and the extremely intense study of entanglement in particular, nonlocality came indeed to be appreciated as a fundamental property of nature. Yet, there is an even more interesting twist in the story. Rohrlich and I5 took the AharonovShimony suggestion seriously and investigated whether or not quantum mechanics can be deduced from the axi-oms of (1) relativistic causality and (2) the existence of nonlocal-ity. In other words, we asked: Is quantum mechanics the unique theory that allows for nonlocal phenomena consistent with special relativity? Surprisingly, we discovered that this is not the case: nature could be even more nonlocal than that quantum mechan-ics predicts, yet be fully consistent with relativity! This immediately raises two questions. Perhaps nature is indeed more nonlocal than is described in quantum mechanics says, but we havent yet observed such a situation experimentally. Alternatively, if such stronger non-local correlations do not exist, why dont they? Is there any deep principle that allows for nonlocality but limits its strength? This Review is dedicated to reporting the very intense present research into thisquestion.

Before going forward, I want to reiterate that the scope of this Review is, by necessity, very limited and what is presented here is only a small part of a much larger effort to understand the founda-tions of quantum mechanics that is going on at present. To start with, I would like to mention the intensive work in characterizing quan-tum nonlocality itself625, where not even the simple algebraic ques-tion of fundamental importance of which nonlocal correlations can be obtained from quantum mechanics is yet completely solved; see seminal works by Tsirelson2629 (Cirelson) as well as others30,31. Another interesting direction is that of generalized probabilistic the-ories3238. I also cannot cover the fascinating flow of ideas back from this research into quantum information theory, where it has led to a variety of new ideas, concepts and applications, out of which I would like to mention the newly emerged area of device-independent phys-ics (including device-independent key distribution3949 and device-independent randomness generation5060). A recent review article61 covers these results and many more in detail. Further afield, I would like to single out the intense activity in searching for natural axioms of quantum mechanics along the lines initiated by Hardy6269. Finally,

H.H.Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK. e-mail: [email protected]

REVIEW ARTICLES | INSIGHTPUBLISHED ONLINE: 1 APRIL 2014|DOI: 10.1038/NPHYS2916



I would like to mention a completely different type of nonlocality, namely dynamic nonlocality70.

Model-independent statements about physics Physics is usually discussed in very concrete terms, indicating the systems of interest and the specific interactions between them. A very important recent development, however, was the realization that physics can also be presented in a model independent way, that is, in a way that is largely independent of the details of the specific underlying theories; this allows one to compare various possibletheories.

For our purpose, it is very convenient to view experiments as inputoutput black-box devices. Every experiment can be viewed as a black box. For example, suppose Alice has a box that accepts inputs x and yields outputs a (Fig. 1). One can imagine that inside the box there is an automated laboratory, containing particles, measuring devices, and so on. The laboratory is prearranged to per-form some specific experiments; the input x simply indicates which experiment is to be performed. Suppose also that for every meas-urement we know in advance the set of the possible outcomes; the output a is simply a label that indicates which of the results has been obtained. In this framework, the entire physics is encapsulated in P(a|x), the probability that output a occurs given that measurement x was made.

In our discussion, we are interested in the constraints that rela-tivistic causality imposes on experiments carried out by two parties, Alice and Bob, who are situated far from each other. The physics is encapsulated in P(a,b|x,y), the joint probability that Alice obtains a and Bob obtains b when Alice inputs x and Bob inputs y. We are interested in the case when the experiments of Alice and Bob are space-like separated, that is, each experiment takes place before any information about the others input and output could reach it. We allow, however, the boxes to have been prepared long in advance, so that they could have been prepared in some correlated way, and they may also be connected by radios, telephone cables and so on. Also, obviously, to determine the joint probability, we need time to collect the entire information in one place.

