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Entanglement of formation
1. Entanglement of formation
for some special two-mode Gaussian states
Paulina Marian & Tudor A. Marian
Centre for Advanced Quantum Physics
University of Bucharest, Romania
QMath10 – p.1/25
2. Outline
Undisplaced two-mode Gaussian states (TMGS’s)
Uncertainty relations
Non-classicality & inseparability
Entanglement of Formation (EoF)
Continuous-variable pure-state decompositions
Gaussian Entanglement of Formation (GEoF)
Factorization of the characteristic functions (CF’s)
Properties of the added classical state
Manipulating the GEoF equations
Explicit evaluations
Conclusions: is GEoF=EoF? QMath10 – p.2/25
3. Two-mode Gaussian states (TMGS’s)
• Density operator ρG.
• Characteristic function (CF):
χG(x) = exp −
1
2
xT
Vx ,
with xT
denoting a real row vector (x1 x2 x3 x4)
and V the 4 × 4 covariance matrix (CM).
• A TMGS is fully described by its CM:
ρG ←→ χG(x) ←→ V.
QMath10 – p.3/25
4. Scaled standard forms
• Equivalence class of locally similar TMGS’s:
S = S1 ⊕ S2, S ∈ Sp(2, R) × Sp(2, R)
→ U(S) = U1(S1) ⊗ U2(S2).
• Consider two independent one-mode squeeze factors
u1 = exp (2r1), u2 = exp (2r2).
• CM of a scaled standard state ρ(u1, u2):
V(u1, u2) =
b1u1 0 c
√
u1u2 0
0 b1/u1 0 d/
√
u1u2
c
√
u1u2 0 b2u2 0
0 d/
√
u1u2 0 b2/u2
.
QMath10 – p.4/25
5. Uncertainty relations
• Standard form I (unscaled): VI := V(1, 1).
• Robertson-Schrödinger uncertainty relations:
V +
i
2
Ω ≥ 0, Ω := i
2
j=1
σ2 equivalent to :
b1 ≥ 1/2, b2(b1b2 − c2
) −
b1
4
≥ 0,
b2 ≥ 1/2, b1(b1b2 − c2
) −
b2
4
≥ 0,
(κ2
− − 1/4)(κ2
+ − 1/4) ≥ 0.
(κ−, κ+ are the symplectic eigenvalues).
QMath10 – p.5/25
6. Non-classicality
• Classicality of a scaled standard state:
V(u1, u2) ≥
1
2
I4 equivalent to
u1 ≤ 2b1, u2 ≤ 2b2,
(b1u1 −
1
2
)(b2u2 −
1
2
) ≥ c2
u1u2,
(b1/u1 −
1
2
)(b2/u2 −
1
2
) ≥ d2
/(u1u2).
•Non-classical state ←→ Matrix condition not fulfilled.
QMath10 – p.6/25
7. Inseparability
R. Simon’s separability criterion (2000):
• TMGS’s with d ≥ 0 are separable.
• for d < 0, one has to check the sign of the invariant
S(ρG) = (b1b2 − c2
)(b1b2 − d2
) −
1
4
(b2
1 + b2
2 + 2c|d|) +
1
16
that can be written as
S(ρG) = (˜κ2
− − 1/4)(˜κ2
+ − 1/4).
Entangled TMGS’s fulfil the condition
S(ρG) < 0.
QMath10 – p.7/25
8. EPR approach
Duan et al.(2000) introduced a scaled standard state for
which the separability and classicality conditions
coincide.
Standard form II of the CM (separability=classicality):
VII := V(v1, v2)
with v1, v2 satisfying the algebraic system
b1(v2
1 − 1)
2b1 − v1
=
b2(v2
2 − 1)
2b2 − v2
,
b1b2(v2
1 − 1)(v2
2 − 1) = (cv1v2 − |d|)2
.
QMath10 – p.8/25
9. Squeezed vacuum states (SVS’s)
The standard-form CM VI of an entangled pure TMGS
has the property
det(VI +
i
2
Ω) = 0
as a product of two vanishing factors −→
b1 = b2 = b, d = −c < 0, b2
− c2
= 1/4.
This state is a SVS and has minimal symplectic
eigenvalues:
κ− = κ+ = 1/2.
The smallest symplectic eigenvalue of ˜V ←→ ρPT2
G is
˜κ− = b − c <
1
2
.
QMath10 – p.9/25
10. Entanglement of formation (EoF)
Pure-state decompositions of a mixed state ρ:
ρ =
k
pk|Ψk Ψk|,
k
pk = 1.
EoF of a mixed bipartite state (Bennett et al., 1996):
EoF(ρ) := inf
{pk}
k
pkE0(|Ψk Ψk|),
where E0(|Ψk Ψk|) is any acceptable measure of pure-
state entanglement.
QMath10 – p.10/25
11. Two-field superpositions
• Glauber (1963): superposition ρS of two fields
ρS = d2
βP2(β)D(β)ρ1D†
(β)
where D(α) := exp (αa†
− α∗
a) is a Weyl displacement
operator,
a is a photon annihilation operator;
ρ1 is a one-mode field state and P2(β) denotes the P
representation of a classical one-mode field state ρ2.
• Equivalent formulation (Marian & Marian, 1996):
χ
(N)
S (λ) = χ
(N)
1 (λ)χ
(N)
2 (λ);
χ(N)
denotes the normally-ordered CF.
QMath10 – p.11/25
12. Gaussian EoF
• A pure-state decomposition of a mixed TMGS is
ρG = d2
β1d2
β2P(β1, β2)D1(β1)D2(β2)ρ0D†
2(β2)D†
1(β1),
where ρ0 is a pure TMGS.
