1.
Bound-entanglement is NOT a rare phenomenon for
continuous variable Gaussian states
R. Simon
The Institute of Mathematical Sciences, Chennai.
January 07, 2013
2.
Convex Subsets of state space
A bipartite state ρAB is separable iﬀ it CAN BE written in the form
ρAB =
j
pj ρAj ⊗ ρBj , pj > 0.
A non-separable (entangled) state may or may not be distillable
All PPT states are provably non-distillable.
But it is not known if all NPT states are distillable.
Non-distillable entanglement is called bound entanglement
Ωsep ⊂ ΩPPT ⊆ ΩND ⊂ Ω.
Like the full state space Ω, all these subsets are CONVEX.
3.
Nowhere dense: measure zero
Horodecki, Cirac, Lewenstein (2001)
“Bound entanglement for continuous variables is a rare phenomenon”
proved that ΩND is nowhere dense in Ω, IN THE CASE of continuous
variable systems (inﬁnite-dimensional Hilbert spaces)
Implies that ΩPPT is nowhere dense in Ω, and so also is Ωsep.
This important negative result would seem to be a no-go-theorem for
experimentalists.
Any attempt to produce a state in ΩND would land outside of ΩND,
with probability one (i.e. almost certainly).
Should experimentalists take this result as a no-go-theorem?
The rest of the talk will examine this issue in the case of Gaussians.
4.
Gaussian states
Gaussian states are those whose quasi-probability has Gaussian form.
P, W , Q are the more popular quasi-probabilities.
If one of them is Gaussian, so are the others. We use W.
Phase-space displacements do not aﬀect our considerations
⇒ WLOG we may assume the state to be a zero-mean state :
ξ = 0, ξ = (q1, p1; q2, p2; · · · , qn, pn).
Then the quasi-probability (state) is fully determined
by the variance matrix V :
Vαβ =
1
2
{ξα, ξβ}, α, β = 1, 2, · · · , 2n;
W (ξ) = A exp[−ξT
V −1
ξ].
A ensures W (ξ)d2nξ = 1 ⇔ trρ = 1.
ρ ≥ 0 ⇔ V +iΛ ≥ 0 (uncertainty principle), Λ = iσ2 ⊕iσ2 ⊕· · ·⊕iσ2.
5.
Canonical transformations : Uncertainty Principle
Linear canonical transformations S ∈ Sp(2n, R), i.e. SΛST = Λ,
act unitarily on the Hilbert space through U(S)
S → U(S) : |ψ → U(S)|ψ , ρ → U(S) ρ U(S)†
.
The induced action on the quasi-probability is
U(S) : W (ξ) → W
′
(ξ) = W (S−1
ξ), V → V
′
= SVST
.
The Uncertainty Principle V + iΛ ≥ 0 ⇔ V ≥ ΛV −1ΛT
⇔ V ≥ SST
, for some S ∈ Sp(2n, R)
Every V > 0 is necessarily of the form
V = SV0ST
, S ∈ Sp(2n, R), V0 = diag (κ1, κ1; κ2, κ2; · · · ; κn, κn)
V + iΛ ≥ 0 ⇔ κ1 ≥ κ2 ≥ · · · ≥ κn ≥ 1.
6.
PPT entangled Gaussians
Bipartite system of k + ℓ = n modes, k with Alice, ℓ with Bob.
Λ = ΛA ⊕ ΛB
ΛA = iσ2 ⊕ iσ2 ⊕ · · · ⊕ iσ2, (2k dimensional)
ΛB = iσ2 ⊕ iσ2 ⊕ · · · ⊕ iσ2, (2ℓ dimensional)
V + iΛ ≥ 0 ⇔ V − iΛ ≥ 0 (transpose or time-reversal).
Separable V necessarily obeys
V + iΛPT
≥ 0, ΛPT
= ΛA ⊕ (−ΛB) (PPT criterion)
V ≥ SAST
A ⊕ SBST
B is a necessary and suﬃcient condition for
separability of Gaussian states, irrespective of k, ℓ
In the special case of k = 1 (or ℓ = 1), the two criteria are equivalent,
and the PPT criterion becomes NS.
i.e., PPT entangled Gaussians can occur only when BOTH k, ℓ ≥ 2.
7.
