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 Robert Spekkens Perimeter Institute July 23, 2010 QAMF workshop, UBC 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) 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 Classical theory Statistical theory for the classical theory Restricted Statistical theory for the classical theory = 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, … noncommutativity collapse Interference Wave-particle duality Teleportation No cloningentanglement Categorizing quantum phenomena Those not arising in a restricted statistical classical theory Those arising in a restricted statistical classical theory Coherent superposition Quantum eraser Pre and post-selection “paradoxes” Bell inequality violations Computational speed-up Key distribution Improvements in metrology Quantized spectra Particle statistics Bell-Kochen-Specker theorem Bell inequality violations Computational speed-up Bell-Kochen-Specker theorem Certain aspects of items on the left Others… Categorizing quantum phenomena Those not arising in a restricted statistical classical theory Those arising in a restricted statistical classical theory Interference Noncommutativity Entanglement Collapse Wave-particle duality Teleportation Quantized spectra? Particle statistics? Others… No cloning Key distribution Improvements in metrology Quantum eraser Coherent superposition Pre and post-selection “paradoxes” Others… Interference Noncommutativity Entanglement Collapse Wave-particle duality Teleportation Bell inequality violations Computational speed-up Bell-Kochen-Specker theorem Certain aspects of items on the left Others… Categorizing quantum phenomena Those not arising in a restricted statistical classical theory Those arising in a restricted statistical classical theory No cloning Key distribution Improvements in metrology Quantum eraser Coherent superposition Pre and post-selection “paradoxes” Others… Quantized spectra? Particle statistics? Others… Not so strange after all! Still surprising! Find more! Focus on these Speculative possibility for an axiomatization of quantum theory A research program 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) 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 Classical theory Statistical theory for the classical theory Restricted Statistical theory for the classical theory = Stabilizer theory for qutrits 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 Classical theory Statistical theory for the classical theory Restricted Statistical theory for the classical theory = 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) Configuration space: Rn ∋ (x1, x2, . . . , xn) Phase space: - ≡ R2n ∋ (x1, p1, x2, p2, . . . , xn, pn) ≡ m F : - → RFunctionals on phase space: Poisson bracket of functionals: Continuous degrees of freedom Xk(m) = xk Pk(m) = pk The linear functionals / canonical variables are: Independent of m Poisson bracket of functionals: Xk(m) = xk Pk(m) = pk Configuration space: Phase space: (Zd)n ∋ (x1, x2, . . . , xn) - ≡ (Zd) 2n ∋ (x1, p1, x2, p2, . . . , xn, pn) ≡ m F : - → ZdFunctionals on phase space: Discrete degrees of freedom Zd = {0, 1, . . . , d− 1} The linear functionals / canonical variables are: Independent of m (F [m + exi] − F [m]) (G[m + epi] −G[m]) −(F [m + epi] − F [m]) (G[m + eqi] −G[m]) A canonically conjugate pair A commuting pair e.g. 