1. Robert Spekkens DAMTP, Cambridge
The power of epistemic restrictions in axiomatizing quantum theory: from trits to qutrits
Robert Spekkens
DAMTP, Cambridge
May 15, 2008
Information primitives and laws of nature
Funding by:
The Royal Society
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2. Classical statistical theory +
fundamental restriction on statistical distributions
A large part of quantum theory
In the sense of reproducing the operational predictions

3. Classical statistical theory +
fundamental restriction on statistical distributions
A large part of quantum theory
In the sense of reproducing the operational predictions
Strong evidence for the view that
quantum states are states of knowledge,
not states of reality

4. Restricted Statistical theory for the classical theory
Liouville mechanics
Mechanics
Restricted Liouville mechanics
= Gaussian quantum mechanics
Bits
Statistical theory of bits
Restricted statistical theory of bits
' Stabilizer theory for qubits
Trits
Statistical theory of trits
Restricted statistical theory of trits
= Stabilizer theory for qutrits

5. These theories include:
Most basic quantum phenomena
e.g. noncommutativity, Interference, coherent superposition, collapse, complementary bases, no-cloning, …
Most quantum information-processing tasks
e.g. teleportation, key distribution, quantum error correction, improvements in metrology, dense coding, …
A large part of entanglement theory
e.g. monogamy, distillation, deterministic and probabilistic single copy entanglement transformation, catalysis, …
A large part of the formalism of quantum theory
e.g. Choi-Jamiolkowski isomorphism, Naimark extension, Stinespring dilation, multiple convex decompositions of states, …
A great deal of what we usually take to be mysteriously nonclassical!

6. Restricted Statistical theory for the classical theory
Liouville mechanics
Mechanics
Restricted Liouville mechanics
= Gaussian quantum mechanics
Bits
Statistical theory of bits
Restricted statistical theory of bits
' Stabilizer theory for qubits
Trits
Statistical theory of trits
Restricted statistical theory of trits
= Stabilizer theory for qutrits
Optics
Statistical optics
Restricted statistical optics
= linear quantum optics
Electrodynamics
Statistical electrodynamics
Restricted statistical electrodynamics
=/' part of QED?
General relativity
Statistical GR
Restricted statistical GR
=/' part of quantum gravity?

7. Categorizing quantum phenomena
Those arising in a restricted statistical classical theory
Those not arising in a restricted statistical classical theory
Wave-particle duality
collapse
Interference
noncommutativity
Teleportation
entanglement
No cloning
Quantized spectra
Coherent superposition
Quantum eraser
Key distribution
Bell inequality violations
+ no-signalling
Bell-Kochen-Specker theorem
Improvements in metrology
Pre and post-selection
“paradoxes”
Computational speed-up
Particle statistics

8. Categorizing quantum phenomena
Those arising in a restricted statistical classical theory
Those not arising in a restricted statistical classical theory
Interference
Noncommutativity
Entanglement Collapse
Wave-particle duality
Teleportation
No cloning
Key distribution
Improvements in metrology
Quantum eraser
Coherent superposition
Pre and post-selection
“paradoxes”
Others…
Bell inequality violations + no-signalling
Computational speed-up (if it exists)
Bell-Kochen-Specker theorem
Certain aspects of items on the left
Others…
Quantized spectra?
Particle statistics?
Others…

9. Still surprising! Find more! Focus on these
Categorizing quantum phenomena
Those arising in a restricted statistical classical theory
Those not arising in a restricted statistical classical theory
Interference
Noncommutativity
Entanglement Collapse
Wave-particle duality
Teleportation
No cloning
Key distribution
Improvements in metrology
Quantum eraser
Coherent superposition
Pre and post-selection
“paradoxes”
Others…
Bell inequality violations + no-signalling
Computational speed-up (if it exists)
Bell-Kochen-Specker theorem
Certain aspects of items on the left
Others…
Still surprising!
Find more!
Focus on these
Not so strange after all!
Quantized spectra?
Particle statistics?
Others…

10. A research program
The success of restricted statistical theories suggests that:
Quantum theory is best understood as a kind of probability theory
Hardy, quant-ph/
Barrett, quant-ph/
Leifer, quant-ph/
Etcetera, etcetera

11. A research program
The success of restricted statistical theories suggests that:
Quantum theory is best understood as a kind of probability theory
and that
Quantum states are states of incomplete knowledge
Caves and Fuchs, quant-ph/
Rovelli, quant-ph/
Hardy, quant-ph/
Brukner and Zeilinger, quant-ph/
Kirkpatrick, quant-ph/
Collins and Popescu, quant-ph/
Fuchs, quant-ph/
Emerson, quant-ph/
etcetera

12. A research program
The success of restricted statistical theories suggests that:
Quantum theory is best understood as a kind of probability theory
and that
Quantum states are states of incomplete knowledge of a deeper reality

