1. 1 / 18 DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Quantum Correlations from Classical Coherence Theory? Daniel F. V. JAMES Department of Physics & Center for Quantum Information and Quantum Control University of Toronto

2. 2 / 18 DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 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. 3 / 18 DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 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. 4 / 18 DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Entanglement and all that… Separable state of two 2-level systems: But in general the state of two 2-level systems is:

5. 5 / 18 DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Quantifying Entanglement of 2 Pure Qubits (Born and Wolf, p.628) Average out qubit B:

6. 6 / 18 DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Extension 1: Two 3-level systems? Average out qutrit B: Omar Gamel & DFVJ, “Measures of quantum state purity and classical degree of polarization,” Phys Rev A 86 033830 (2012) (Two competing claimants due to Friberg et al, and Wolf et al.…)

7. 7 / 18 DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Extension 2: Mixed States Mixed states: roll the dice, and create a pure state (with some probability): “Average” Tangle: depends on decomposition “Minimized Average Tangle”: –Reduces to T(  ) 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! (“convex hull”)

8. 8 / 18 DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Two Qubit Mixed State Concurrence* *W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998) “spin flip matrix” Eigenvalues of R (in decreasing order) “there remains a basic question concerning the interpretation …that has not yet been resolved.” Transpose (in computational basis)

9. 9 / 18 DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 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 *friendly guide: Quesada, Al-Qasimi & DFVJ, J. Mod. Opt. 59, 1322 (2012) Schmidt Number -Defined for Pure and Mixed states -connection with “ancilla assisted process tomography” (Altpeter et al. 2003)

10. 10 / 18 DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 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 Other Quantities (cont…) Entropic Measures of Quantum Correlations… (Lang, Caves, Shaji, 2011) -six in total (MID and Discord are extremal cases…)

11. 11 / 18 DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 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. 12 / 18 DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Tangle and Purity Munro, DFVJ, Kwiat, White, Phys Rev A 64, 030302 (2001) Discord and Purity Al-Qasimi, DFVJ, Phys Rev A 83, 032101(2011) Examples: Tangle

13. 13 / 18 DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 What do these all mean? Doesn’t it rather make you nostalgic for good old fashioned classical coherence theory?

14. 14 / 18 DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 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: - Observable quantities (actually what is measured in tomography*) *DFVJ, Kwiat, Munro, White, Phys Rev A 64, 052312 (2001) - “Two-photon Stokes Parameters”

15. 15 / 18 DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 Properties of c i,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. 16 / 18 DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 “sinister” states orthonormal, yes, but they do not necessarily form a right handed system… Properties: 1. All entangled states (pure and mixed) are sinister. “sinister states”:= if one of the SVD bases is left- handed, and one is right-handed* *Term introduced by Joe Altepeter. 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.

17. 17 / 18 DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 00.20.40.60.81.0 0.2 0.4 0.6 0.8 1.0 Maybe is just as good a quantifier of “quantumness” as any of those other measures?* * Al-Qasimi and DFVJ, in preparation easy to calculate physically motivated ~

18. 18 / 18 DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7 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…