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arXiv:1504.06459 · Random Matrices Theory and Application

k-Extendibility of high-dimensional bipartite quantum states

Cécilia Lancien

Published Jul 1, 2016• 8 citations

Abstract

The idea of detecting the entanglement of a given bipartite state by searching for symmetric extensions of this state was first proposed by Doherty et al. The complete family of separability tests it generates, often referred to as the hierarchy of [Formula: see text]-extendibility tests, has already proved to be most promising. The goal of this paper is to try and quantify the efficiency of this separability criterion in typical scenarios. For that, we essentially take two approaches. First, we compute the average width of the set of [Formula: see text]-extendible states, in order to see how it scales with the one of separable states. And second, we characterize when random-induced states are, depending on the ancilla dimension, with high probability violating or not the [Formula: see text]-extendibility test, and compare the obtained result with the corresponding one for entanglement vs separability. The main results can be precisely phrased as follows: on [Formula: see text], when [Formula: see text] grows, the average width of the set of [Formula: see text]-extendible states is equivalent to [Formula: see text], while random states obtained as partial traces over an environment [Formula: see text] of uniformly distributed pure states are violating the [Formula: see text]-extendibility test with probability going to [Formula: see text] if [Formula: see text]. Both statements converge to the conclusion that, if [Formula: see text] is fixed, [Formula: see text]-extendibility is asymptotically a weak approximation of separability, even though any of the other well-studied separability relaxations is outperformed by [Formula: see text]-extendibility as soon as [Formula: see text] is above a certain (dimension independent) value.

DOI: 10.1142/s2010326316500118

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The idea of detecting the entanglement of a given bipartite state by searching for symmetric extensions of this state was first proposed by Doherty et al. The complete family of separability tests it generates, often referred to as the hierarchy of [Formula: see text]-extendibility tests, has already proved to be most promising. The goal of this paper is to try and quantify the efficiency of this separability criterion in typical scenarios. For that, we essentially take two approaches. First, we compute the average width of the set of [Formula: see text]-extendible states, in order to see how it scales with the one of separable states. And second, we characterize when random-induced states are, depending on the ancilla dimension, with high probability violating or not the [Formula: see text]-extendibility test, and compare the obtained result with the corresponding one for entanglement vs separability. The main results can be precisely phrased as follows: on [Formula: see text], when [Formula: see text] grows, the average width of the set of [Formula: see text]-extendible states is equivalent to [Formula: see text], while random states obtained as partial traces over an environment [Formula: see text] of uniformly distributed pure states are violating the [Formula: see text]-extendibility test with probability going to [Formula: see text] if [Formula: see text]. Both statements converge to the conclusion that, if [Formula: see text] is fixed, [Formula: see text]-extendibility is asymptotically a weak approximation of separability, even though any of the other well-studied separability relaxations is outperformed by [Formula: see text]-extendibility as soon as [Formula: see text] is above a certain (dimension independent) value.

Motivated by the notions of [Formula: see text]-extendability and complete extendability of the state of a finite level quantum system as described by Doherty et al. [Complete family of separability criteria, Phys. Rev. A 69 (2004) 022308], we introduce parallel definitions in the context of Gaussian states and using only properties of their covariance matrices, derive necessary and sufficient conditions for their complete extendability. It turns out that the complete extendability property is equivalent to the separability property of a bipartite Gaussian state. Following the proof of quantum de Finetti theorem as outlined in Hudson and Moody [Locally normal symmetric states and an analogue of de Finetti’s theorem, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 33(4) (1975/76) 343–351], we show that separability is equivalent to complete extendability for a state in a bipartite Hilbert space where at least one of which is of dimension greater than 2. This, in particular, extends the result of Fannes, Lewis, and Verbeure [Symmetric states of composite systems, Lett. Math. Phys. 15(3) (1988) 255–260] to the case of an infinite dimensional Hilbert space whose C* algebra of all bounded operators is not separable.

Moment-based relaxations of the positive partial transpose (PPT) criterion have been recently introduced, as a hierarchy of entanglement criteria involving only experimentally accessible quantities of a given bipartite state. The goal of this work is to study their typical detection performance on high-dimensional bipartite systems. Concretely, we investigate whether random bipartite mixed states on

\mathbb C^d\otimes\mathbb C^d

, obtained as the marginal over an environment

\mathbb C^s

of a uniformly distributed pure state, generically satisfy or violate them. For each fixed level

m\in\mathbb N

in this hierarchy of moment-based PPT criteria, we are able to identify a threshold environment dimension

s=λ_md^2

at which the behavior of the associated random state switches from violating to satisfying it, with probability going to

1

as

d

grows. The proof combines combinatorics of permutations techniques to estimate the average value of moments of partially transposed random states and concentration of measure arguments to bound the probability of deviating from such average, when the underlying local dimension

d

is large. We additionally need tools from the theory of Hankel determinant evaluation via orthogonal polynomials.

Entanglement detection in high dimensional systems is a NP-hard problem since it is lacking an efficient way. Given a bipartite quantum state of interest free entanglement can be detected efficiently by the PPT-criterion (Peres-Horodecki criterion), in contrast to detecting bound entanglement, i.e. a curious form of entanglement that can also not be distilled into maximally (free) entangled states. Only a few bound entangled states have been found, typically by constructing dedicated entanglement witnesses, so naturally the question arises how large is the volume of those states. We define a large family of magically symmetric states of bipartite qutrits for which we find [Formula: see text] to be free entangled, [Formula: see text] to be certainly separable and as much as [Formula: see text] to be bound entangled, which shows that this kind of entanglement is not rare. Via various machine learning algorithms we can confirm that the remaining [Formula: see text] of states are more likely to belonging to the set of separable states than bound entangled states. Most important we find via dimension reduction algorithms that there is a strong two-dimensional (linear) sub-structure in the set of bound entangled states. This revealed structure opens a novel path to find and characterize bound entanglement towards solving the long-standing problem of what the existence of bound entanglement is implying.

We investigate classification and detection of entanglement of multipartite quantum states in a very general setting, and obtain efficient

k

-separability criteria for mixed multipartite states in arbitrary dimensional quantum systems. These criteria can be used to distinguish

n-1

different classes of multipartite inseparable states and can detect many important multipartite entangled states such as GHZ states, W states, anti W states, and mixtures thereof. They detect

d

-nonseparable

d

-partite quantum states which have previously not been identified. Here

k=2,3,\cdots,n

. No optimization or eigenvalue evaluation is needed, and our criteria can be evaluated by simple computations involving components of the density matrix. Most importantly, they can be implemented in today's experiments by using at most

\mathcal{O}(n^2)

local measurements.