Quantum Correlations from Classical Coherence Theory?

1. 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, Rochester

2. My group at Toronto Asma Al-Qasimi (postdoc) Christian Weedbrook (postdoc) Omar El-Gamel (PhD student) Hoi-Kawn Lau (PhD student) Nicolas Quesada (PhD student) Arnab Dewanjee (PhD student) Jaspreet Sahota (PhD student) Kevin Marshall (MSc student) Asma Al-Qasimi (postdoc) Christian Weedbrook (postdoc) Omar El-Gamel (PhD student) Hoi-Kawn Lau (PhD student) Nicolas Quesada (PhD student) Arnab Dewanjee (PhD student) Jaspreet Sahota (PhD student) Kevin Marshall (MSc student)

3. Outline • Entanglement of pure quantum states. • The menagerie of non-classical mixed state quantum correlations. • Can Classical Coherence theory help? The sinisterness of entanglement.

4. Entanglement and all that… •  Separable state of two 2-level systems: •  But in general the state of two 2-level systems is:

5. Quantifying Entanglement of 2 Pure Qubits Average out qubit B: (Born and Wolf, p.628)

6. 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)

7. Extension 2: Mixed States • • Mixed states: roll the dice, and create a pure state (with some probability): (“convex hull”) • “Average” Tangle: depends on decomposition • “Minimized Average Tangle”: • 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!

8. 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)

9. 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)

10. 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 “DQC1” 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…)

11. 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?

12. Examples: Tangle Tangle and Purity Munro, DFVJ, Kwiat, White, Phys Rev A 64, 030302 (2001) Discord and Purity Al-Qasimi, DFVJ, Phys Rev A83, 032101(2011)

13. • What do these all mean? • Doesn’t it rather make you nostalgic for good old fashioned classical coherence theory?

14. 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)

15. Properties of ci,j • 3x3 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)

16. “sinister” states • orthonormal, yes, but they do not necessarily form a right handed system… • “sinister states”:= if one of the SVD bases is left-handed, 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.

17. 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 0.2 0.4 0.6 0.8 1.0 * Al-Qasimi and DFVJ, in preparation

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…