13. A research program
The success of restricted statistical theories suggests that:
Quantum theory is best understood as a kind of probability theory
and that
Quantum states are states of incomplete knowledge of a deeper reality
Speculative possibility for an axiomatization of quantum theory
Principle 1: There is a fundamental restriction on observers capacities to know and control the systems around them
Principle 2: ??? (Some change to the classical picture of the world)
We need to explore possibilities for principle 2, even if only in toy theories
Ultimately, we need to derive principle 1

14. Restricted statistical theory of trits
Joint work with Olaf Schreiber
Building upon:
Spekkens, quant-ph/ [Phys. Rev. A 75, (2007)]
S. van Enk, arxiv: [Found. Phys. 37, 1447 (2007)]
D. Gross, quant-ph/ [J. Math. Phys. 47, (2006)]
Bartlett, Rudolph, Spekkens, unpublished

15. Does quantum theory suggest a good statistical restriction?
For finite-d system, no good analogue of
Instead, note simply that:
Jointly-measurable observables = a commuting set of observables
This suggests
Jointly-knowable variables = a set of canonically nonconjugate variables
Continuous case: variables with vanishing Poisson bracket
where

16. Discrete configuration space
Discrete phase space
General canonical variable
Vector in symplectic vector space associated with variable
Discrete case: nonconjugate variables are those such that
where

17. The discrete classical uncertainty principle:
The only distributions that can be prepared are those that correspond to fixing the values of a set of canonical and nonconjugate variables
An observer can know at most half of the full set of variables
(but only certain halves)
Contrast the proposal by van Enk, arxiv:

18. A single trit Canonical variables Canonically nonconjugate sets:
The singleton sets
the empty set
Suppose X is known to be 0 2 1
1/3
1/3
1/3 1 2

19. A single trit Canonical variables Canonically nonconjugate sets:
The singleton sets
the empty set
Suppose X is known to be 0 2 1 1 2

20. Epistemic states of maximal knowledge
known
known
known
known 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

21. Epistemic states of non-maximal knowledge
Nothing known 1 2

22. A pair of trits Canonical variables 1st system variables
2nd system variables
Joint variables
Canonically nonconjugate sets :
2 variables known:
1 variable known
The singleton sets
Nothing known

23. How to represent this graphically2 1 22 21 20 12 11 10 02 01 00

24. 1 variable known known known 22 22 21 21 20 20 12 12 11 11 10 10 02 0201 00 22 21 20 12 11 10 02 01 00

25. 2 variables known and known 22 21 20 12 11 10 02 01 00

26. Uncorrelated pure epistemic states
X2=0
P2=0
X2+P2=0
X2-P2=0
X1=0
P1=0
X1+P1=0
X1-P1=0

27. 1 variable known known known 22 22 21 21 20 20 12 12 11 11 10 10 02 0201 00 22 21 20 12 11 10 02 01 00

28. 2 variables known and known 22 21 20 12 11 10 02 01 00

29. Correlated pure epistemic states
X1+X2=0
P1-P2=0
X1+X2=0
X1+P1-P2=0
X1+X2=0
X1-P1+P2=0
X1-X2=0
P1+P2=0
X1-X2=0
X1-P1-P2=0
X1-X2=0
P1+X2+P2=0
X1-P2=0
P1-X2=0
X1-P2=0
P1-X2+P2=0
X1-P2=0
P1-X2-P2=0
P1-X2=0
X1+P1-P2=0
X1+P1-X2=0
X1+X2-P2=0
P1-P2=0
X1-P1+X2=0
P1-X2=0
X1-P1-P2=0
X1+P1-X2=0
X1-P1-X2-P2=0
X1-X2-P2=0
X1-P1+X2=0
X1+P2=0
P1+X2=0
X1+P2=0
P1+X2+P2=0
X1+P2=0
P1+X2-P2=0
P1+X2=0
X1+X2+P2=0
X1+P1-P2=0
P1-X2+P2=0
X1+P1+P2=0
P1-P2=0
P1+X2=0
X1+P1+P2=0
P1+P2=0
X1-X2+P2=0
X1-P1-P2=0
P1-X2-P2=0

30. Valid measurements Sets of indicator functions
= probability of k given (x,p)

31. On a single trit On a pair of trits 22 22 21 21 20 20 12 12 11 11 10
On a pair of trits 22 21 20 12 11 10 02 01 00 22 21 20 12 11 10 02 01 00

