Bound-entanglement is NOT a rare phenomenon for

continuous variable Gaussian states

R. Simon

The Institute of Mathematical Sciences, Chennai.

January 07, 2013

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.

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 continuousvariable 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 forexperimentalists.

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.

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 determinedby the variance matrix V :

Vαβ = 〈12ξα, ξβ, 〉 α, β = 1, 2, · · · , 2n;

W (ξ) = A exp[−ξTV−1ξ].

A ensures∫

W (ξ)d2nξ = 1 ⇔ trρ = 1.

ρ ≥ 0 ⇔ V + iΛ ≥ 0 (uncertainty principle), Λ = iσ2⊕ iσ2⊕· · ·⊕ iσ2.

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) ρU(S)†.

The induced action on the quasi-probability is

U(S) : W (ξ) → W′

(ξ) = W (S−1ξ), 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.

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 ≥ SASTA ⊕ SBS

TB 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.

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 00 0 0 −10 0 −1 00 −1 0 0

= RT = R−1.

trR = 0, R2 = 11 ⇒ doubly degenerate eigenvalues ±1.

UP : V ≥ ΛV−1ΛT ⇔ |c | ≤√

a2 −√

2a2 − 1

⇒ V is an acceptable variance matrix iff c2 ≤ a2 −√2a2 − 1.

Effect of Partial Transpose

Under partial transpose :

R =

1 0 0 00 0 0 −10 0 −1 00 −1 0 0

→

1 0 0 00 0 0 10 0 −1 00 1 0 0

= SA0 R(S

B0 )

T ,

where SA0 = SB

0 = diag(−1,−1, 1, 1).

Thus, partial transposition on V corresponds to V → S0VST0 , 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).

Separability

Since the state is essentially invariant under PT, the partial transposecriterion is of no use. So we resort to the stronger NS criterion

V ≥ SASTA ⊕ SBS

TB .

Consider the four dimensional projection

Ω =

(

1 −RT

−R 1

)

.

For separable V , one necessarily has

tr(VΩ) ≥ trSASTA + trSBS

TB ≥ 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 SASTA + SBS

TB ,

the condition a− 1 ≥ |c | is SUFFICIENT as well.

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 entangledregion of extent c = 0.164.

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 toa 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.

Laboratory Feasibility

At the minumum : We have shown that inseparable PPT variancematrix is not a rare phenomenon.

Restricted to Gaussian states of the state space, this impliesPPT-entangled Gaussian state is not a rare phenomenon.

The considered state can be produced using linear optics and a pair ofinseparable two-mode Gaussian states.

Arbitrarily small squeezing is sufficient and, in particular, currentlyavailable levels of squeezing suffice.

It is true that there could be non-Gaussian states arbitrarily close tothe considered Gaussian state.

Any number of papers talk about separable Gaussian states andentanglement death. No need to be more apologetic!

Finally, there are physical processes whose output is guaranteed to beGaussian states.