Quantum Correlations from
Classical Coherence Theory?
Daniel F. V. JAMES
Department of Physics &
Center for Quantum Information and Quantum Control
University of Toronto
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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)
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Outline
• Entanglement of pure quantum states.
• The menagerie of non-classical mixed state
quantum correlations.
• Can Classical Coherence theory help? The
sinisterness of entanglement.
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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
Average out
qubit B:
(Born and Wolf, p.628)
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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)
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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!
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Two Qubit Mixed State Concurrence*
r
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)
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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)
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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…)
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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
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)
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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:
- “Two-photon Stokes Parameters”
- Observable quantities (actually what is measured in tomography*)
*DFVJ, Kwiat, Munro, White, Phys Rev A 64, 052312 (2001)
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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)
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“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.
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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
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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…
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