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Quantum Measurements, ICTP, 2019
How to observe and quantify quantum discord in electronic systems - - PowerPoint PPT Presentation
How to observe and quantify quantum discord in electronic systems - - PowerPoint PPT Presentation
How to observe and quantify quantum discord in electronic systems Igor Lerner & Matt Hunt (Birmingham) Igor Yurkevich (Aston Uni, UK) Yuval Gefen (Weizmann Inst, Israel) Overview Quantum correlations in bipartite systems: entanglement
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Quantum Measurements, ICTP, 2019
Bipartite system: pure state
Von Neumann entropy S = - Tr r ln r =0 - no uncertainty Quantumness (or its absence) reveals in partitioning the system
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Quantum Measurements, ICTP, 2019
Entanglement of pure state
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Quantum Measurements, ICTP, 2019
Bipartite system: mixed state
Is AB entangled or not? – not necessarily obvious, e.g. Werner state: Generically, a mixed state of a bipartite system is not entangled iff
(Werner, ’89)
However, such a separable mixed state can still have quantumness exemplified by quantum discord.
(Ollivier and Zurek, ’01; Henderson and Vedral, ‘01)
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Quantum Measurements, ICTP, 2019
Classical mutual information
The concepts of quantum discord comes from comparing quantum and classical conditional entropies in bipartite systems.
Shannon entropy of system AB with joint probability distribution p(a,b) :
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Quantum Measurements, ICTP, 2019
Quantum mutual information
However, JA(rAB) is more tricky as a basis-independent definition of conditional entropy requires optimization over all possible measurements
- ver ‘passive’ subsystem B. So the more precise definition is
Quantum analogue: I,J(A:B) → I,J(AB) with H→S:
minimizing ignorance about A, i.e. picking the best measurement basis
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Quantum Measurements, ICTP, 2019
Quantum discord
(Ollivier and Zurek, ’01; Henderson and Vedral, ‘01)
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Quantum Measurements, ICTP, 2019
Quantum discord
Alternative expression for quantum discord
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Quantum Measurements, ICTP, 2019
Pure state: discord ≡ entanglement
If a mixed state is entangled, it is always discorded – D adds little. Hence, our main interest is in discord of separable – unentangled – states.
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Quantum Measurements, ICTP, 2019
Discord of mixed separable state
Here DA=0 for any q. For q=0,p the subsystems are totally uncorrelated and DB=0 For q=p/2, the classical mutual information is maximal but it is entirely classical and DB=0 again.
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Quantum Measurements, ICTP, 2019
Discord of mixed separable state
This state is A-non-discorded in either a trivial case – all |an > coincide,
- r when all |an > are orthogonal.
Otherwise, they are discorded independently of B Our aim: to find linear in r characteristics of a bipartite system that detect and quantify discord
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Quantum Measurements, ICTP, 2019
Entanglement witness
How to detect and quantify the remaining quantumness – discord – linearly, as Tr (r …), without full or partial quantum tomography.
Q.
Way around: repeated measurements of certain correlations. Theorem: no linear witness of discord (R. Rahimi and A. SaiToh, 2010)
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Discord via correlations
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Quantum Measurements, ICTP, 2019
Visibility as discord witness
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Quantum Measurements, ICTP, 2019
Measuring setup
System preparation
Evolution & measurement
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Quantum Measurements, ICTP, 2019
Input state
nondiscorded discorded
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Quantum Measurements, ICTP, 2019
Visibility pattern
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Quantum Measurements, ICTP, 2019
Discord witnesses
4p 3p 2p 0p 00 0 p 2p 3p 4p
b (b) a
4p 3p 2p 0p 00 0 p 2p 3p 4p
b a
4p 3p 2p 0p 00 0 p 2p 3p 4p
b (b) a
nondiscorded discorded
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Quantum Measurements, ICTP, 2019
Quantifying discord
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Quantum Measurements, ICTP, 2019
- Discord is hard to measure. Alternatives
(geometric discord) are based on full or partial quantum tomography – hardly extendable to condensed matter systems
- The proposed discord witness – the visibility in
(linear in r) interference pattern
- The proposed quantifier gives results similar to