Canadian Summer School on Quantum Information (CSSQI) (10th : 2010)

Why the quantum? Insights from classical theories with a statistical restriction Spekkens, Robert Jul 23, 2010

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Why the quantum? Insights from classical theories with a statistical restriction  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 ψ ψ'  i.e. quantum states emerge as statistical distributions (epistemic states)  Classical theory  Statistical theory for the classical theory  Restricted Statistical theory for the classical theory  Mechanics  Liouville 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, …  Categorizing quantum phenomena Those not arising in a restricted statistical classical theory  Those arising in a restricted statistical classical theory Wave-particle duality noncommutativity  Teleportation  entanglement Quantized spectra Key distribution  collapse  Interference  No cloning  Coherent superposition  Bell inequality violations  Quantum eraser  Bell-Kochen-Specker theorem  Improvements in metrology Computational speed-up Particle statistics  Pre and post-selection “paradoxes”  Categorizing quantum phenomena Those arising in a restricted statistical classical theory  Those not arising in a restricted statistical classical theory  Interference Bell inequality violations Noncommutativity Computational speed-up Entanglement Bell-Kochen-Specker theorem Collapse Certain aspects of items on the left Wave-particle duality Others… Teleportation No cloning Key distribution Improvements in metrology Quantum eraser Coherent superposition Pre and post-selection “paradoxes” Others… Quantized spectra? Particle statistics? Others…  Categorizing quantum phenomena Those arising in a restricted statistical classical theory  Those not arising in a restricted statistical classical theory  Interference Bell inequality violations Noncommutativity Computational speed-up Entanglement Bell-Kochen-Specker theorem Collapse Certain aspects of items on the left Wave-particle duality Others… Teleportation Still surprising! No cloning Key distribution Find more! Not so strange after all! Improvements in metrology Focus on these Quantum eraser Coherent superposition Pre and post-selection “paradoxes” Others… Quantized spectra? Particle statistics? Others…  A research program 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)  Classical theory  Statistical theory for the classical theory  Restricted Statistical theory for the classical theory  Mechanics  Liouville 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  Classical theory  Statistical theory for the classical theory  Restricted Statistical theory for the classical theory  Mechanics  Liouville mechanics  Restricted Liouville mechanics = Quadrature 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  Classical complementarity as a statistical restriction with broad applicability Joint work with Olaf Schreiber Building upon: RS, 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)] S. Bartlett, T. Rudolph, RS, unpublished  A fact about operational quantum theory: Jointly-measurable observables  =  a commuting set of observables (relative to matrix commutator)  This suggests a restriction on a classical statistical theory: Jointly-knowable variables  =  a commuting set of variables (relative to Poisson bracket)  Continuous degrees of freedom Configuration space: Rn ∋ ( x1 , x2 , . . . , xn) Phase space: - ≡ R2n ∋ ( x1 , p1 , x2 , p2 , . . . , xn, pn) ≡ m Functionals on phase space: F : - → R  Xk ( m) = xk Pk ( m) = pk Poisson bracket of functionals:  The linear functionals / canonical variables are:  Independent of m  Discrete degrees of freedom  Zd = {0, 1, . . . , d − 1}  Configuration space:( Zd) n ∋ ( x1 , x2 , . . . , xn) Phase space: - ≡ ( Zd) 2n ∋ ( x1 , p1 , x2 , p2 , . . . , xn, pn ) ≡ m Functionals on phase space: F : - → Zd  Xk ( m) = xk Pk ( m) = pk Poisson bracket of functionals: ( F [m + exi ] − F [m]) ( G[m + epi ] − G[m]) −( F [m + epi ] − F [m]) ( G[m + eqi ] − G[m]) The linear functionals / canonical variables are:  Independent of m  A canonically conjugate pair e.g. A commuting pair e.g.  The principle of classical complementarity: An observer can only have knowledge of the values of a commuting set of canonical variables and is maximally ignorant otherwise.  