### A simple quantum experiment Computing probabilities in a sensible way (½)(½)+(½)(½) = ½ ½ ½

```A simple quantum experiment
Computing probabilities in a sensible way
(½)(½)+(½)(½) = ½
½
½
Computing probabilities in a sensible way
½
½
The experimental result
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100%
A quantum explanation of this result
A quantum explanation of this result
A quantum explanation of this result
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100%
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
Measuring photon polarization
42
°
polarizing filter
Measuring photon polarization
42
°
polarizing filter
There is no such device.
Measuring photon polarization
polarizing filter
polarizing beam splitter
Measuring photon polarization
polarizing filter
polarizing beam splitter
Measuring photon polarization
polarizing filter
By measuring many photons,
we can estimate the probability
of the vertical outcome.
This tells us about the angle.
The standard account of probability vs angle
Squaring the
amplitude
gives
the probability:
p
q
amplitude
for vertical
q
A completely different explanation for that curve:
Optimal information transfer?
polarizing filter
The angle varies continuously.
But the measurement is probabilistic
with only two possible outcomes.
Is the communication optimal?
A Communication Puzzle
q
Alice is going to think of a number q between 0 and p/2.
A Communication Puzzle
q
Alice is going to think of a number q between 0 and p/2.
She will construct a coin, with her number encoded in
A Communication Puzzle
q
Alice is going to think of a number q between 0 and p/2.
She will construct a coin, with her number encoded in
the probability of heads. She will send the coin to Bob.
A Communication Puzzle
q
Alice is going to think of a number q between 0 and p/2.
She will construct a coin, with her number encoded in
the probability of heads. She will send the coin to Bob.
To find q, Bob will flip the coin...
A Communication Puzzle
q
Alice is going to think of a number q between 0 and p/2.
She will construct a coin, with her number encoded in
the 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.
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.)
1
probability
0
0
Alice’s number q
p/2
Modified Puzzle—Bob Gets Two Flips
q
The coin self-destructs after two flips.
(It’s like sending two photons with the same polarization.)
An Optimal Encoding (2 flips)
1
probability
0
0
Alice’s number q
p/2
New Modification—Bob Gets 25 Flips
q
The coin self-destructs after 25 flips.
(It’s like sending 25 photons with the same polarization.)
An Optimal Encoding (25 flips)
1
probability
0
0
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
1
probability
0
0
Alice’s number q
This is exactly what photons do!
p/2
Why this works: Wider deviation matches greater slope
1
n/N
N = number of tosses.
n/N
0
0
Alice’s number
p/2
Another way of seeing the same thing
Δ(n/N) pictured
on the probability
interval.
1
1
p2
p2
0
0
0
p1
1
same size
0
p1
Using square roots of probability equalizes
the spread in the binomial distribution.
1
A Good Story
In quantum theory, it’s impossible to have a perfect
correspondence 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 misses
a 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.
p3
a3
a2
p2
a1
p1
The rule pk ak2 again maximizes the information gained
about a, compared with other conceivable probability rules,
in the limit of an infinite number of trials.
Making statistical fluctuations uniform and isotropic
p3
p3
p2
p2
p1
p1
In this sense real square roots of probability arise
naturally.
From Am. J. Human Genetics
(1967).
But again, information transfer is not optimal in
standard quantum theory with complex amplitudes.
In d dimensions, a pure state holds
2(d1) real parameters, but there are
only d1 independent probabilities for
a complete orthogonal measurement.
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
where
Then the real theory in
2d dimensions becomes
equivalent to the complex
theory in d dimensions.
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
local
system
We can get an effective theory similar to standard quantum
Roughly, the ubit plays the role of the phase factor.
The ubit’s state space:
|i
|1
But treating it as an
actual physical system
makes a difference.
|1
|i
ubit
The dynamics
environment E
local
system A
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.
Effect of the environment on the ubit
The state of UA has components
proportional to JU and XU and ZU.
coefficient of JU
ubit
environment E
local
system A
coefficient of XU
time
time
We assume (i) rapid rotation of the ubit and (ii) strong
interaction with the environment, so that the above
processes happen much faster than any local process.
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:
z
Precessing qubit
x
(i) Retardation of the evolution
(second order in s/w)
x component of Bloch vector
s/w = 0.3
time
z
Precessing qubit
x
(i) Retardation of the evolution
(second order in s/w)
x component of Bloch vector
Not sure how to look
for this effect.
time
There’s another special
axis that is not special
in standard quantum
theory.
z
Precessing qubit
x
(ii) Flattening of the Bloch sphere
(second order in s/w)
x component of Bloch vector
s/w = 0.1
time
z
Precessing qubit
x
(ii) Flattening of the Bloch sphere
(second order in s/w)
x component of Bloch vector
This effect vanishes
if we include part
of the environment
in an “effective ubit.”
time
Precessing qubit
(iii) Decoherence
(third 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)
length of the Bloch vector
tcoh ≈ (period)(w /s)3
To be consistent with
experimental results
we must have
s/w < 106
time in precession periods, for s/w = 0.1
(Chou et al, 2011)
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 with
Stueckelberg’s rule: all operators commute with J.
Or we may assume a universal rebit. In a certain limit we seem to
recover standard quantum theory, but short of this limit the model
predicts spontaneous decoherence.
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