The most general ρ0, which is a scaled SVS, was
employed by Wolf et al. (2003)−→Gaussian EoF
(GEoF):
GEoF(ρG) = E(ρoptimal
0 ).
• Main problem: find the optimal decomposition
(=determine ρ0 having the minimal entanglement).
• Giedke et al. (2003) evaluated the exact EoF
for symmetric TMGS’s (b1 = b2).
QMath10 – p.12/25
13. Our work
• exploit the factorization formula of the CF’s
χG(λ1, λ2) = χ0(λ1, λ2)χcl(λ1, λ2) exp −
|λ1|2
2
−
|λ2|2
2
.
• choose χ0(λ1, λ2) to be a SVS with the CM
V0 =
x 0 y 0
0 x 0 −y
y 0 x 0
0 −y 0 x
, x2
− y2
= 1/4.
QMath10 – p.13/25
14. Relation between CM’s
Reason for V0: A SVS is the pure state with minimal
entanglement at a given EPR correlation (Giedke et al.,
2003).
∆EPR = 2(x − y)
• Gaussian χ0(λ1, λ2) ←→ Gaussian χcl(λ1, λ2).
• For any pure-state decomposition of the TMGS ρ
V = V0 + Vcl −
1
2
I4.
QMath10 – p.14/25
15. Variables
For any entangled TMGS d = −|d| < 0.
V =
b1u1 0 c
√
u1u2 0
0 b1/u1 0 −|d|/
√
u1u2
c
√
u1u2 0 b2u2 0
0 −|d|/
√
u1u2 0 b2/u2
.
• Given parameters: b1, b2, c, |d|, (|d| ≤ c).
• Any measure of pure-state entanglement is a
monotonous function of x −→ we have to find the min-
imal value of x as a function of the variables u1, u2.
QMath10 – p.15/25
16. Analytical method
• concentrate on the added classical state Vcl.
First step: Towards the optimal pure-state
decomposition, Vcl should reach the classicality
threshold
det(Vcl −
1
2
I4) = 0
as a product of two vanishing factors:
(b1u1 − x)(b2u2 − x) = (c
√
u1u2 − y)2
,
(b1/u1 − x)(b2/u2 − x) = (|d|/
√
u1u2 − y)2
.
(derived by Wolf et al. (2003) on different grounds).
QMath10 – p.16/25
17. Nature of Vcl
Second step: By minimization of the function x(u1, u2),
we proved that in the optimal pure-state decomposition,
Vcl is also at the separability threshold:
S(ρcl) = 0,
i.e., Vcl has the standard form II:
b1u1 − x
b1/u1 − x
=
b2u2 − x
b2/u2 − x
,
b1b2(u2
1 − 1)(u2
2 − 1) = (cu1u2 − |d|)2
.
QMath10 – p.17/25
18. EoF equations
System of algebraic equations with four unknowns:
(b1w1 − x)(b2w2 − x) = (c
√
w1w2 − y)2
,
(b1/w1 − x)(b2/w2 − x) = (|d|/
√
w1w2 − y)2
,
b1w1 − x
b1/w1 − x
=
b2w2 − x
b2/w2 − x
,
b1b2(w2
1 − 1)(w2
2 − 1) = (cw1w2 − |d|)2
,
x2
− y2
= 1/4.
Solution only in some particular cases.
QMath10 – p.18/25
19. Symmetric TMGS’s
b1 = b2 = b −→ ˜κ− = (b − c)(b − |d|).
Results
w1 = w2 =
b − |d|
b − c
,
x =
˜κ2
− + 1/4
2˜κ−
;
x − y = ˜κ−.
x is a function of ˜κ− only that coincides with its expres-
sion for the exact EoF (Giedke et al., 2003).
QMath10 – p.19/25
20. STS’s
c = |d| −→ ˜κ− =
1
2
[b1 + b2 − (b1 − b2)2 + 4c2].
• important mixed states used as two-mode resource in
quantum teleportation of one-mode states.
• proved to have the maximal negativity at fixed local
purities: Adesso et al. (2004,2005).
Results
w1 = w2 = 1,
x =
(b1 + b2)(b1b2 − c2
+ 1/4) − 2c det(V + i
2Ω)
(b1 + b2)2 − 4c2
:
x not depending on ˜κ− only.
QMath10 – p.20/25
21. States with κ− = 1/2
• Mixed TMGS’s with
det(V +
i
2
Ω) = 0 ←→ κ− = 1/2;
• proved to have minimal negativity at fixed local and
global purities: Adesso et al. (2004,2005).
Results depending on a parameter inequality,
as follows.
QMath10 – p.21/25
23. States with κ− = 1/2
II. b1 > b2, c > |d|, b2c > b1|d| :
w1 = 2
b1
b2
(b1b2 − d2),
w2 = 2
b2
b1
(b1b2 − d2),
x =
1
2
b1b2
b1b2 − d2
.
QMath10 – p.23/25
24. Conclusions I
We have reformulated the problem of GEoF in terms
of CF’s and CM’s.
The added classical state is at the classicality and
separability threshold as well: its CM has the standard
form II.
We have retrieved in a unitary way previous results for
some important classes of entangled TMGS’s.
General case hard to be exploited analytically.
Work in progress.
QMath10 – p.24/25
25. Conclusions II
The GEoF built with the Bures metric is proved to
coincide with the Bures entanglement for symmetric
TMGS’s, as well as for STS’s.
Main question: Is GEoF=EoF?
Answer: Yes.
This is based on the above-mentioned theorem of
Giedke et al. (2003): A SVS is the pure state with
minimal entanglement at a given EPR correlation
∆EPR = 2(x − y).
QMath10 – p.25/25