arXiv subjects

Cécilia Lancien

Publications and source records attributed to Cécilia Lancien.

25 records · Page 2Linked to original sources

arXiv2016

Flexible constrained de Finetti reductions and applications

De Finetti theorems show how sufficiently exchangeable states are well-approximated by convex combinations of i.i.d. states. Recently, it was shown that in many quantum information applications a more relaxed de Finetti reduction (i.e. only a matrix inequality between the symmetric state and one of de Finetti form) is enough, and that it leads to more concise and elegant arguments. Here we show several uses and general flexible applicability of a constrained de Finetti reduction in quantum information theory, which was recently discovered by Duan, Severini and Winter. In particular we show that the technique can accommodate other symmetries commuting with the permutation action, and permutation-invariant linear constraints. We then demonstrate that, in some cases, it is also fruitful with convex constraints, in particular separability in a bipartite setting. This is a constraint particularly interesting in the context of the complexity class $\mathrm{QMA}(2)$ of interactive quantum Merlin-Arthur games with unentangled provers, and our results relate to the soundness gap amplification of $\mathrm{QMA}(2)$ protocols by parallel repetition. It is also relevant for the regularization of certain entropic channel parameters. Finally, we explore an extension to infinite-dimensional systems, which usually pose inherent problems to de Finetti techniques in the quantum case.

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arXiv2016

High dimension and symmetries in quantum information theory

If a one-phrase summary of the subject of this thesis were required, it would be something like: miscellaneous large (but finite) dimensional phenomena in quantum information theory. That said, it could nonetheless be helpful to briefly elaborate. Starting from the observation that quantum physics unavoidably has to deal with high dimensional objects, basically two routes can be taken: either try and reduce their study to that of lower dimensional ones, or try and understand what kind of universal properties might precisely emerge in this regime. We actually do not choose which of these two attitudes to follow here, and rather oscillate between one and the other. In the first part of this manuscript, our aim is to reduce as much as possible the complexity of certain quantum processes, while of course still preserving their essential characteristics. The two types of processes we are interested in are quantum channels and quantum measurements. Oppositely, the second part of this manuscript is specifically dedicated to the analysis of high dimensional quantum systems and some of their typical features. Stress is put on multipartite systems and on entanglement-related properties of theirs. In the third part of this manuscript we eventually come back to a more dimensionality reduction state of mind. This time though, the strategy is to make use of the symmetries inherent to each particular situation we are looking at in order to derive a problem-dependent simplification.

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arXiv2016

$k$-extendibility of high-dimensional bipartite quantum states

The idea of detecting the entanglement of a given bipartite state by searching for symmetric extensions of this state was first proposed by Doherty, Parrilo and Spedialeri. The complete family of separability tests it generates, often referred to as the hierarchy of $k$-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 $k$-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 $k$-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 $\mathbf{C}^d\otimes\mathbf{C}^d$, when $d$ grows, the average width of the set of $k$-extendible states is equivalent to $(2/\sqrt{k})/d$, while random states obtained as partial traces over an environment $\mathbf{C}^s$ of uniformly distributed pure states are violating the $k$-extendibility test with probability going to $1$ if $s<((k-1)^2/4k)d^2$. Both statements converge to the conclusion that, if $k$ is fixed, $k$-extendibility is asymptotically a weak approximation of separability, even though any of the other well-studied separability relaxations is outperformed by $k$-extendibility as soon as $k$ is above a certain (dimension independent) value.

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arXiv2015

Relaxations of separability in multipartite systems: Semidefinite programs, witnesses and volumes

While entanglement is believed to be an important ingredient in understanding quantum many-body physics, the complexity of its characterization scales very unfavorably with the size of the system. Finding super-sets of the set of separable states that admit a simpler description has proven to be a fruitful approach in the bipartite setting. In this paper we discuss a systematic way of characterizing multiparticle entanglement via various relaxations. We furthermore describe an operational witness construction arising from such relaxations that is capable of detecting every entangled state. Finally, we also derive an analytic upper-bound on the volume of biseparable states and show that the volume of the states with a positive partial transpose for any split rapidly outgrows this volume. This proves that simple semi-definite relaxations in the multiparticle case cannot be an equally good approximation for any scenario.

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arXiv2015

Zonoids and sparsification of quantum measurements

In this paper, we establish a connection between zonoids (a concept from classical convex geometry) and the distinguishability norms associated to quantum measurements, or POVMs (Positive Operator-Valued Measures), recently introduced in quantum information theory. This correspondence allows us to state and prove the POVM version of classical results from the local theory of Banach spaces about the approximation of zonoids by zonotopes. We show that on $\mathbf{C}^d$, the uniform POVM (the most symmetric POVM) can be sparsified, i.e. approximated by a discrete POVM, the latter having only $O(d^2)$ outcomes. We also show that similar (but weaker) approximation results actually hold for any POVM on $\mathbf{C}^d$. By defining an appropriate notion of tensor product for zonoids, we are then able to extend our results to the multipartite setting: we show, roughly speaking, that local POVMs may be sparsified locally. In particular, the local uniform POVM on $\mathbf{C}^{d_1}\otimes\cdots\otimes\mathbf{C}^{d_k}$ can be approximated by a discrete POVM which is local and has $O(d_1^2\times\cdots\times d_k^2)$ outcomes.

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arXiv2015

Locally restricted measurements on a multipartite quantum system: data hiding is generic

We study the distinguishability norms associated to families of locally restricted POVMs on multipartite systems. These norms (introduced by Matthews, Wehner and Winter) quantify how quantum measurements, subject to locality constraints, perform in the task of discriminating two multipartite quantum states. We mainly address the following question regarding the behaviour of these distinguishability norms in the high-dimensional regime: On a bipartite space, what are the relative strengths of standard classes of locally restricted measurements? We show that the class of PPT measurements typically performs almost as well as the class of all measurements whereas restricting to local measurements and classical communication, or even just to separable measurements, implies a substantial loss. We also provide examples of state pairs which can be perfectly distinguished by local measurements if (one-way) classical communication is allowed between the parties, but very poorly without it. Finally, we study how many POVMs are needed to distinguish almost perfectly any pair of states on $\mathbf{C}^d$, showing that the answer is $\exp(Θ(d^2))$.

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arXiv2013

Distinguishing multi-partite states by local measurements

We analyze the distinguishability norm on the states of a multi-partite system, defined by local measurements. Concretely, we show that the norm associated to a tensor product of sufficiently symmetric measurements is essentially equivalent to a multi-partite generalisation of the non-commutative 2-norm (aka Hilbert-Schmidt norm): in comparing the two, the constants of domination depend only on the number of parties but not on the Hilbert spaces dimensions. We discuss implications of this result on the corresponding norms for the class of all measurements implementable by local operations and classical communication (LOCC), and in particular on the leading order optimality of multi-party data hiding schemes.

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