The document discusses two consequences of using pseudorandomness instead of true randomness in experimental quantum physics: 1) It is possible for a local model to violate Bell inequalities if Alice and Bob's inputs are computable functions, and 2) Mixed states prepared by computably sampling pure states retain information about the preparation process that could be detectable.

1. Consequences of the Physical CTT over
experimental setups in quantum physics
Antonio Acín1, Ariel Bendersky1, Gonzalo de La Torre1,
Santiago Figueira2 y Gabriel Senno2
1ICFO-Institut de Ciencies Fotoniques, Barcelona, España
2Departamento de Computación, FCEN, Universidad de Buenos Aires
Workshop on Information and Physics,
Paris, 2015

2. The Physical Church-Turing Thesis
Physical CTT
Every function calculable by a physical system is
Turing-computable.
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3. Outline
We present two consequences of the use of pseudorandomness
instead of randomness in experimental quantum physics:
A local model for Bell-like experiments in which Alice and
Bob use computable (but unknown to an eavesdropper)
inputs.
3

4. Outline
We present two consequences of the use of pseudorandomness
instead of randomness in experimental quantum physics:
A local model for Bell-like experiments in which Alice and
Bob use computable (but unknown to an eavesdropper)
inputs.
When preparing mixed states by computably sampling
pure states, the final preparations retains information on
how it was mixed.
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5. Bell inequality computability loophole
4