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 ω : - × - → RSymplectic inner product ω(m,m′) = mTJm′ where ω(m,m′) = ∑i ( qip ′ i − piq ′ i )Thus Poisson bracket of functionals = symplectic inner product of vectors F= ∑ i (aiXi + biPi) The linear functionals form a dual space {X1, P1, . . . ,Xn, Pn} is dual to {ex1 , ep1 , . . . , exn, epn} A set of known variables V ∀F,G ∈ V : [F,G] = 0 These are specified by: A valuation of the known variables v : V → R(Zd) V = {X1, P2} v(X1) = 2, v(P2) = 2 Valid epistemic states: Example: An isotropic subspace V ⊆ Ω∗ ∀F,G ∈ V : ω(F,G) = 0 A valuation vector v ∈ V *⊆ Ω v : ∀F ∈ V, FTv = v(F ) Equivalently, V v ≡ V ⊥ + v = {m ∈ - | ∀F ∈ V : FTm = FTv} = {m ∈ - | PVm = v} The ontic states consistent with the epistemic state (V,v) are {m ∈ - | ∀F ∈ V : F (m) = v(F )} ( ) = 1 ( ) The associated distribution is (Dirac-delta / Kronecker delta) 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} Example pV,v m N δV ⊥+ v m “Heisenberg picture” and “Schrodinger picture” v The group of symplectic affine transformations (Clifford group) where and Valid reversible transformations: Those that preserve the Poisson bracket / symplectic inner product: Symplectic Affine (Heisenberg-Weyl) Valid reproducible measurements: Any commuting set of canonical variables Restricted Liouville mechanics - = R2n X P XP X Pv V V v Valid epistemic states for a single degree of freedom X P V v X PV=∅ Valid epistemic states for a pair of degrees of freedom Restricted statistical theory of trits - = (Z3) 2n 0 1 2 0 1 2 Canonical variables Commuting sets: The singleton sets the empty set Valid epistemic states for a single trit known known known known 0   1   2 0 1 2 0   1   2 0 1 2 0   1   2 0 1 2 0   1   2 0 1 2 0   1   2 0 1 2 Nothing known 0   1   2 0 1 2 0   1   2 0 1 2 0   1   2 0 1 2 0   1   2 0 1 2 0   1   2 0 1 2 0   1   2 0 1 2 0   1   2 0 1 2 0   1   2 0 1 2 22 21 How to represent this graphically Canonical variables Valid epistemic states for a pair of trits a1, b1, a2, b2 ∈ Z3 00 01 02 10 11 12 20 21 22 20 12 11 10 02 01 00 22 21 20 12 11 10 known 22 21 20 12 11 10 known 1 variable known 02 01 00 02 01 00 00 01 02 10 11 12 20 21 22 00 01 02 10 11 12 20 21 22 knownand 22 21 20 12 11 10 2 variables known 02 01 00 00 01 02 10 11 12 20 21 22 22 21 20 12 11 10 known 22 21 20 12 11 known 1 variable known 02 01 00 10 02 01 00 00 01 02 10 11 12 20 21 22 00 01 02 10 11 12 20 21 22 22 21 20 12 and known 2 variables known 11 10 02 01 00 00 01 02 10 11 12 20 21 22 On a single trit On a pair of trits 0   1   2 0 1 2 0   1   2 0 1 2 0   1   2 0 1 2 0   1   2 0 1 2 22 22 Valid reproducible measurements 21 20 12 11 10 02 01 00 21 20 12 11 10 02 01 00 00 01 02 10 11 12 20 21 2200 01 02 10 11 12 20 21 22 etc. Restricted statistical theory of bits - = (Z2) 2n 0      1 0 1 Canonical variables A single bit known known known Epistemic states of maximal knowledge 0   1 0 1 0   1 0 1 0   1 0 1 a, b ∈ Z2 0   1 0 1 0   1 0 1 0   1 0 1 0   1 0 1 Nothing known Epistemic states of non-maximal knowledge Canonical variables A pair of bits 11 10 01 00 a1, b1, a2, b2 ∈ Z2 00 01 10 11 known known 1 variable known 00 01 10 11 11 10 01 00 00 01 10 11 11 10 01 00 2 variables known knownand 00 01 10 11 11 10 01 00 1 variable known 00 01 10 11 11 10 01 00 00 01 10 11 11 10 01 00 2 variables known known known 00 01 10 11 11 10 01 00 and known 0   1 0 1 0   1 0 1 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 - = R2n X P Quadrature quantum mechanics F̂ : L2(Rn) → L2(Rn)Hermitian operators: Commutator: The quadrature operators are: Quadrature states are eigenstates of a commuting set of quadrature operators (Quadrature transformations and measurements take quadrature states to quadrature states) Specified by an isotropic subspace V and a valuation vector v∈V Wigner representation of quantum mechanics Weyl operator Quantum state ρ Characteristic function Wigner function WV,v(m) = 1N δV ⊥+ v(m) For quadrature state associated with V, v Theorem: Restricted statistical