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Quantum versus Classical Correlations in Gaussian States
Gerardo Adesso
joint work with Animesh Datta (Imperial College / Oxford)
School of Mathematical Sciences
Imperial College London 10/08/2010
2
Quantum versus Classical Correlations in Gaussian States
Outline
•Quantum versus classical correlations
•Quantum discord
•Gaussian quantum discord
•Structure of Gaussian correlations
•Open problems
Imperial College London 10/08/2010
3
Correlations
Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010
A B
Classical correlations
Quantum correlations
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Correlations
Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010
•Pure global composite states:
▫entanglement = nonlocality = nonclassicality (quantum
correlations)
•Mixed global composite states:
▫Werner 1989: separable = classically correlated
A B
5
Quantum versus Classical Correlations in Gaussian States
Quantumness in separable states
Nonorthogonal separable states cannot be discriminated exactly
Measuring a local observable on a separable bipartite state will perturb
the stateThe eigenvectors of a separable state can be entangled superpositions
…
In general separable states have not a purely classical nature
Imperial College London 10/08/2010
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Quantum versus Classical Correlations in Gaussian States
A new paradigm
Imperial College London 10/08/2010
M. Piani, P. Horodecki, R. Horodecki, PRL 2008
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Quantum versus Classical Correlations in Gaussian States
Quantum discord• A measure that strives at capturing all quantum
correlations, beyond entanglement, which can be nonzero also in separable states
• Introduced a decade ago in two independent works (Ollivier/Zurek and Henderson/Vedral)
• Recently became very popular: stats from arXiv:quant-ph…
Imperial College London 10/08/2010
05101520253035
# pr
eprin
ts
year
8
Quantum versus Classical Correlations in Gaussian States
Quantum discord• Almost all bipartite states have nonzero quantum discord
(purely classically correlated states are of zero measure) A. Ferraro et al. PRA 2010
• Reduces to the entropy of entanglement on pure bipartite states
• Quantum discord without entanglement may allow for a computational speed-up in the DQC1 model of quantum computation A. Datta et al. 2008-2010; experiment: M. Barbieri et al. PRL 2008 discord
entanglement
Imperial College London 10/08/2010
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Mutual information: classical
measuring
total correlations…
( )H A ( )H B
( : ) ( ) ( ) ( , )I A B H A H B H A B= + -
( : ) ( ) ( | )
( : ) ( ) ( | )
J A B H A H A B
J B A H B H B A
= -
= -
all equal (Bayes’
rule)
Imperial College London 10/08/2010
Quantum versus Classical Correlations in Gaussian States
10
what are these ??
Mutual information: quantum
( )A
S ñ ( )B
S ñ
( ) ( ) ( ) ( )AB A B AB
I S S S= + -ñ ñ ñ ñ
Imperial College London 10/08/2010
Quantum versus Classical Correlations in Gaussian States
( ) Tr[ log ]H S® = -ñ ñ ñ
( ) ( ) ( | )
( ) ( ) ( | )AB A
BA B
J S S A B
J S S B A
¬
®
= -
= -
ñ ñ
ñ ñ
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Conditional entropy
( )A
S ñ ( )B
S ñ
Imperial College London 10/08/2010
Quantum versus Classical Correlations in Gaussian States
• Introduce POVM on B:
• Posterior state of A after B has been measured:
{ },B Bi i
i
P P =å 1
|
Tr [ ],
with r[ ]T
BB i AB
A ii
Bi i AB
pp P=
P=
ññ
ñ
|( | ) inf ( )
Bi
i A ii
S A B pSP
º å ñ• looking for the “least disturbing measurement”:
( )AB
I ñ
12
Quantum versus Classical Correlations in Gaussian States
Bipartite correlations
•Total correlation
•One-way classical correlation Henderson, Vedral, JPA 2001
•Quantum discord Ollivier, Zurek, PRL 2001
Imperial College London 10/08/2010
( ) ( ) ( ) ( )AB A B AB
I S S S= + -ñ ñ ñ ñ
|( ) ( ) ( | ) ( ) inf ( )
Bi
AB A A i A ii
S S A B S pS¬
P= - = - åJ ñ̂ ñ ñ ñ
|
( ) ( ) ( )
( ) ( ) inf ( )Bi
AB AB AB
B AB i A ii
I
S S pS
¬ ¬
P
= -
= - + åD ñ̂ ñ J ñ̂
ñ ñ ñ
A B
13
Quantum versus Classical Correlations in Gaussian States
Quantum discord• For classical states (classical probability distribution embedded
into density matrices) I=J hence the quantum discord vanishes
• Zurek introduced it in the context of environment-induced selection, identifying classical states with the pointer states
• The optimization involved in the conditional entropy is hard. Closed analytical formulas are available only for special families of two-qubit staes (X-shaped), not even for arbitrary states of two qubits