Symplectic geometry Symplectic inner product  ω( m, m′) = mT Jm′  ω : - ×- →R where  Thus  ω( m, m′) =  ′ ′ i q i pi − pi q i  The linear functionals  F=  i ( aiXi + bi Pi ) form a dual space  {X1 , P1 , . . . , Xn, Pn} is dual to {ex1 , ep1 , . . . , exn , epn }  Poisson bracket of functionals = symplectic inner product of vectors  Valid epistemic states: These are specified by: Example:  A set of known variables V  V = {X1 , P2 }  ∀F, G ∈ V : [F, G] = 0 A valuation of the known variables  v( X1 ) = 2, v( P2 ) = 2  v : V → R( Z d ) Equivalently, An isotropic subspace V ⊆ Ω∗  v  ∀F, G ∈ V : ω( F, G) = 0 A valuation vector v ∈  V*⊆  Ω  v : ∀F ∈ V, F T v = v( F )  V  The ontic states consistent with the epistemic state (V,v) are  {m ∈ - | ∀F ∈ V : F ( m) = v( F ) } = {m ∈ - | ∀F ∈ V : F T m = F T v} = {m ∈ - | PV m = v} ≡V⊥+ v  (Dirac-delta / Kronecker delta)  The associated distribution is 1δ pV,v ( m)) = N V ⊥ + v ( m))  Example  V = {X1 , X2 } v( X1 ) = 1, v( X2 ) = 2  V ⊥ + v = {m ∈ - | X1 ( m) = 1, X2 ( m) = 2} = {( 1, s, 2, t) | s, t ∈ R} “Heisenberg picture” and “Schrodinger picture”  v  Valid reversible transformations: Those that preserve the Poisson bracket / symplectic inner product: The group of symplectic affine transformations (Clifford group)  where and  Symplectic Affine (Heisenberg-Weyl)  Valid reproducible measurements: Any commuting set of canonical variables  Restricted Liouville mechanics  -  2n = R  X  P  Valid epistemic states for a single degree of freedom X  X  V v P  v  X  V  P  X  V  v  P  V=∅  P  Valid epistemic states for a pair of degrees of freedom  Restricted statistical theory of trits - = ( Z3 ) 2n 2 1 0 0  1  2  Valid epistemic states for a single trit Commuting sets: The singleton sets the empty set  Canonical variables  known  known 2 1 0  2 1 0  0 1 2  2 1 0  2 1 0 0 1 2  2 1 0 0 1 2  0 1 2  2 1 0 0 1 2  0 1 2  2 1 0 0 1 2  Nothing known 2 1 0  0 1 2  0 1 2  2 1 0  2 1 0  known 2 1 0  2 1 0 0 1 2  0 1 2  2 1 0  known  0 1 2  0 1 2  Valid epistemic states for a pair of trits a1 , b1 , a2 , b2 ∈ Z3  Canonical variables  How to represent this graphically 22 21 20 12 11 10 02 01 00 00 01 02 10 11 12 20 21 22  1 variable known known 22 21 20 12 11 10 02 01 00  known 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  1 variable known  known 22 21 20 12 11 10 02 01 00  known 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  Valid reproducible measurements On a single trit 2 1 0  2 1 0 0 1 2  2 1 0  2 1 0 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  etc.  00 01 02 10 11 12 20 21 22  Restricted statistical theory of bits - = ( Z2 ) 2n  1 0 0  1  A single bit a, b ∈ Z2  Canonical variables  Epistemic states of maximal knowledge known 1 0  known 1 0  0 1  1 0  1 0 0 1  1 0 0 1  known  0 1  1 0 0 1  0 1  Epistemic states of non-maximal knowledge Nothing known 1 0 0 1  A pair of bits a1 , b1 , a2 , b2 ∈ Z2  Canonical variables  11 10 01 00 00 01 10 11  1 variable known known  known  11 10 01 00  11 10 01 00 00 01 10 11  00 01 10 11  2 variables known and  known  11 10 01 00 00 01 10 11  1 variable known known  known  11 10 01 00  11 10 01 00 00 01 10 11  00 01 10 11  2 variables known and 11 10 01 00 00 01 10 11  known  1 0  1 0 0 1  1 0 0 1  0 1  Equivalence of these restricted statistical theories to “subtheories” of quantum theory Look to a representation of quantum theory on phase space – the Wigner representation  Restricted Liouville mechanics = Quadrature Quantum Mechanics  -  2n = R  X  P  Quadrature quantum mechanics ^ : L2 ( Rn) → L2 ( Rn) Hermitian operators: F Commutator:  The quadrature operators are:  Quadrature states are eigenstates of a commuting set of quadrature operators Specified by an isotropic subspace V and a valuation vector v∈V (Quadrature transformations and measurements take quadrature states to quadrature states)  Wigner representation of quantum mechanics Weyl operator Quantum state ρ Characteristic function Wigner function  For quadrature state associated with V, v 1δ WV,v ( m) = N V ⊥ + v ( m)  Equivalence of states implies equivalence of measurements and transformations Therefore Theorem: Restricted statistical Liouville mechanics is empirically equivalent to quadrature quantum mechanics  Restricted statistical theory of trits = Stabilizer theory for qutrits  - = ( Z3 ) 2n 2 1 0 0  1  2  C  C3  Equivalence of states implies equivalence