arXiv subjects

Mark W. Girard

Publications and source records attributed to Mark W. Girard.

6 recordsLinked to original sources

arXiv2019

Entanglement manipulation and distillability beyond LOCC

When a quantum system is distributed to spatially separated parties, it is natural to consider how the system evolves when the parties perform local quantum operations with classical communication (LOCC). However, the structure of LOCC channels is exceedingly complex leaving many important physical problems unsolved. In this paper we consider generalized resource theories of entanglement based on different relaxations to the class of LOCC. The behavior of various entanglement measures is studied under non-entangling channels, as well as the newly introduced classes of dually non-entangling and PPT-preserving channels. In an effort to better understand the nature of LOCC bound entanglement, we study the problem of entanglement distillation in these generalized resource theories. We first show that unlike LOCC, general non-entangling maps can be superactivated, in the sense that two copies of the same non-entangling map can nevertheless be entangling. On the single-copy level, we demonstrate that every NPT entangled state can be converted into an LOCC-distillable state using channels that are both dually non-entangling and having a PPT Choi representation and that every state can be converted into an LOCC-distillable state using operations belonging to any family of polytopes that approximate LOCC. We then turn to the stochastic convertibility of multipartite pure states and show that any two states can be interconverted by any polytope approximation to the set of separable channels. Finally, as an analog to $k$-positive maps, we introduce and analyze the set of $k$-non-entangling channels.

quant-ph↗

arXiv2018

On directional derivatives of trace functionals of the form $A\mapsto\Tr(Pf(A))$

Given a function $f:(0,\infty)\rightarrow\RR$ and a positive semidefinite $n\times n$ matrix $P$, one may define a trace functional on positive definite $n\times n$ matrices as $A\mapsto \Tr(Pf(A))$. For differentiable functions $f$, the function $A\mapsto \Tr(Pf(A))$ is differentiable at all positive definite matrices $A$. Under certain continuity conditions on~$f$, this function may be extended to certain non-positive-definite matrices $A$, and the \emph{directional} derivatives of $\Tr(Pf(A)$ may be computed there. This note presents conditions for these directional derivatives to exist and computes them. These conditions hold for the function $f(x)=\log(x)$ and for the functions $f_p(x)=x^p$ for all $p>-1$. The derivatives of the corresponding trace functionals are computed here.

math.FA↗

arXiv2017

The binegativity of two qubits

We introduce an entanglement-related quantity that we call the binegativity. Based on numerical evidence, we conjecture that the binegativity is an entanglement measure for two-qubit states. The binegativity is compared to the concurrence and negativity, the only known entanglement measures that can be computed analytically for all two-qubit states. The ordering of entangled two-qubit mixed states induced by the binegativity is distinct from the ordering determined by previously known measures, yielding possible new insights into the structure of entangled two-qubit mixed states.

quant-ph↗

arXiv2017

Entanglement monotones and transformations of symmetric bipartite states

The primary goal in the study of entanglement as a resource theory is to find conditions that determine when one quantum state can or cannot be transformed into another via local operations and classical communication. This is typically done through entanglement monotones or conversion witnesses. Such quantities cannot be computed for arbitrary quantum states in general, but it is useful to consider classes of symmetric states for which closed-form expressions can be found. In this paper, we show how to compute the convex roof of any entanglement monotone for all Werner states. The convex roofs of the well-known Vidal monotones are computed for all isotropic states, and we show how this method can generalize to other entanglement measures and other types of symmetries as well. We also present necessary and sufficient conditions that determine when a pure bipartite state can be deterministically converted into a Werner state or an isotropic state.

quant-ph↗

arXiv2016

Computable entanglement conversion witness that is better than the negativity

The primary goal of entanglement theory is to determine convertibility conditions for two quantum states. Up until now, this has always been done with the use of entanglement monotones. With the exception of the negativity, such quantities tend to be rather uncomputable. We instead promote the idea of conversion witnesses in this paper. A conversion witness is a function on pairs of states and whose value determines whether a state can be converted into another. We construct a conversion witness that can be efficiently computed for arbitrary states in systems of any size. This conversion witness is always better than the negativity at detecting when two entangled states are not interconvertible. Furthermore, when considering states of two-qubit systems, this new conversion witness is sometimes better than the entanglement of formation. This shows that the study of conversion witness is in fact useful, and may have applications in resource theories beyond that of entanglement.

quant-ph↗

arXiv2014

On convex optimization problems in quantum information theory

Convex optimization problems arise naturally in quantum information theory, often in terms of minimizing a convex function over a convex subset of the space of hermitian matrices. In most cases, finding exact solutions to these problems is usually impossible. As inspired by earlier investigations into the relative entropy of entanglement [Phys. Rev. A 78 032310 (2008)], we introduce a general method to solve the converse problem rather than find explicit solutions. That is, given a matrix in a convex set, we determine a family of convex functions that are minimized at this point. This method allows us find explicit formulae for the relative entropy of entanglement and the Rains bound, two well-known upper bounds on the distillable entanglement, and yields interesting information about these quantities, such as the fact that they coincide in the case where at least one subsystem of a multipartite state is a qubit.

quant-ph↗