NonlocalityConsider the simple case when x, y, a and b have only two pos-sible values, conventionally denoted 0 and 1. Suppose that Alice and Bob would like to construct some boxes that will yield outputs a and b such that:

a b=xy (1)

where denotes addition modulo 2 (that is, a b=0 if a=0 and b=0 or a=1 and b=1 and a b=1 if a=0 and b=1 or a=1 and b=0). In simple terms, what the above equation says is that when the inputs are x = y = 1, the outputs must be different from each other, whereas for any other pair of inputs, the outputs must be equal to each other. The question is, how well can they succeed?

Suppose, without loss of generality, that Alice and Bob pre-arrange that if x=0 then Alices box yields a=0. Now, to ensure that they win the game when the inputs are x=0 and y=0, they obviously must arrange that when y=0 Bobs box should yield b=0. Furthermore, to ensure success when x=0 and y=1, they must also arrange that when y=1 Bobs box must also yield b=0. Now, as Bobs box will yield b=0 when y=0, to ensure success if the inputs are x = 1 and y = 0, Alices box must be such that it yields a = 0 when x = 1. But by now we have fixed the behaviour of both boxes for all the inputs. And we have a problem: if x=1 and y=1 the outputs will be a=0 and b=0, which constitutes a failure. Hence for one in four inputs, Alice and Bob fail. If the inputs x and y are given at random, 0 and 1 with equal probability, then Alice and Bobs probability of success is at most 3/4.

Of course, if the boxes could communicate with each other, then they could always succeed: Alices box tells Bobs something like, My input was x=0, I output a=0, take care what you do!. But, the whole point of the set-up was that Alice and Bobs experi-ments are space-like separated from each other, so any such sig-nal would have to propagate faster than light. The upper bound of 3/4 on the probability of success derives from locality (that is, no superluminal communication between the boxes), and it is called a Bell inequality2. There are many different Bell inequalities, describing constraints derived from locality in similar tasks; the particular one discussed here is the ClauserHorneShimonyHolt (CHSH)inequality71.

JohnBells seminal discovery2 was that if the boxes contain quan-tum particles prepared in an appropriate entangled quantum state, and if appropriate measurements are performed on them, one can arrange a situation such that the probability of success of the above game is larger than3/4.

Quantum particles, therefore, somehow communicate with each other superluminally. One could wonder if this doesnt immediately contradict Einsteins relativity. Here is precisely where the probabil-istic nature of quantum mechanics comes into play. All that Alice and Bob can immediately see are the probabilities of their experi-ments; to learn the joint probabilities takes time. Suppose, for example, that for Alice the outcomes a=0 and a=1 are equally probable, regardless of what x and y are. Then Bob has no way of sig-nalling superluminally to Alice: all he can do is to choose the value of y, but this doesnt affect the probabilities of Alices outcomes. Similarly, Alice could also be prevented from signalling to Bob. So, in Shimonys words, the probabilistic nature of quantum mechanics allows for the peaceful co-existence of relativity and nonlocality: the particles could communicate to each other superluminally, but the experimentalists cannot use them to communicate superlumi-nally with eachother.

Nonlocality beyond quantum mechanicsAs discussed above, quantum mechanics allows for a probability of success larger than 3/4 in the correlation game, meaning that the boxes (or the particles contained within) somehow commu-nicate superluminally with each other. This is now recognized as being one of the most important aspects of quantum mechanics. However, quantum mechanics cannot always win in the game the quantum probability of success is at most (2+

-2)/4, as proved by Cirelson29. That this is the case is a simple consequence of the Hilbert-space structure of quantum mechanics. But the deeper question is, why?5 Is there a deep principle of nature that limits the amount ofnonlocality?

The first guess was that stronger nonlocal correlations would be forbidden by relativistic causality; perhaps the randomness that provides the umbrella under which nonlocality can coexist with relativistic causality is not enough to allow for stronger nonlocality. So the very first question to ask is: could theoretically nonlocal

ba

yx

Figure 1 | The black-box model of two experiments. Each black box is a whole laboratory. The inputs, x and y, are instructions indicating the experiment to be performed in the box and a and b are the outcomes of theexperiments.