of measurements and transformations Therefore Theorem: The restricted statistical theory of trits is empirically equivalent to the Stabilizer theory for qutrits  Restricted statistical theory of bits ≃ Stabilizer theory for qubits  - = ( Z2 1 0 0  1  2n )  Analogously to what we did for trits, one can: Define stabilizer theory for qubits Define Gross’ discrete Wigner function for qubits Find: Wigner function can be negative for qubit stabilizer states  The restricted statistical theory of bits is not equivalent but very close to the Stabilizer theory for qubits  Knowledge balance vs. classical complementarity Contrast: The principle of classical complementarity: An observer can only have knowledge of the values of a commuting set of canonical variables and otherwise is maximally ignorant. The knowledge-balance principle: The only distributions that can be prepared are those that correspond to knowing at most half the information From: RS, quant-ph/0401052 [Phys. Rev. A 75, 032110 (2007)]  The same epistemic states are found to be valid, but the logic is different… Example:  11 10 01 00  is forbidden 00 01 10 11  Knowledge-balance principle: It is forbidden by an assumption of locality and the existence of nontrivial measurements: Find X1=1  Principle of epistemic complementarity: It is forbidden because it corresponds to X2 = 0 and X1 + P2 = 0 but [X2, X1 + P2] ≠ 0  What about applying knowledge-balance to trits? (See S. van Enk, arxiv:0705.2742) Valid epistemic states for a pair of systems are different! and  known  22 21 20 12 11 10 02 01 00 00 01 02 10 11 12 20 21 22  Allowed by knowledge-balance, but corresponding to nothing in QM!  Knowledge-Balance Principle 2003-2008  Long live Symplectic Structure!  Beyond classical complementarity: could a different statistical restriction get us closer to quantum theory? NO for discrete degrees of freedom Supplementing the unitary representation of the Clifford group with a single non-Clifford unitary yields all unitaries  YES for continuous degrees of freedom In addition to rotations and displacements in phase space, one can add squeezing – one gets all the quadratic Hamiltonians (Bartlett, Rudolph, Spekkens, unpublished)  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  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  According to Knowledge-Balance Valid epistemic states for a single system 1 0  1 0 0 1  1 0  1 0 0 1  1 0 0 1  0 1  1 0 0 1  0 1  Valid epistemic states for a pair of systems  Plus permutations of rows and columns  The same epistemic states are found to be valid, but the logic is different… Example:  11 10 01 00  is forbidden 00 01 10 11  Knowledge-balance principle: It is forbidden by an assumption of locality and the existence of nontrivial measurements: Find X1=1 Are the two theories always equivalent? Principle of epistemic complementarity: It is forbidden because it corresponds to X2 = 0 and X1 + P2 = 0 but [X2, X1 + P2] ≠ 0  What about applying knowledge-balance to trits? (See S. van Enk, arxiv:0705.2742) Valid epistemic states for a single system are the same Valid epistemic states for a pair of systems are slightly different! and  and  known  22 21 20 12 11 10 02 01 00  known  22 21 20 12 11 10 02 01 00 00 01 02 10 11 12 20 21 22  Ruled out by locality  00 01 02 10 11 12 20 21 22  Allowed by locality, but corresponding to nothing in QM!  Valid epistemic states for a pair of degrees of freedom  Quantum theory Classical theory  Category Theory Framework Possibilistic Theories  Classical Statistical Theories with epistemic restriction  Convex theories C* algebraic theories Convex theories with maximal dual cone  How to represent this graphically 2 1 0  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  Uncorrelated pure epistemic states X2=0  X1=0  P1=0  X1+P1=0  X1-P1=0  P2=0  X2+P2=0  X2-P2=0  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 reversible transformations  1 trit example:  2 trit example:  Valid reproducible measurements On a single trit 2 1 0  2 1 0 0 1 2  2 1 0  2 1 0 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  etc.  00 01 02 10 11 12 20 21 22  Uncorrelated pure epistemic states X2=0  X1=0  P1=0  X1+P1=0  P2=0  X2+P2=0  Correlated pure epistemic states  1 0 0 1  11 10 01 00 00 01 10 11  1 0  1 0 0 1  1 0 0 1  0 1  

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