what is the origin of complex probability amplitudes? william k. wootters williams college
TRANSCRIPT
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A simple quantum experiment
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Computing probabilities in a sensible way
(½)(½)+(½)(½) = ½
½
½
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½
½
Computing probabilities in a sensible way
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The experimental result
0%
100%
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A quantum explanation of this result
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A quantum explanation of this result
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0%
100%
A quantum explanation of this result
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Questions
(i) Why do we have to work with “square roots” of probability? Is there a deeper explanation?
(ii) And why are these “square roots” complex?
I will try to answer the first question—why square roots?But my answer will make the second question worse.
Then I address the second question—why complex amplitudes?—and ask in particular whether the real-amplitude theory could conceivably be correct.
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Photon polarization
polarizing filter
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Measuring photon polarization
polarizing filter
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polarizing filter
42°
Measuring photon polarization
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polarizing filter
42°
There is no such device.
Measuring photon polarization
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polarizing filter
Measuring photon polarization
polarizing beam splitter
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polarizing filter
Measuring photon polarization
polarizing beam splitter
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polarizing filter
Measuring photon polarization
By measuring many photons,we can estimate the probabilityof the vertical outcome. This tells us about the angle.
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The standard account of probability vs angle
amplitudefor vertical
Squaring the amplitudegives the probability:
q
p
q
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polarizing filter The angle varies continuously.But the measurement is probabilisticwith only two possible outcomes.
Is the communication optimal?
A completely different explanation for that curve: Optimal information transfer?
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A Communication Puzzle
Alice is going to think of a number q between 0 and p/2.
q
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A Communication Puzzle
Alice is going to think of a number q between 0 and p/2.
She will construct a coin, with her number encoded inthe probability of heads.
q
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A Communication Puzzle
Alice is going to think of a number q between 0 and p/2.
She will construct a coin, with her number encoded inthe probability of heads. She will send the coin to Bob.
q
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A Communication Puzzle
Alice is going to think of a number q between 0 and p/2.
She will construct a coin, with her number encoded inthe probability of heads. She will send the coin to Bob.
To find q, Bob will flip the coin...
q
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A Communication Puzzle
Alice is going to think of a number q between 0 and p/2.
She will construct a coin, with her number encoded inthe probability of heads. She will send the coin to Bob.
To find q, Bob will flip the coin, but it self-destructs after one flip.
q
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The Goal: Find the optimal encoding p(q )
q
Maximize the mutual information:
Here n is the number of heads Bob tosses (n = 0 or 1), and q is distributed uniformly between 0 and p/2.
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An Optimal Encoding (1 flip) (Information-maximizing for a uniform a priori distribution.)
00
1
probabilityof heads
Alice’s number q p/2
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Modified Puzzle—Bob Gets Two Flips
The coin self-destructs after two flips.
(It’s like sending two photons with the same polarization.)
q
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An Optimal Encoding (2 flips)
00
1
probabilityof heads
Alice’s number q p/2
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New Modification—Bob Gets 25 Flips
The coin self-destructs after 25 flips.
(It’s like sending 25 photons with the same polarization.)
q
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An Optimal Encoding (25 flips)
00
1
probabilityof heads
Alice’s number q p/2
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Taking the limit of an infinite number of flips
For any given encoding pheads(q ), consider the following limit.
We ask what encodings maximize this limit.
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An optimal encoding in the limit of infinitely many flips
00
1
probabilityof heads
Alice’s number q
This is exactly what photons do!
p/2
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Why this works: Wider deviation matches greater slope
00
1n/N
Alice’s number
n/N
p/2
N = number of tosses.
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Another way of seeing the same thing
Δ(n/N) picturedon the probabilityinterval.
same size
p1
p2
p1
p2
Using square roots of probability equalizesthe spread in the binomial distribution.
0 10
1
0 10
1
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A Good Story
In quantum theory, it’s impossible to have a perfectcorrespondence between past and future (in measurement).
But the correspondence is as close as possible, given the limitations of a probabilistic theory with discrete outcomes.
This fact might begin to explain why we have to use “square roots of probability.”
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But this good story is not true!
Why not?
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But this good story is not true!
Why not?
Because probability amplitudes are complex.
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No information maximization in the complex theory.
|
| |
|
An orthogonal measurement completely missesa whole degree of freedom (phase).
pvertical = cos2(/2),
but is not uniformly distributed.
