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Experimental quantum foundations Robert Spekkens
Solstice of foundations June 19, 2017
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Realism It aims at a true description of physical objects and their attributes, and aims to provide successively better approximations to the truth over time. The realist endorses a correspondence theory of truth.
Empiricism It aims at an efficient summary of our experience. The empiricist seeks to avoid false belief by building on top of what we cannot be mistaken about, such as statements about what we’ve observed directly.
Pragmatism It drops the notion of truth as correspondence with reality altogether, and aims only to be useful to us in achieving various goals.
What does a scientific theory aim to do?
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Empiricism/realism/pragmatism as a philosophy of science
vs.
Empiricism/realism/pragmatism as a methodological principle for devising new
theories
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What is the historical scorecard for realism vs. empiricism vs pragmatism as methodological principles for devising new theories? - Thermodynamics
- The atomic hypothesis
- Relativity theory
- Quantum theory
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Empiricist
Realist
Pragmatist
Causal structure Rep’n of symmetries
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Axiomatization from pragmatic principles à experimental consequences of pragmatic principles
Pragmatic principles such as: - Second law - No superluminal signalling - Data processing inequality are unlikely to be violated, so one would like to know the scope of physical theories that respect them Variation of axioms a good way to probe alternatives to QT (contrast w/ Weinberg’s proposed modification of QT)
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Experimental metaphysics à Experimental consequences of ontological principles
Provide constraint on ontological possibilities for all future theories of physics This is a precise sense in which experimental quantum foundations distinguishes itself from experiments in the rest of physics
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Empiricist
Realist
Pragmatist
Device-independent
paradigm
Interventionist Causal models
Theory of Bayesian inference
Ontological models
Generalized probabilistic
theories
GPTs w/ symmetries
Thermodynamic
Process theories
Frameworks for describing theories
Information processing
Resource theories
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Empiricist
Realist
Pragmatist
Causal structure Rep’n of symmetries
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The framework of generalized probabilistic
theories (GPTs) See: L. Hardy, quant-ph/0101012 J. Barrett, PRA 75, 032304 (2007)
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Preparation Measurement
The framework of generalized probabilistic theories
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P2
P4 P3
P6 P5
P1
M1
M5
M3
M10 M8
P9 P8
P7
M2
M6
M4
M7
M9
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Preparation Measurement
The framework of generalized probabilistic theories
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Preparation Measurement
The framework of generalized probabilistic theories
pass or fail
Suppose there are K measurements in a tomographically complete set (pass-fail mmts from which one can infer the statistics for all mmts)
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State tomography for a single qubit
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Preparation Measurement
Suppose there are K measurements in a tomographically complete set (pass-fail mmts from which one can infer the statistics for all mmts)
What can we say about f?
“operational state”
The framework of generalized probabilistic theories
pass or fail
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Operational states form a convex set
(w,1-w)
Convex linear
Also true for mmts in tomo. complete set, so
Closed under convex combination –> a convex set
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Convex linearity implies linearity
Therefore
If f is convex linear on GPT states
Then f is linear on GPT states
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Preparation Measurement
“operational effects” “operational states”
S = Convex set in R = Interval of positive cone in
S and R characterize the GPT theory!
The framework of generalized probabilistic theories
pass or fail
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S = a simplex
R = the unit hypercube
s can be any probability distribution
r can be any vector of conditional probabilities
(0,1)
(1,0)
(0,1)
(1,0)
(1,1)
GPT characterization of classical theory
(0,0)
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GPT characterization of classical theory
S = a simplex
R = the unit hypercube
s can be any probability distribution
r can be any vector of conditional probabilities
(0,0,1)
(1,0,0)
(0,1,0)
(0,0,1)
(1,0,0) (0,1,0)
(1,1,0)
(1,1,1)
(0,1,1)
(1,0,1)
(0,0,0)
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GPT characterization of classical theory
S = a simplex
R = the unit hypercube
s can be any probability distribution
r can be any vector of conditional probabilities
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GPT characterization of quantum theory
S = the convex set of such operators
R = an interval of the positive cone of such operators
s can represent any trace one positive operator
r can be any positive operator less than identity
Recall: The Hermitian operators on a Hilbert space of dimension d form a real Euclidean vector space of dimension d2
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Canonical variables
known known known
Probability distributions allowed by the epistemic restriction
0 1 0 1
0 1 0 1
0 1 0 1
0 1 0 1
0 1 0 1
0 1 0 1 0 1
0 1
Nothing known
0 1
0 1
Addition is mod2
Toy theory RWS, PRA 75, 032110 (2007)
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The valid space of GPT states
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The valid space of GPT states