• Two recent independent works, including this one, defined a Gaussian version of the quantum discord for bipartite Gaussian states, where the optimization is restricted to Gaussian measurements P. Giorda & M.G.A. Paris PRL 2010; GA & A. Datta PRL 2010
• We have solved the optimization problem and obtained a simple formula for the Gaussian quantum discord of arbitrary two-mode Gaussian states
Imperial College London 10/08/2010
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Quantum versus Classical Correlations in Gaussian States
Gaussian states
Imperial College London 10/08/2010
Very natural: ground and thermal states of all physical systems in the harmonic approximation regime (M.S.Kim: like orange juice and sunshine)
Relevant theoretical testbeds for the study of structural properties of entanglement and correlations, thanks to the symplectic formalism
Preferred resources for experimental unconditional implementations of continuous variable protocols
Crucial role and remarkable control in quantum optics- coherent states- squeezed states- thermal states
Gaussian operationsGaussian states can beefficiently:
displaced (classical currents)
squeezed (nonlinear crystals)
rotated (phase plates, beam splitters)
measured (homodyne detection)
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Imperial College London 10/08/2010
Quantum versus Classical Correlations in Gaussian States
Gaussian operationsGaussian states can beefficiently:
displaced (classical currents)
squeezed (nonlinear crystals)
rotated (phase plates, beam splitters)
measured (homodyne detection)
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Imperial College London 10/08/2010
Quantum versus Classical Correlations in Gaussian States
Gaussian operationsGaussian states can beefficiently:
displaced (classical currents)
squeezed (nonlinear crystals)
rotated (phase plates, beam splitters)
measured (homodyne detection)
17
Imperial College London 10/08/2010
Quantum versus Classical Correlations in Gaussian States
Gaussian operationsGaussian states can beefficiently:
displaced (classical currents)
squeezed (nonlinear crystals)
rotated (phase plates, beam splitters)
measured (homodyne detection)
18
Imperial College London 10/08/2010
Quantum versus Classical Correlations in Gaussian States
Gaussian operationsGaussian states can beefficiently:
displaced (classical currents)
squeezed (nonlinear crystals)
rotated (phase plates, beam splitters)
measured (homodyne detection)
19
Imperial College London 10/08/2010
Quantum versus Classical Correlations in Gaussian States
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Quantum versus Classical Correlations in Gaussian States
Gaussian states: formalism
• Up to local unitaries, Gaussian states are completely specified by the covariance matrix…
• … or equivalently by thefour symplectic invariants
Imperial College London 10/08/2010
standardform
TAB
a c
a d
c b
d b
a gs s
g b
æ ö÷ç ÷ç ÷çæ ö ÷ç ÷÷ç ç ÷÷ç= = = ç ÷÷ç ç ÷÷ç ÷÷ç çè ø ÷ç ÷ç ÷ç ÷çè ø
det , det , det , detA B C Da b g s= = = =
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Quantum versus Classical Correlations in Gaussian States
Gaussian POVMs
Imperial College London 10/08/2010
• All the measurements that can be done by linear optics (appending Gaussian ancillas, manipulating with symplectic transformations, plus homodyne detection):
• The posterior state of A after measuring B has a covariance matrix (independent of the measurement outcome)
1 0 † †
1 2 0
0
is the density
ˆ
matrix of a
single-mode Gaussian state with covariance ma
ˆ ˆ ˆ ˆ( ) ( ) ( ), where ( ) exp( )
trix
,
( ) , a
n
dB B B B B
B B
W W W b b
d
h p h h h h h
p h h
s
- *
-
P = P = -
P = Pò 1
|A hñ
e1
0( ) Te a g b s g-= - +
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Quantum versus Classical Correlations in Gaussian States
Gaussian quantum discord
• The Gaussian quantum discord is the quantum discord of a bipartite Gaussian state where the optimization in the conditional entropy is restricted to Gaussian POVMs
• and can be rewritten as
▫ where the symplectic eigenvalues are
Imperial College London 10/08/2010
( |)( ) ( ) ( ) inf ( ) ( )
BAB B AB B A
D S S d p Shh
h h¬
P= - + òñ ñ ñ ñ
0
( ) ( ) ( ) ( ) inf ( det )AB
D f B ff fs
s n n e¬- +
= - - +
2 22 4 , 2D A B Cn±
= D ± D - D = + +
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Quantum versus Classical Correlations in Gaussian States
Gaussian quantum discord
• Optimal POVM: heterodyne for squeezed thermal states, homodyne for another class of states, something in-between for the other two-mode Gaussian states
Imperial College London 10/08/2010
( )( ) ( )( )( )
( ) ( ) ( )
( ) ( )0
2 22 2
2
22 4 2
2 1 2| | 1, 1 ;
1inf det( )
2 , .