A class of PPT variance matrices : Uncertainty Principle
First example of a PPT entangled Gaussian state : Werner and Wolf
(2001)
Consider the 2 + 2-mode Gaussians deﬁned by
V =
a cR
cRT a
, a > 1, c real,
R =
1 0 0 0
0 0 0 −1
0 0 −1 0
0 −1 0 0
= RT
= R−1
.
trR = 0, R2 = 11 ⇒ doubly degenerate eigenvalues ±1.
UP : V ≥ ΛV −1
ΛT
⇔ |c| ≤ a2 − 2a2 − 1
⇒ V is an acceptable variance matrix iﬀ c2 ≤ a2 −
√
2a2 − 1.
8.
Eﬀect of Partial Transpose
Under partial transpose :
R =
1 0 0 0
0 0 0 −1
0 0 −1 0
0 −1 0 0
→
1 0 0 0
0 0 0 1
0 0 −1 0
0 1 0 0
= SA
0 R(SB
0 )T
,
where SA
0 = SB
0 = diag(−1, −1, 1, 1).
Thus, partial transposition on V corresponds to V → S0VST
0 , where
S0 = diag (−1, −1, 1, 1, −1, −1, 1, 1).
⇒ PT of V0 is UNITARILY EQUIVALENT to V .
⇒ The Gaussian state V is PPT, for all (a, c).
9.
Separability
Since the state is essentially invariant under PT, the partial transpose
criterion is of no use. So we resort to the stronger NS criterion
V ≥ SAST
A ⊕ SBST
B .
Consider the four dimensional projection
Ω =
1 −RT
−R 1
.
For separable V , one necessarily has
tr(VΩ) ≥ trSAST
A + trSBST
B ≥ 8.
For our V we have tr(VΩ) = 8(a − c)
⇒ a − 1 ≥ |c| is a NECESSARY condition for separability.
Since this implies V ≥ 1, and since 1 is of the form SAST
A + SBST
B ,
the condition a − 1 ≥ |c| is SUFFICIENT as well.
10.
Recapitulation
Since V is invariant under PT, it can never be NPT entangled.
Positivity of ρ ⇔ c2 ≤ a2 −
√
2a2 − 1, Separability of ρ ⇔ a − |c| ≥ 1
With a = 2 : ρ is a state if |c| ≤ 4 −
√
7 = 1.164. And ρ is
separable iﬀ |c| ≤ 1. If |c| > 1.164, then ρ is not even a state!
There is a FINITE parameter range
a − 1 < |c| ≤ a2 − 2a2 − 1 which is 1 < |c| ≤ 1.164 for a = 2.
in which the state is PPT entangled.
With a = 2, c0 = 1.08 is ‘deep’ in the interior of the PPT entangled
region of extent △c = 0.164.
11.
NOT a rare phenomenon
When the uncertainty inequality |c| ≤ a2 −
√
2a2 − 1 is saturated, two
of the invariant κ’s equal unity, and the other two equal a2 − c2. Let V0
correspond to (a, c0) in the ‘deep’ interior.
Then, every κ ≥ 1 + ǫ for some ǫ > 0. And tr(ΩV0) = 8(1 − 0.08).
With Γ real symmetric and trΓ2 ≤ 1, V0 + rΓ corresponds to
a solid ball of radius r in Rn(2n+1) [n = 4 for us] around V0;
Choose r << ǫ
The entire ball V0 + rΓ comprises bonaﬁde variance matrices
trΩ(V0 + rΓ) < 8 for the entire ball hence all states non-separable
Since the ball itself is PT symmetric, it has no NPT entangled state.
12.
Laboratory Feasibility
At the minumum : We have shown that inseparable PPT variance
matrix is not a rare phenomenon.
Restricted to Gaussian states of the state space, this implies
PPT-entangled Gaussian state is not a rare phenomenon.
The considered state can be produced using linear optics and a pair of
inseparable two-mode Gaussian states.
Arbitrarily small squeezing is suﬃcient and, in particular, currently
available levels of squeezing suﬃce.
It is true that there could be non-Gaussian states arbitrarily close to
the considered Gaussian state.
Any number of papers talk about separable Gaussian states and
entanglement death. No need to be more apologetic!
Finally, there are physical processes whose output is guaranteed to be
Gaussian states.
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