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• W2196383449 abstract "Dvoretzky's theorem and the complexity of entanglement detection, Discrete Analysis 2017:1, 20 pp. Let $H$ be a Hilbert space. A _state_ on $H$ is a linear operator $rho:Hto H$ such that tr$(rho)=1$ and tr$(rho P)geq 0$ for every orthogonal projection $P$. A linear operator satisfying just the second condition is called _positive_. If $H=H_1otimes H_2$ then an important piece of information about $rho$ is the extent to which it can be split up into a part that acts on $H_1$ and a part that acts on $H_2$. An appropriate formal definition is the following: we say that $rho$ is _separable_ if it can be approximated in the trace-class norm by operators of the form $sum_ic_irho^1_iotimesrho^2_i$, where each $rho^1_i$ is a state on $H_1$, each $rho^2_i$ is a state on $H_2$, and the $c_i$ are positive constants. An operator that is not separable is called _entangled_. Owing to the importance of entanglement, it is useful to have a criterion that detects it. A simple necessary condition is that if $Phi$ is a linear map from operators to operators that takes positive operators to positive operators, and if $rho$ is separable, then $(Phiotimes I)rho$ is a positive operator: that is simply because $(Phiotimes I)(sum_ic_irho^1_iotimesrho^2_i)=sum_ic_iPhirho^1_iotimesrho^2_i$ and each $Phirho^1_i$ is positive. Interestingly, however, the converse is true: if $rho$ is entangled, then there exists a positive $Phi$ such that $(Phiotimes I)(rho)$ is _not_ positive. This equivalence is called the _Horodecki criterion_ for entanglement. The Horodecki criterion gives us a witness $Phi$ for each entangled state $rho$. The starting question for this paper is how many different witnesses one needs to detect all entangled states (as a function of the dimensions of $H_1$ and $H_2$). More precisely, since this number is known to be infinite, the paper considers the stronger notion of _robustly_ entangled states, which are states that remain entangled even when you average them with a suitable multiple of the identity. The main result of the paper is that the number needed is large: the authors obtain a bound of $exp(cd^3/log d)$. A key observation that enables them to prove this is that there is a resemblance between the problem they are considering and a result from a famous paper of Figiel, Lindenstrauss and Milman concerning Dvoretzky's theorem [1]. It follows from the analysis in the FLM paper that the product of the logarithm of the number of faces of a symmetric convex body with the logarithm of the number of vertices has to be at least $cn$, so it is not possible for a symmetric convex body to have few faces and few vertices (in strong contrast with a general convex body, since an $n$-dimensional simplex has $n+1$ of each). Something like that occurs here. Roughly speaking, the set of separable states can be regarded as having few extreme points, and therefore many faces. A single witness $Phi$ cannot help with too many faces, so the number of witnesses needed must be large. These interesting ideas are new to the field of quantum information. [1] T. Figiel, J. Lindenstrauss and V. D. Milman, _The dimension of almost spherical sections of convex bodies,_ Acta Math. 139 (1-2) (1977), pp. 53-94." @default.
• W2196383449 created "2016-06-24" @default.
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• W2196383449 date "2017-01-01" @default.
• W2196383449 modified "2023-09-25" @default.
• W2196383449 title "Dvoretzky's theorem and the complexity of entanglement detection" @default.
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• W2196383449 doi "https://doi.org/10.19086/da.1242" @default.
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