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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, R ochester. My group at Toronto. Asma Al-Qasimi (postdoc) Christian Weedbrook (postdoc) - PowerPoint PPT Presentation
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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:
5/18DEPARTMENT 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:
6/18DEPARTMENT 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.…)
7/18DEPARTMENT 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”)
8/18DEPARTMENT 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…)
11/18DEPARTMENT 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?
12/18DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
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
13/18DEPARTMENT 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?
14/18DEPARTMENT 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”
15/18DEPARTMENT 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)
16/18DEPARTMENT 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.
17/18DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7
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…