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

Quantum Correlations from Classical Coherence Theory?

Daniel F. V. JAMESDepartment of Physics &Department of Physics &

Center for Quantum Information and Quantum ControlCenter for Quantum Information and Quantum ControlUniversity of TorontoUniversity of Toronto

2/18DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7

My group at TorontoAsma 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)

3/18DEPARTMENT 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.

4/18DEPARTMENT 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:

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Quantifying Entanglement of 2 Pure Qubits

(Born and Wolf, p.628)

Average out qubit B:

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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.…)

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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”)

8/18DEPARTMENT 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)

9/18DEPARTMENT 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)

10/18DEPARTMENT 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…)

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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?

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Tangle and PurityMunro, DFVJ, Kwiat, White,

Phys Rev A 64, 030302 (2001)

Discord and PurityAl-Qasimi, DFVJ,

Phys Rev A 83, 032101(2011)

Examples:T

angl

e

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• What do these all mean?

• Doesn’t it rather make you nostalgic for good old fashioned classical coherence theory?

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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”

15/18DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7

Properties of ci,j

• 3x3 matrix (uh-oh…)

• “sort of” a tensor…*

• real, but not necessarily symmetric

• Singular Value DecompositionSingular 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)

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“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.

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0 0.2 0.4 0.6 0.8 1.0

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

~

18/18DEPARTMENT 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…