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Quantum theory with d orthogonal states:With real amplitudes, information transfer is again
optimal.
a1
a2
a3
p1
p2
p3
The rule pk ak2 again maximizes the information gainedabout a, compared with other conceivable probability rules,in the limit of an infinite number of trials.
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Making statistical fluctuations uniform and isotropic
p2
p1
p3
p2
p1
p3
In this sense real square roots of probability arise naturally.
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From Am. J. Human Genetics (1967).
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In d dimensions, a pure state holds 2(d1) real parameters, but there are only d1 independent probabilities for a complete orthogonal measurement.
But again, information transfer is not optimal in standard quantum theory with complex amplitudes.
Why complex? Why this factor of 2?
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Conceivable answers to “Why complex amplitudes?”
Want an uncertainty principle (Stueckelberg; Lahti & Maczynski)
Want local tomography (Hardy; Chiribella et al; Müller & Masanes et al; Dakić & Brukner; me)
Want complementarity (Goyal et al)
Want square roots of transformations (Aaronson)
Want algebraic closure (many people)
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A Different Approach:Maybe the real-amplitude theory is correct.
Require that every real operator commute with
Then the real theory in2d dimensions becomesequivalent to the complextheory in d dimensions.
where
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Our take on Stueckelberg’s idea: The ubit model(with Antoniya Aleksandrova and Victoria Borish)
Assume:
Real-amplitude quantum theory A special universal rebit (ubit)—doubles the dimension The ubit interacts with everything
ubit
environment localsystem
We can get an effective theory similar to standard quantum theory.
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Roughly, the ubit plays the role of the phase factor.
The ubit’s state space:
|1-|1
-|i
|i
But treating it as anactual physical systemmakes a difference.
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The dynamics
Generated by an antisymmetric operator S:
w is the ubit’s rotation rate.
s is the strength of the ubit-environment interaction.BEU is chosen randomly.
We do perturbation theory, with s/w as our small parameter.
ubit
environment E localsystem A
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Effect of the environment on the ubit
The state of UA has components proportional to JU and XU and ZU.
ubit
environment E
coefficient of JU coefficient of XU
time
time
We assume (i) rapid rotation of the ubit and (ii) stronginteraction with the environment, so that the aboveprocesses happen much faster than any local process.
localsystem A
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What we assume in our analysis:
Both s (ubit-environment interaction strength) and w (ubit
rotation rate) approach infinity with a fixed ratio. Infinite-dimensional state space of the
environment; thus, infinitely many random parameters in BEU.
What we find for the effective theory of the UA system:
We recover Stueckelberg’s rule: all operators commute with JU. There is no signaling through the ubit (for a UAB
system). As s/w approaches zero, we seem to recover
standard quantum theory. Short of this limit, we see deviations from the
standard theory. We see three distinct effects:
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Precessing qubit
x component of Bloch vector
times/w = 0.3
x
z
(i) Retardation of the evolution (second order in s/w)
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Precessing qubit
x component of Bloch vector
time
Not sure how to lookfor this effect.
There’s another specialaxis that is not specialin standard quantum theory.
x
z
(i) Retardation of the evolution (second order in s/w)
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Precessing qubit
s/w = 0.1
x component of Bloch vector
time
x
z
(ii) Flattening of the Bloch sphere (second order in s/w)
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Precessing qubit
x component of Bloch vector
time
This effect vanishesif we include partof the environmentin an “effective ubit.”
x
z
(ii) Flattening of the Bloch sphere (second order in s/w)
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length of the Bloch vector
time in precession periods, for s/w = 0.1
Precessing qubit
(iii) Decoherence (third order in s/w)
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tcoh ≈ (period)(w /s)3
To be consistent with experimental resultswe must have
s/w < 10-6
(Chou et al, 2011)
length of the Bloch vector
time in precession periods, for s/w = 0.1
Precessing qubit
(iii) Decoherence (third order in s/w)
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Summary
If probability amplitudes were real, we could tell a nice story:information is transferred optimally from preparation to measurement, and this fact could begin to explain “square roots of probability.”
But there is no such optimization if amplitudes are complex.
Many ideas have been proposed to explain complex amplitudes.
We may use the real-amplitude theory, if we supplement it withStueckelberg’s rule: all operators commute with J.
Or we may assume a universal rebit. In a certain limit we seem torecover standard quantum theory, but short of this limit the modelpredicts spontaneous decoherence.