AN

NA

I. P

OPE

SCU

INSIGHT | REVIEW ARTICLESNATURE PHYSICS DOI: 10.1038/NPHYS2916



correlations stronger than quantum mechanical ones exist, without violating relativity? When Bell discovered nonlocality, the problem was not formulated in a model-independent way but by using the specific language of quantum mechanics: entangled quantum states, Hermitian operators, eigenvalues and so on. From this point of view, the very question of whether or not nonlocal correlations stronger than the quantum mechanical ones could exist was very difficult to even envisage, let alone to answer. In the above box framework, however, the question and its answer are almost trivial: as long as locally a and b are 0 or 1 with equal probability, there is nothing that prevents the game from being won with certainty. These particular correlations are now known as PopescuRohrlich (PR)boxes5,72.

Super-quantum correlationsThe existence of super-quantum nonlocal correlations shows that quantum mechanics cannot be deduced from the two axioms of (1) relativistic causality and (2) the existence of nonlocal correla-tions. Something else is needed. But what? What could be a supple-mentary, very natural, axiom that could rule out suchcorrelations?

The statement that super-quantum correlations could in principle exist is very far from a fully fledged physical theory. Therefore, it may seem very unlikely that one could make further progress in answering the above question before such a full theory, which could explain all the known results hydrogen atoms and so on but also incorporate super-quantum correlations, is for-mulated. Surprisingly enough, it turns out that there is a lot one can do with even just the above particular example. Help came at first from computer science, and now this is one of the hottest areas in the foundations of physics. Various very interesting situations have been discussed, including communication complexity73,74, nonlocal computation75, information causality76, macroscopic locality77, local orthogonality78 and nonlocality swapping79. In this Review, I will discuss only a fewexamples.

Communication redundancyAlmost all of our communication is redundant, and that is not only because some of us like to talk too much, but also because it is a law of nature. Indeed, consider the following problem. Suppose Alice and Bob would like to meet, but are both very busy. They speak on the telephone and try to find a day this year when they could meet. To make the problem more interesting, suppose that they do not want to find out a precise day, but first they want to establish whether the number of days when they could meet is even or odd (zero counting as even). To make the problem simpler, suppose it is only Bob that sends information to Alice, and Alice has to decide the result. The question is, how much information must Bob send toAlice?

We have now a problem in which the result is a single bit, a sin-gle yes or no answer: yes=even, no=odd. On the other hand, it is obvious that Bob needs to inform Alice about the status of each day of the year in his calendar. Indeed, one of the possible situa-tions is that Alice is free only one single day. To decide whether they can meet or not, she has to know whether Bob is free that day; as Bob doesnt know anything about Alices calendar, he has to tell her about each of his days. He has therefore to send Alice 365bits of information, a yes=Im free or no=Im not free for each day of the year; all this for Alice to find out a single bit of information. Very redundantindeed.

Clearly, in the process Alice learns much more than what she wanted to know. Indeed, not only will she find out if the total num-ber of days when they could meet is even or odd, but also she will know the precise days they can meet. She didnt want to learn that, but there is no otherway.

Wim van Dam73 observed in his PhD thesis, however, that if Alice and Bob have access to PR boxes, they could reduce the com-munication to a single bit, eliminating therefore the entire redun-dancy. They can do this by not attempting to directly communicate information about their calendars, but using this as input to their boxes and communicating information about theiroutputs.

In particular, all Alice and Bob have to do is to associate with each day i a variable xi (yi) that is equal to 0if the day is busy and to 1if the day is free and use them as inputs for their PR boxes. The sum of their outputs is even (odd) if the number of days when they can meet is even (odd). For Alice to find out whether the sum of their outputs is even or odd, Bob only needs to inform her whether the sum of his outputs is even or odd, that is, a single bit of com-munication (Box 1).

The result is particularly important, as the above calendar problem is not just some silly communication task; it is in fact the most difficult communication task possible (technically called the inner product problem). Ind

nature physics insight. foundations of quantum mechanics. april 2014 volume 10. no 4

Documents