Quantum Correlations from Classical Coherence Theory Daniel F

Quantum Correlations from Classical Coherence Theory? Daniel F. V. JAMES Department of Physics & Center for Quantum Information and Quantum Control University of Toronto CQO-X, ochester Rochester DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M 5 S 1 A 7 1/18

My group at Toronto Asma Al-Qasimi (postdoc) Christian Weedbrook (postdoc) Omar El-Gamel (Ph. D student) Hoi-Kawn Lau (Ph. D student) Nicolas Quesada (Ph. D student) Arnab Dewanjee (Ph. D student) Jaspreet Sahota (Ph. D student) Kevin Marshall (MSc student) DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M 5 S 1 A 7 2/18

Outline • Entanglement of pure quantum states. • The menagerie of non-classical mixed state quantum correlations. • Can Classical Coherence theory help? The sinisterness of entanglement. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M 5 S 1 A 7 3/18

Entanglement and all that… • Separable state of two 2 -level systems: • But in general the state of two 2 -level systems is: DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M 5 S 1 A 7 4/18

Quantifying Entanglement of 2 Pure Qubits Average out qubit B: (Born and Wolf, p. 628) DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M 5 S 1 A 7 5/18

Extension 1: Two 3 -level systems? Average out qutrit B: (Two competing claimants due to Friberg et al, and Wolf et al. …) Omar Gamel & DFVJ, “Measures of quantum state purity and classical degree of polarization, ” Phys Rev A 86 033830 (2012) DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M 5 S 1 A 7 6/18

Extension 2: Mixed States • Mixed states: roll the dice, and create a pure state (with some probability): (“convex hull”) • “Average” Tangle: decomposition • “Minimized Average Tangle”: depends on –Reduces to T(y) for pure states –T=0 implies separable state –T=1 implies maximally entangled state (e. g. Bell states) –Analytic expression (Wootters, 1998) makes things very convenient! DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M 5 S 1 A 7 7/18

Two Qubit Mixed State Concurrence* Transpose (in computational basis) “spin flip matrix” Eigenvalues of R (in decreasing order) “there remains a basic question concerning the interpretation …that has not yet been resolved. ” *W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998) DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M 5 S 1 A 7 8/18

Other Quantities (non-exhaustive list)* • Negativity of the partial transpose (Peres, 1996; Vidal and Werner, 2001) -if the partial transpose has negative eigenvalues it is entangled (IFF for two qubits or qubit-qutrit). -readily computable for larger systems. • Fully Entangled Fraction (Bennett et al. , 1996) -Maximum overlap with a fully entangled state -connection with optimum teleportation fidelity -not a monotone • Schmidt Number -Defined for Pure and Mixed states -connection with “ancilla assisted process tomography” (Altpeter et al. 2003) *friendly guide: Quesada, Al-Qasimi & DFVJ, J. Mod. Opt. 59, 1322 (2012) DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M 5 S 1 A 7 9/18

Other Quantities (cont…) • Discord (Olivier and Zurek, 2001) -Another notion of quantumness: how does measurement of system A affect the state of system B? -Connection with “DQC 1” model of quantum computing (accidental? ) -Difficult to compute (exact formula for states with maximally mixed marginals; approximate formula for “X states”) • Measurement Induced Disturbance (Luo, 2008) -Related to discord, easier to calculate • Entropic Measures of Quantum Correlations… (Lang, Caves, Shaji, 2011) -six in total (MID and Discord are extremal cases…) DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M 5 S 1 A 7 10/18

Connections? • Geometric Relations (Modi et al. , 2010): “distances” between product states, classically correlated states, separable states and entangled states • Algebraic Relations: does entanglement limit discord? How does entanglement or discord change with purity? DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M 5 S 1 A 7 11/18

Tangle Examples: Tangle and Purity Discord and Purity Munro, DFVJ, Kwiat, White, Phys Rev A 64, 030302 (2001) Al-Qasimi, DFVJ, Phys Rev A 83, 032101(2011) DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M 5 S 1 A 7 12/18

• What do these all mean? • Doesn’t it rather make you nostalgic for good old fashioned classical coherence theory? DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M 5 S 1 A 7 13/18

Another Approach? • The Bloch Vectors are a wonderful way of thinking about individual 2 -level quantum systems: we should not abandon it when dealing with pairs of 2 -level systems. • What would Albert Michelson or Fritz Zernike do? Bloch-vector correlation matrix: - “Two-photon Stokes Parameters” - Observable quantities (actually what is measured in tomography*) *DFVJ, Kwiat, Munro, White, Phys Rev A 64, 052312 (2001) DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M 5 S 1 A 7 14/18

Properties of ci, j • 3 x 3 matrix (uh-oh…) • “sort of” a tensor…* • real, but not necessarily symmetric • Singular Value Decomposition Singular values (real, positive) orthonormal vectors: • Why not use these orthonormal vectors to define a special basis for each qubit? *Englert and Metwally, Kinematics of qubit pairs (2002) DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M 5 S 1 A 7 15/18

“sinister” states • orthonormal, yes, but they do not necessarily form a right handed system… • “sinister states”: = if one of the SVD bases is lefthanded, and one is right-handed* • Properties: 1. All entangled states (pure and mixed) are sinister. 3. All sinister states have discord (i. e. it’s a discord witness) 4. Separable states with Werner decomposition of length N= 3 or less are never sinister. *Term introduced by Joe Altepeter. DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M 5 S 1 A 7 16/18

Maybe is just as good a quantifier of “quantumness” as any of those other measures? * • easy to calculate • physically motivated ~ 1. 0 0. 8 0. 6 0. 4 0. 2 0. 4 0. 6 0. 8 1. 0 * Al-Qasimi and DFVJ, in preparation DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M 5 S 1 A 7 17/18

Conclusion • “Quantumness” is elusive and frustrating. • “Quantum phenomena do not occur in a Hilbert space. They occur in a laboratory. ” (Asher Peres, 1995) Suggested Corollary: Maybe we should start approaching quantum mechanics is a branch of physics, not of information theory… DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M 5 S 1 A 7 18/18