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The valid space of GPT states |0〉
|1〉
|-i〉 |+i〉
|+〉
|-〉 ½I
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Valid measurements:
Any commuting set of canonical variables
0 1 0 1
0 1 0 1
0 1 0 1
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Valid GPT effects
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GPT characterization of convex closure of toy theory
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GPT characterization of boxworld (Popescu-Rohrlich box correlations)
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Quantum
Boxworld
Convex hull of toy theory Generic GPT
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Interesting question about a given GPT:
Does it satisfy No-restriction hypothesis: the space of GPT effects in a theory include all effects that assign positive probabilities to every GPT state in the theory
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GPT characterization of convex closure of toy theory
Dual of effect space
Dual of state space
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Convex theories with maximal dual cone
C* algebraic theories
Quantum theory
Classical theory
Classical Statistical Theories with epistemic restriction
Boxworld
Toy theory
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Deviations from quantum theory in the landscape of
generalized probabilistic theories: Direct constraints from experimental data
Joint work with: Matthew Pusey Mike Mazurek Kevin Resch
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Determining GPT from infinite-run experimental statistics
Preparation Measurement
pass or fail
Pre
para
tions
Measurements
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p(0|P2,M4) =⇣1 P (2)
2 · · · P (k)2
⌘·⇣M (1)
4 · · · M (k)4
⌘
Determining GPT from infinite-run experimental statistics
Preparation Measurement
pass or fail
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Determining GPT from infinite-run experimental statistics
GPT states GPT effects
Preparation Measurement
pass or fail
Use singular value decomposition: k = rank of data matrix
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Quantum
Boxworld
Convex hull of toy theory Generic GPT
GPT states GPT effects
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Determining GPT from finite-run experimental statistics
Raw data Because of statistical
noise, the matrix of raw data is always full rank
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Determining GPT from finite-run experimental statistics
=
Raw data
Find GPT model of best fit of
rank k
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=
Determining GPT from finite-run experimental statistics
Raw data
Find GPT model of best fit of
rank k for Poissonian noise This is the “weighted low-rank approximation problem”
and factorizing appropriately, minimize
In variation over Kij satisfying
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=
Determining GPT from finite-run experimental statistics
Raw data
Find GPT model of best fit of
rank k
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Experimental set-up
Heralded SinglePhoton Source
State Preparation
PBS
Measurement
GT-PBS
Coupler
Mirror
IF
HWP
QWP
Dh
PPKTP
Dt
Drmeascompprep
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Experimental data 100 measurements on 100 preparations
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2 3 4 5 6 7 8 9 107000
8000
9000
10000
2 3 4 5 6 7 8 9 10103
104
105
106
107
10899% rangeFit
Model Rank
(a)
(b) (c)
50 100
Test
Measurement
50
10050 100
Pre
para
tion
Training
Measurement
0.0
0.2
0.4
0.6
0.8
1.0
χ2statistic
Characterize quality of fit by Â2 statistic
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Characterize tradeoff between quality of fit and overfitting by Akaike information criterion
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Characterize tradeoff between quality of fit and overfitting by Akaike information criterion
2 3 4 5 6 7 8 9 100
1
Rel
ativ
e M
odel
Lik
elih
ood
Model Rank
Score(rank-k model) = Exp{-1/2 [AIC(rank-k model) - AIC_min]}, and the relative likelihood of each rank-k model is its score divided by the sum of all the model scores. AIC_min is the minimum AIC value among the 9 model ranks I'm considering, and in this case it is the AIC for the rank-4 model.
rank 4: 0.9998 rank 5: 1.99£10-4
rank 6: 1.6£10-13 all others: <10-25
Use a GPT of rank 4!
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Rank-4 GPT of best fit for the experimental data 100 measurements on 100 preparations
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Rank-4 GPT of best fit for the experimental data 100 measurements on 100 preparations
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2 3 4 5 6 7 8 9 107000
8000
9000
10000
2 3 4 5 6 7 8 9 10103
104
105
106
107
10899% rangeFit
Model Rank
(a)
(b) (c)
50 100
Test
Measurement
50
10050 100
Pre
pa
ratio
nTraining
Measurement
0.0
0.2
0.4
0.6
0.8
1.0χ2statistic
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1006 measurements on 6 preparations 6 measurements on 1006 preparations
More experimental data
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1006 measurements on 6 preparations 6 measurements on 1006 preparations
More experimental data
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VSmin
/VSmax
= 0.968± 0.001
1006 measurements on 1006 preparations
Rank-4 GPT of best fit for the experimental data
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Experimental constraints on violations of Tsirelson bound
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Empiricist
Realist
Pragmatist
Causal structure Rep’n of symmetries
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Experimental constraints on violations of Tsirelson bound
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Some morals of the story
The GPT framework provides a means of analyzing experimental data that does not presume the correctness of quantum theory. Use it for any experiment that seeks to look for deviations from QT! Tomography for states and measurements can be achieved in a bootstrap manner Don’t worry only about underfitting. Worry also about overfitting.