2
C B A D C C B A DD AB B C A D
B
AB C D C AB D C AB Dotherwise
B
se
ìï + - + - + + + - + - +ïïï - £ + +ïï - +ï= íïïï - + - + - + - +ïïïïî
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Quantum versus Classical Correlations in Gaussian States
Discord/separability/entanglement
• By relating the nullity of discord with saturation of strong subadditivity of entropy, we demonstrated that (for finite mean energies) the only two-mode Gaussian states with zero Gaussian discord are product states
• All correlated Gaussian states (including all entangled states and all non-product separable mixed states) are quantumly correlated!
• This proves the truly quantum nature of Gaussian states despite their positive Wigner function…
Imperial College London 10/08/2010
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Quantum versus Classical Correlations in Gaussian States
• Consider this class of states (box=two-mode squeezing)▫ s: initial entanglement; r: entanglement
degradation
Imperial College London 10/08/2010
Discord/separability/entanglement
sA
B
C
r
ABs *
when ,
0
1
s r¬
®
® ¥
®
®
D
D
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Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010
max discord is limited
if 1 en
to
tangled
1¬ > ÞD
Discord/separability/entanglement
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Quantum versus Classical Correlations in Gaussian States Imperial College London 10/08/2010
Discord/separability/entanglement
( )AB
s® *D
( )AB
s¬ *D
pure
: Gaussian Entanglement of FormationG
E
1
28
Quantum versus Classical Correlations in Gaussian States
Other results & comments
• Via the Koashi-Winter duality between entanglement and one-way classical correlations we can derive a closed formula for the Gaussian EoF of a family of three-mode Gaussian states
• Only in very special cases we can prove that the Gaussian quantum discord realizes the absolute minimum in the conditional entropy optimization not constrained to Gaussian POVMs (this is related to the problem of additivity of bosonic channel capacity etc…)
• It would be interesting to prove, or show counterexamples to it, that Gaussian POVMs are always optimal among all continuous variable measurements (including photodetection etc.)
Imperial College London 10/08/2010
29
Quantum versus Classical Correlations in Gaussian States
Summary
• The concept of quantum correlations goes beyond entanglement
• Quantum discord is a bona fide measure of such general quantum correlations
• Quantum discord can be computed for Gaussian states under Gaussian measurements
• All correlated Gaussian states have quantum correlations
• They are limited for separable states• They admit upper and lower bounds as a
function of the entanglement, for entangled states
Imperial College London 10/08/2010
30
Quantum versus Classical Correlations in Gaussian States
Open problems
•Maximum discord for separable states in any dimension.
▫known for qubits,numerically, to be 1/3
Al-Qasimi & James, arXiv:1007.1814
Imperial College London 10/08/2010
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Quantum versus Classical Correlations in Gaussian States
Open problems
• Operational interpretation of discord• Usefulness of quantum correlations in separable
states for quantum information processing• Understanding connection with other
nonclassicality indicators in continuous variable systems (e.g. in terms of P function)
• Producing a theory of quantum correlations, with axioms to be satisfied by any valid measure of quantum correlations (e.g. nonincreasing under local operations and classical communication…)
• …
Imperial College London 10/08/2010
Quantum versus Classical Correlations in Gaussian States
Thank you
Imperial College London 10/08/2010