[POSTER] Certifying Separability of Mixed States, and Superradiance

Abstract

Poster summarizing the Certifying Separability paper.

Full text

Ambiguous Set Structure pre-analysis M OT I VAT I O N S Precise Set Equivalence post-analysis
Ambiguous Set Structure pre-analysis PPT is sufficient to establish separability for N≤3 for symmetric states.
DOI: 10.1006/APhy.2002.6268
Symmetric PPT entangled states exist for N≥4.
DOI: 10.1103/PhysRevA.85.060302
Sometimes a mixture of entangled states can itself be separable.
How can we tell for sure? We need a sufficient separability criterion. Ex:
A method exists to certify partial separability when the mixture can be
decomposed into a finite number of pure separable vectors.
DOI: 10.1103/PhysRevA.61.062302
We, however, consider full separability with continuous decompositions.
Precise Set Equivalence post-analysis
D E F I N I T I O N S R E S U LT S
The Dicke states are the superpositions of the N-particle
equal-energy states. We consider 2-level systems, where the
Dicke states are defined as
and form a complete eigenbasis for symmetric states. Dicke
states are typically highly entangled, for example we have
We study states which are diagonal in the Dicke state basis.
The General Diagonal Symmetric state can be parameterized
by N+1populations, as
Dicke Model Superradiance can be shown to evolve in this
manifold; the time-dependent populations can be shown to
obey
Lemma 1) A GDS state “fitting” the constructed SDS form is
not just sufficient to demonstrate separability
but is also necessary;all separable GDS states
are of the SDS form.
Lemma 2) The PPT criterion (positivity under all partial
transpositions) although usually only necessary
for separability, is specially sufficient on the GDS
states.
We prove both-at-once by demonstrating that the volume of
states compatible with each criterion is identical*; volume is
given by a multidimensional integral where we treat the
populations as coordinates and apply the criteria as
indicator functions.
Since our criterion is tight, then even its looser forms are still
necessary for separability, ie
Furthermore, one can show that superradiant populations
always satisfy the PPT criterion, leading to the unexpected
result that entanglement cannot be generated by
superradiance time evolution.
* Volume equality only numerically demonstrated for N=4.
But why should it be any different for N>4?
METHODS
One way to certify separability is to fit the state into some a-priori
separable parameterization. The completely general single-qubit pure
state is
An N-copy tensor product
defines the symmetric N-qubit pure state. We choose to mix uniformly
over all phases and to mix discretely over various parametric amplitudes
with arbitrary but convex weights to obtain
which we call the Separable Diagonally Symmetric states. They are
valuable because the have the same form as the GDS states, but are
innately separable. In the supplementary online materials show that
these states can equivalently be expressed as
Matching terms we obtain the sufficient separability criterion that
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