A simple quantum experiment

Computing probabilities in a sensible way

(½)(½)+(½)(½) = ½

½

½

½

½

Computing probabilities in a sensible way

The experimental result

0%

100%

A quantum explanation of this result

A quantum explanation of this result

0%

100%

A quantum explanation of this result

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.

Photon polarization

polarizing filter

Measuring photon polarization

polarizing filter

polarizing filter

42°

Measuring photon polarization

polarizing filter

42°

There is no such device.

Measuring photon polarization

polarizing filter

Measuring photon polarization

polarizing beam splitter

polarizing filter

Measuring photon polarization

polarizing beam splitter

polarizing filter

Measuring photon polarization

By measuring many photons,we can estimate the probabilityof the vertical outcome. This tells us about the angle.

The standard account of probability vs angle

amplitudefor vertical

Squaring the amplitudegives the probability:

q

p

q

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?

A Communication Puzzle

Alice is going to think of a number q between 0 and p/2.

q

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

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

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

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

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.

An Optimal Encoding (1 flip) (Information-maximizing for a uniform a priori distribution.)

00

1

probabilityof heads

Alice’s number q p/2

Modified Puzzle—Bob Gets Two Flips

The coin self-destructs after two flips.

(It’s like sending two photons with the same polarization.)

q

An Optimal Encoding (2 flips)

00

1

probabilityof heads

Alice’s number q p/2

New Modification—Bob Gets 25 Flips

The coin self-destructs after 25 flips.

(It’s like sending 25 photons with the same polarization.)

q

An Optimal Encoding (25 flips)

00

1

probabilityof heads

Alice’s number q p/2

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.

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

Why this works: Wider deviation matches greater slope

00

1n/N

Alice’s number

n/N

p/2

N = number of tosses.

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

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.”

But this good story is not true!

Why not?

But this good story is not true!

Why not?

Because probability amplitudes are complex.

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.

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.

Making statistical fluctuations uniform and isotropic

p2

p1

p3

p2

p1

p3

In this sense real square roots of probability arise naturally.

From Am. J. Human Genetics (1967).

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?

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)

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

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.

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.

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

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

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:

Precessing qubit

x component of Bloch vector

times/w = 0.3

x

z

(i) Retardation of the evolution (second order in s/w)

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)

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)

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)

length of the Bloch vector

time in precession periods, for s/w = 0.1

Precessing qubit

(iii) Decoherence (third order in s/w)

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)

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.