32. Valid transformations
Maps
validity-preserving

33. Valid transformations
Maps
Completely validity-preserving
This rules out, for example

34. Deterministic transformations
The group of symplectic affine transformations (Clifford group)
where
and

35. Symplectic transformations for a single trit
P → X
X → -P
P → X
X → X-P
P → X
X → -X-P
P → -X
X → P
P → -X
X → X+P
P → -X
X → P-X
P → P
X → X
P → P
X → X+P
P → P
X → X-P
P → X+P
X → X
P → X+P
X → P-X
P → X+P
X → -P
P → P-X
X → X
P → P-X
X → P
P → P-X
X → -X-P
P → -P
X → -X
P → -P
X → P-X
P → -P
X → -X-P
P → X-P
X → -X
P → X-P
X → X+P
P → X-P
X → -P
P → -X-P
X → -X
P → -X-P
X → P
P → -X-P
X → X-P

36. A symplectic transformation for a pair of trits22 21 20 12 11 10 02 01 00

37. Measurement-induced transformations
Measure X in a reproducible way
Prepare P=0 1 2 2 2 1 1

38. Measurement-induced transformations
Measure X in a reproducible way
Prepare P=0 1 2 2 2 1 1
A reproducible measurement of X leads to an unknown disturbance to P
Root of collapse and noncommutativity

39. Unknown disturbance to P1
Note: the evolution is deterministic if the apparatus is treated internally
Internal apparatus
External apparatus
Measure X2
Prepare X2
Measure X1 1 2
Unknown disturbance to P1
Implement
Final X2 reflects initial X1
Final P1 reflects initial P2
The position of the internal-external cut doesn’t matter

44. known
known
and 22 21 20 12 11 10 02 01 00 22 21 20 12 11 10 02 01 00

45. Implies equivalence for measurements and transformations
Therefore
Theorem: The restricted statistical theory of trits is empirically equivalent to the Stabilizer theory for qutrits

46. Stabilizer qutrit theory includes:
Most basic quantum phenomena
e.g. noncommutativity, Interference, coherent superposition, collapse, complementary bases, no-cloning, …
Most quantum information-processing tasks
e.g. teleportation, key distribution, quantum error correction, improvements in metrology, dense coding, …
A large part of entanglement theory
e.g. monogamy, distillation, deterministic and probabilistic single copy entanglement transformation, catalysis, …
A large part of the formalism of quantum theory
e.g. Choi-Jamiolkowski isomorphism, Naimark extension, Stinespring dilation, multiple convex decompositions of states, …

47. Categorizing quantum phenomena
Those arising in a restricted statistical classical theory
Those not arising in a restricted statistical classical theory
Interference
Noncommutativity
Entanglement Collapse
Wave-particle duality
Teleportation
No cloning
Key distribution
Improvements in metrology
Quantum eraser
Coherent superpositions
Pre and post-selection
“paradoxes”
Others…
Bell inequality violations + no-signalling
Computational speed-up (if it exists)
Bell-Kochen-Specker theorem
Certain aspects of items on the left
Others…
Quantized spectra?
Particle statistics?
Others…

48. Restricted Liouville mechanics
Bartlett, Rudolph, Spekkens, unpublished
Liouville distribution on phase space
e.g.
What is a good statistical restriction to apply ?
Classical analogue of Heisenberg uncertainty principle

49. The classical uncertainty principle:
The only Liouville distributions that can be prepared are those satisfying
and that have maximal entropy for a given set of second-order moments.
The theory is empirically equivalent to Gaussian quantum mechanics

50. Restricted statistical theory of bits ( “the toy theory”)
Spekkens, quant-ph/ [Phys. Rev. A 75, (2007)]
The knowledge-balance principle:
The only distributions that can be prepared are those that correspond to knowing at most half the information (Specifically, half of a set of binary variables sufficient to specify the state)

51. Why the restricted statistical theory of bits
Is not equivalent to qubit stabilizer theory
Even number of correlations
Odd number of correlations
Qubit stabilizer theory is nonlocal and contextual (e.g. GHZ)
Restricted statistical theory of bits is local and noncontextual
The theory is close to but not precisely the Stabilzer theory for qubits

52. bits Continuous variables Trits
Comparison of successful epistemic restrictions
bits
The knowledge-balance principle:
The only distributions that can be prepared are those that correspond to knowing at most half the information (Specifically, half of a set of binary variables sufficient to specify the state)
Continuous variables
The classical uncertainty principle:
The only distributions that can be prepared are those satisfying
and that have maximal entropy for a given set of 2nd-order moments.
Trits
The discrete classical uncertainty principle:
The only distributions that can be prepared are those that correspond to fixing the values of a set of canonical and nonconjugate variables

53. Is there a general form for these restrictions?
Note: no theory “between” Stabilizer/Gaussian (Clifford operations)
and full quantum theory (SU(d) operations)
For bits
knowledge-balance principle
 discrete uncertainty principle?
For bits and trits
discrete uncertainty principle
 “discrete covariance matrix” constraint + entropy maximization?