Liouville mechanics is empirically equivalent to quadrature quantum mechanics Equivalence of states implies equivalence of measurements and transformations Therefore Restricted statistical theory of trits = Stabilizer theory for qutrits - = (Z3) 2n 0 1 2 0 1 2 C C3    Theorem: The restricted statistical theory of trits is empirically equivalent to the Stabilizer theory for qutrits Equivalence of states implies equivalence of measurements and transformations Therefore Restricted statistical theory of bits ≃ Stabilizer theory for qubits - = (Z2) 2n 0      1 0 1 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 The knowledge-balance principle: Knowledge balance vs. classical complementarity 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. Contrast: 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)] Knowledge-balance principle: It is forbidden by an assumption of locality and the existence of nontrivial measurements: The same epistemic states are found to be valid, but the logic is different… is forbidden 00 01 10 11 11 10 01 00 Example: X2 = 0 and X1 + P2 = 0   but [X2, X1 + P2] ≠ 0 Principle of epistemic complementarity: It is forbidden because it corresponds to Find X1=1 What about applying knowledge-balance to trits? (See S. van Enk, arxiv:0705.2742) Valid epistemic states for a pair of systems are different! 22 21 20 and known 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 Even number of Why the restricted statistical theory of bits Is not equivalent to qubit stabilizer theory correlations Odd number of correlations Qubit stabilizer theory is nonlocal and contextual (e.g. GHZ) Restricted statistical theory of bits is local and noncontextual 0   1 0 1 0   1 0 1 0   1 0 1 0   1 0 1 0   1 0 1 0   1 0 1 According to Knowledge-Balance Valid epistemic states for a single system Plus permutations of rows and columns Valid epistemic states for a pair of systems Knowledge-balance principle: It is forbidden by an assumption of locality and the existence of nontrivial measurements: The same epistemic states are found to be valid, but the logic is different… is forbidden 00 01 10 11 11 10 01 00 Example: X2 = 0 and X1 + P2 = 0   but [X2, X1 + P2] ≠ 0 Principle of epistemic complementarity: It is forbidden because it corresponds to Are the two theories always equivalent? Find X1=1 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! 22 21 and known 22 21 and known 20 12 11 10 02 01 00 00 01 02 10 11 12 20 21 22 20 12 11 10 02 01 00 00 01 02 10 11 12 20 21 22 Ruled out by locality Allowed by locality, but corresponding to nothing in QM!   Valid epistemic states for a pair of degrees of freedom Convex theories Quantum theory Category Theory Framework Classical theory Convex theories with maximal dual cone C*  algebraic theories Possibilistic Theories Classical Statistical Theories with epistemic restriction 00 01 02 10 11 12 20 21 22 0   1   2 0 1 2 22 21 How to represent this graphically 00 01 02 10 11 12 20 21 22 20 12 11 10 02 01 00 X1=0 P1=0 X2=0 P2=0 X2+P2=0 X2-P2=0 Uncorrelated pure epistemic states X1+P1=0 X1-P1=0 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 Correlated pure epistemic states 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: On a single trit On a pair of trits 0   1   2 0 1 2 0   1   2 0 1 2 0   1   2 0 1 2 0   1   2 0 1 2 22 22 Valid reproducible measurements 21 20 12 11 10 02 01 00 21 20 12 11 10 02 01 00 00 01 02 10 11 12 20 21 2200 01 02 10 11 12 20 21 22 etc. X1=0 P1=0 X2=0 P2=0 X2+P2=0 Uncorrelated pure epistemic states X1+P1=0 Correlated pure epistemic states 0   1 0 1 11 10 01 00 00 01 10 11 0   1 0 1 0   1 0 1 0   1 0 1

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