6. Bell’s experiments
A source (S) prepares and distributes two physical systems to
distant observers Alice and Bob.
Upon receiving their systems, each observer performs a
measurement on it.
The object of interest is
p(a, b|x, y)
the joint probability distribution of obtaining outcomes a and b
when Alice and Bob choose measurements x and y.
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7. Locality
In general,
p(a, b|x, y) = p(a|x)p(b|y)
Local explanation for distant correlations: past common cause,
λ.
p(a, b|x, y, λ) = p(a|x, λ)p(b|y, λ)
λ may not be constant over all runs.
Hence, in general, we say that a probability distribution is local
if it can be written as:
p(a, b|x, y) =
Λ
dλq(λ)p(a|x, λ)p(b|y, λ)
Measurement independence assumption:
q(λ|x, y) = q(λ)
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8. CHSH inequality
Suppose x, y ∈ {0, 1} and a, b ∈ {−1, +1}, and consider
axby =
a,b
ab p(a, b|x, y)
Let
S = a0b0 + a0b1 + a1b0 − a1b1
Theorem (Clauser, Horne, Shimony, Holt)
Any local probability distribution has to satisfy,
S ≤ 2
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9. Quantum theory is non-local
Predictions for the outcomes of some distant measurements on
entangled systems violate the previous inequality.
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10. Quantum theory is non-local
Predictions for the outcomes of some distant measurements on
entangled systems violate the previous inequality.
For example, if the systems are prepared in the singlet state and
Alice and Bob measure in the following spin directions,
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11. Quantum theory is non-local
Predictions for the outcomes of some distant measurements on
entangled systems violate the previous inequality.
For example, if the systems are prepared in the singlet state and
Alice and Bob measure in the following spin directions,
We have that
S = a0b0 + a0b1 + a1b0 − a1b1
S =
√
2
2
+
√
2
2
+
√
2
2
−
−
√
2
2
= 2
√
2
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12. Loopholes
A loophole, in this context, is an experimental situation
allowing for local devices to generate experimental data
violating a Bell’s inequality.
Examples:
Detection loophole.
Finite statistics loophole.
Locality loophole.
In this work we present, the computability loophole.
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13. The computational loophole
It is convenient for what follows to rephrase the standard Bell
scenario in cryptographic terms. In this approach, Alice and
Bob get their systems from a non-trusted provider Eve.
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14. The computational loophole
It is convenient for what follows to rephrase the standard Bell
scenario in cryptographic terms. In this approach, Alice and
Bob get their systems from a non-trusted provider Eve.
We will allow Eve access to the inputs and outputs of previous
rounds of the experiment.
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15. The computational loophole
It is convenient for what follows to rephrase the standard Bell
scenario in cryptographic terms. In this approach, Alice and
Bob get their systems from a non-trusted provider Eve.
We will allow Eve access to the inputs and outputs of previous
rounds of the experiment. This memory scenario still allows to
see non-locality (Barret et al PRA 66:042111, Pironio et al
Nature 464(7291):1021-1024, Pironio et al PRA 87:012336).
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16. The computational loophole
It is convenient for what follows to rephrase the standard Bell
scenario in cryptographic terms. In this approach, Alice and
Bob get their systems from a non-trusted provider Eve.
We will allow Eve access to the inputs and outputs of previous
rounds of the experiment. This memory scenario still allows to
see non-locality (Barret et al PRA 66:042111, Pironio et al
Nature 464(7291):1021-1024, Pironio et al PRA 87:012336).
We show that,
Theorem
If Alice and Bob choose their measurements following an algorithm,
Eve can prepare devices that locally violate CHSH inequality.
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17. Predicting computable functions
Definition
A class of total computable functions C is identifiable by next
value (C ∈ NV) if there exists a computable function g (called a
predictor for C) such that for every f ∈ C,
(∃n0)(∀n ≥ n0) f(n) = g( f(0), . . . , f(n − 1) ) (1)
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18. Predicting computable functions
Definition
A class of total computable functions C is identifiable by next
value (C ∈ NV) if there exists a computable function g (called a
predictor for C) such that for every f ∈ C,
(∃n0)(∀n ≥ n0) f(n) = g( f(0), . . . , f(n − 1) ) (1)
Proposition (Putnam)
The class of all total computable functions is not identifiable by next
value.
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19. Predicting computable functions
Definition
A class of total computable functions C is identifiable by next
value (C ∈ NV) if there exists a computable function g (called a
predictor for C) such that for every f ∈ C,
(∃n0)(∀n ≥ n0) f(n) = g( f(0), . . . , f(n − 1) ) (1)
Proposition (Putnam)
The class of all total computable functions is not identifiable by next
value.
Theorem (Adleman)
A class of total computable functions is in NV if and only if it is a
subclass of a complexity class (w.r.t. some complexity measure).
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20. Explanation of the loophole
If
Alice and Bob choose their measurements using
computable functions,
Eve doesn’t know the functions, but knows a time
complexity class C that contains them and
on every round, Eve receives the choices of Alice and Bob
of previous rounds..
Then, Eve can prepare devices that
1 on every round, using the choices of previous rounds,
execute a predictor for C and guess what the new choices
will be and
2 output the necessary values for the statistic to violate
CHSH inequality.
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21. Importance of the loophole
Since every computable function belongs to some
complexity class, Alice and Bob can never rule out the
possibility of Eve predicting their functions.
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22. Importance of the loophole
Since every computable function belongs to some
complexity class, Alice and Bob can never rule out the
possibility of Eve predicting their functions.
Therefore no computable pseudo randomness criterion
will suffice for a proper Bell inequality violation.
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23. Importance of the loophole
Since every computable function belongs to some
complexity class, Alice and Bob can never rule out the
possibility of Eve predicting their functions.
Therefore no computable pseudo randomness criterion
will suffice for a proper Bell inequality violation.
Other sources of randomness:
1 Quantum coins. Not desirable to assume a non-local theory,
like quantum mechanics, in order to test non-locality.
2 Free will. Can humans generate non-computable
sequences?
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24. Mixing states computably
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25. Two kinds of mixed states
Proper mixed states
Describe ensembles of pure states of which we have classical
uncertainty.
Improper mixed states
Describe systems which form part of bigger quantum system in
a pure state.
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26. Case 1
R2D2 chooses from each box. The observer only knows
that R2D2 will pick half times each state but not how he’ll
pick each time.
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27. Case 1
R2D2 chooses from each box. The observer only knows
that R2D2 will pick half times each state but not how he’ll
pick each time.
The state, as described by the observer is ρ = I
2 .
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28. Case 2
C3PO chooses from each box. The observer only knows
that C3PO will pick half times each state but not how he’ll
pick each time.
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29. Case 2
C3PO chooses from each box. The observer only knows
that C3PO will pick half times each state but not how he’ll
pick each time.
The state, as described by the observer is ρ = I
2 .
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30. Undistiguishable
Both situations seem to be undistinguishable.
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31. However...
... they are robots, so they can only choose in a computable
manner.
Any classical system used to choose only yields computable
choices.
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32. Therefore
We will be able to distinguish both situations.
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33. Distinguishing computable preparations
Assumption
We have a black box containing one of the two previous
situations and we want to know which one it is.
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34. Distinguishing computable preparations
Assumption
We have a black box containing one of the two previous
situations and we want to know which one it is.
Procedure
We measure every odd qubit on the basis of eigenstates of σX
and every even qubit on the basis of eigenstates of σZ
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35. Distinguishing computable preparations
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36. Distinguishing computable preparations
Now what?
We obtain two sequences.
When we measure in the same basis as the preparation, the
sequence obtained is computable.
When we measure in the other basis, the sequence
obtained is a fair coin.
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37. Distinguishing computable preparations
Now what?
We obtain two sequences.
When we measure in the same basis as the preparation, the
sequence obtained is computable.
When we measure in the other basis, the sequence
obtained is a fair coin.
Let’s go classical
Can we distinguish a computable sequence from one arising
from a fair coin with high probability of success?
We proved this to be true.
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38. The distinguishing protocol
Idea
Almost all sequences arising from independent tosses of fair
coins have incompressible prefixes.
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39. The distinguishing protocol
Input: k ∈ N and X, Z ∈ {0, 1}ω, two bit sequences with the
promise that one of them is computable.
Output: ‘X’ or ‘Z’ as the candidate for being computable;
wrong answer with probability bounded by O(2−k).
for t = 0, 1, 2 . . . do
for p = 0, . . . , t do
if Ut(p) = X k|p| then
output ‘X’ and halt
if Ut(p) = Z k|p| then
output ‘Z’ and halt
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40. Probability of misrecognition
Probability of error
Perror ≤
>0
2
2k
=
2−(k−1)
1 − 2−(k−1)
which goes to 0 as k goes to ∞.
We pick a k such that the error is lower than what we want, and
then we run the recognition algorithm.
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41. Some subtelties
Our algorithm, although it runs in finite time, is infeasible.
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42. Some subtelties
Our algorithm, although it runs in finite time, is infeasible.
Still, the state has the information on how it was mixed.
This is surprising from a fundamental point of view.
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43. Some subtelties
Our algorithm, although it runs in finite time, is infeasible.
Still, the state has the information on how it was mixed.
This is surprising from a fundamental point of view.
A slight variation on the algorithm makes it noise tolerant
for noise rates up to 0,21.
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44. Assuming the Physical CTT
Physical CTT
Every function calculable by a physical system is
Turing-computable.
So, deterministic physical processes, if we accept the
impossibility of preparing non-computable initial conditions,
won’t do it.
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45. Assuming the Physical CTT
Physical CTT
Every function calculable by a physical system is
Turing-computable.
So, deterministic physical processes, if we accept the
impossibility of preparing non-computable initial conditions,
won’t do it.
Consequences
We are left with quantum randomness. This means that:
Only kind of mixed state is those being part of larger
system in a pure state.
Need quantum randomness to test quantum non-locality.
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