The Quantum Separability Problem for Gaussian States

Determining whether a quantum state is separable or entangled is a problem of fundamental importance in quantum information science. This is a brief review in which we consider the problem for states in infinite dimensional Hilbert spaces. We show how the problem becomes tractable for a class of Gaussian states.Comment: 8 page

Gaussian entanglement revisited

We present a novel approach to the problem of separability versus entanglement in Gaussian quantum states of bosonic continuous variable systems, as well as a collection of closely related results. We derive a simplified necessary and sufficient separability criterion for arbitrary Gaussian states of $m$ vs $n$ modes, which relies on convex optimisation over marginal covariance matrices on one subsystem only. We further revisit the currently known results stating the equivalence between separability and positive partial transposition (PPT) for specific classes of multimode Gaussian states. Using techniques based on matrix analysis, such as Schur complements and matrix means, we then provide a unified treatment and compact proofs of all these results. In particular, we recover the PPT-separability equivalence for Gaussian states of $1$ vs $n$ modes, for arbitrary $n$. We then proceed to show the novel result that Gaussian states invariant under partial transposition are separable.
 Next, we provide a previously unknown extension of the PPT-separability equivalence to arbitrary Gaussian states of $m$ vs $n$ modes that are symmetric under the exchange of any two modes belonging to one of the parties. Further, we include a new proof of the sufficiency of the PPT criterion for separability of isotropic Gaussian states, not relying on their mode-wise decomposition. In passing, we also provide an alternative proof of the recently established equivalence between separability of an arbitrary Gaussian state and its complete extendability with Gaussian extensions. Finally, we prove that Gaussian states which remain PPT under passive optical operations cannot be entangled by them either; this is not a foregone conclusion per se (since Gaussian bound entangled states do exist) and settles a question that had been left unanswered in the existing literature on the subject.
 This paper, enjoyable by both the quantum optics and the matrix analysis communities, overall delivers technical and conceptual advances which are likely to be useful for further applications in continuous variable quantum information theory, beyond the separability problem

Gaussian entanglement witness and refined Werner-Wolf criterion for continuous variables

We use matched quantum entanglement witnesses to study the separable criteria of continuous variable states. The witness can be written as an identity operator minus a Gaussian operator. The optimization of the witness then is transformed to an eigenvalue problem of a Gaussian kernel integral equation. It follows a separable criterion not only for symmetric Gaussian quantum states, but also for non-Gaussian states prepared by photon adding to or/and subtracting from symmetric Gaussian states. Based on Fock space numeric calculation, we obtain an entanglement witness for more general two-mode states. A necessary criterion of separability follows for two-mode states and it is shown to be necessary and sufficient for a two mode squeezed thermal state and the related two-mode non-Gaussian states. We also connect the witness based criterion with Werner-Wolf criterion and refine the Werner-Wolf criterion.Comment: 11pages, 2 figure

Gaussian resource theories and semidefinite programming hierarchies for quantum information

Determining which quantum tasks we can perform with currently available tools and devices is one of the most important goals of quantum information science today. To achieve this requires careful investigation of the capability of current quantum tools as well as development of classical protocols which can assist quantum tasks and amplify their abilities. In this thesis, we approach this problem through two different topics in quantum information theory: Gaussian resource theories and semidefinite programming hierarchies. In the first part of this thesis, we examine the possibility of implementing quantum information processing tasks in the Gaussian platform through the eyes of quantum resource theories. Gaussian states and operations are primary tools for the study of continuous-variable quantum information processing due to their easy accessibility and concise mathematical descriptions, although it has been discovered that they are subject to a number of limitations for advanced quantum information processing tasks. We explore the capability of the Gaussian platform further in the first part of this thesis. Firstly, we investigate whether introducing convex structure to the Gaussian framework can circumvent the known no-go theorem of Gaussian resource distillation. Surprisingly, we find that resource distillation becomes possible — albeit in a limited fashion — when convexity is introduced. Then, we consider the quantum resource theory of Gaussian thermal operations when catalysts are allowed, and examine the abilities of catalytic Gaussian thermal operations by characterising all possible state transformations under them. In the second part of this thesis, we address the problem of characterising quantum cor- relations via semidefinite programming hierarchies. In particular, we focus on characterising quantum correlations of fixed dimension, which is practically relevant to the field of semi- device-independent quantum information processing. Semidefinite programming is a special type of mathematical optimisation, and it is known that some important but difficult problems in quantum information theory admit semidefinite programming relaxations; these include the characterisation of general quantum correlations in the context of non-locality and the distinction of quantum separable states from entangled states. In this second part, we show how to construct a hierarchy of semidefinite programming relaxations for quantum correlations of fixed dimension and derive analytical bounds on the convergence speed of the hierarchy. For the proof, we make a connection to a variant of quantum separability problem and employ multipartite quantum de Finetti theorems with linear constraints.Open Acces