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 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAA

Classical statistical theory + fundamental restriction on statistical distributions A large part of quantum theory In the sense of reproducing the operational predictions

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

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

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!

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?

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

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…

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…

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/0101012 Barrett, quant-ph/0508211 Leifer, quant-ph/0611233 Etcetera, etcetera

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/9601025 Rovelli, quant-ph/9609002 Hardy, quant-ph/9906123 Brukner and Zeilinger, quant-ph/0005084 Kirkpatrick, quant-ph/0106072 Collins and Popescu, quant-ph/0107082 Fuchs, quant-ph/0205039 Emerson, quant-ph/0211035 etcetera

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

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

Restricted statistical theory of trits Joint work with Olaf Schreiber Building upon: Spekkens, quant-ph/0401052 [Phys. Rev. A 75, 032110 (2007)] S. van Enk, arxiv:0705.2742 [Found. Phys. 37, 1447 (2007)] D. Gross, quant-ph/0602001 [J. Math. Phys. 47, 122107 (2006)] Bartlett, Rudolph, Spekkens, unpublished

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

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

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:0705.2742

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

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

Epistemic states of maximal knowledge known known known known 0 1 2 1 2 0 1 2 1 2 0 1 2 1 2 0 1 2 1 2 0 1 2 1 2 0 1 2 1 2 0 1 2 1 2 0 1 2 1 2 0 1 2 1 2 0 1 2 1 2 0 1 2 1 2 0 1 2 1 2

Epistemic states of non-maximal knowledge Nothing known 0 1 2 1 2

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

How to represent this graphically 2 1 00 01 02 10 11 12 20 21 22 0 1 2 22 21 20 12 11 10 02 01 00 00 01 02 10 11 12 20 21 22

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

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

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

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

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

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

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

On a single trit On a pair of trits 22 22 21 21 20 20 12 12 11 11 10 0 1 2 0 1 2 0 1 2 0 1 2 On a pair of trits 22 21 20 12 11 10 02 01 00 22 21 20 12 11 10 02 01 00 00 01 02 10 11 12 20 21 22 00 01 02 10 11 12 20 21 22

Valid transformations Maps validity-preserving

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

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

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

A symplectic transformation for a pair of trits 22 21 20 12 11 10 02 01 00 00 01 02 10 11 12 20 21 22

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

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

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 0 1 2 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

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

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

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, …

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…

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

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

Restricted statistical theory of bits ( “the toy theory”) Spekkens, quant-ph/0401052 [Phys. Rev. A 75, 032110 (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)

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

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

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?