Entanglement Thresholds for Random Induced States

Aubrun, Guillaume; Szarek, Stanisław J.; Ye, Deping

doi:10.1002/cpa.21460

Cited by 61 publications

(104 citation statements)

References 58 publications

(120 reference statements)

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“…The striking feature of these values is the fact that the (bounds for the) thresholds for the set SEP, obtained in the important work [ASY14], are one order of magnitude above the thresholds for the various entanglement criteria. This means that, from a volume perspective, the set SEP is much smaller than the set of states satisfying the different entanglement criteria.…”

Section: 3mentioning

confidence: 84%

“…More precise estimates have been obtained in [ASY14] in the case of the induced measures. In order to present these results, we need first to introduce the concept of thresholds.…”

Section: 3mentioning

confidence: 99%

“…We state now the main result in [ASY14], regarding the threshold for the sets SEP n,n . The following statement corresponds to [ASY14, Theorem 2.3], which deals with the so-called balanced regime k = n. For the unbalanced regime k = n, see [ASY14, Section 7.2].…”

Section: 3mentioning

confidence: 99%

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Random matrix techniques in quantum information theory

Collins

Nechita

2015

Journal of Mathematical Physics

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Abstract. The purpose of this review article is to present some of the latest developments using random techniques, and in particular, random matrix techniques in quantum information theory. Our review is a blend of a rather exhaustive review, combined with more detailed examples -coming from research projects in which the authors were involved. We focus on two main topics, random quantum states and random quantum channels. We present results related to entropic quantities, entanglement of typical states, entanglement thresholds, the output set of quantum channels, and violations of the minimum output entropy of random channels. Contents

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Section: 3mentioning

confidence: 84%

“…More precise estimates have been obtained in [ASY14] in the case of the induced measures. In order to present these results, we need first to introduce the concept of thresholds.…”

Section: 3mentioning

confidence: 99%

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Random matrix techniques in quantum information theory

Collins

Nechita

2015

Journal of Mathematical Physics

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“…Such an inequality is arguably more important than the classical isoperimetric inequality. Consequently, the Blaschke-Santaló inequality has numerous applications in (affine) convex geometry and many other related fields, for instance, in quantum information theory [2,38]. The family of affine isoperimetric inequalities continues to grow (see, for example [6,10,11,24,25]).…”

Section: Introduction and Overview Of Resultsmentioning

confidence: 99%

Lp Geominimal Surface Areas and their Inequalities

Ye¹

2014

International Mathematics Research Notices

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In this paper, we introduce the L p geominimal surface area for all −n = p < 1, which extends the classical geominimal surface area (p = 1) by Petty and the L p geominimal surface area by Lutwak (p > 1). Our extension of the L p geominimal surface area is motivated by recent work on the extension of the L p affine surface area -a fundamental object in (affine) convex geometry. We prove some properties for the L p geominimal surface area and its related inequalities, such as, the affine isoperimetric inequality and a Santaló style inequality. Cyclic inequalities are established to obtain the monotonicity of the L p geominimal surface areas. Comparison between the L p geominimal surface area and the p-surface area is also provided.2010 Mathematics Subject Classification: 52A20, 53A15 Introduction and Overview of ResultsThe classical isoperimetric problem asks: what is the minimal area among all convex bodies (i.e., convex compact subsets with nonempty interior) K ⊂ R n with volume 1? The solution of this old problem is now known as the (classical) isoperimetric inequality, namely, the minimal area is attained at and only at Euclidean balls with volume 1. The isoperimetric inequality is an extremely powerful tool in geometry and related areas. Note that the classical isoperimetric inequality does not have the "affine invariant" flavor, because the area may change under linear transformations (even) with unit (absolute value of) determinant.However, many objects in (affine) convex geometry are invariant under invertible linear transformations. A typical example is the Mahler volume product M(K) = |K||K• |, the product of the volume of K and its polar body K• . That is, M(K) = M(T K) for all invertible linear transformations T on R n . One can ask a question for M(K) similar * Keywords: affine surface area, L p affine surface area, geominimal surface area, L p geominimal surface area, L p -Brunn-Minkowski theory, affine isoperimetric inequalities, the Blaschke-Santaló inequality, the Bourgain-Milman inverse Santaló inequality.

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“…In a previous paper [1], a family of formulas was obtained for the (total) separability probabilities ( , ) of generalized two-qubit states ( = 4) endowed with Hilbert-Schmidt ( = 0) [2] or more generally random induced measure [3,4]. In this regard, we note that the natural, rotationally invariant measure on the set of all pure states of a × composite system ( = − ) induces a unique measure in the space of × mixed states [3, eq.…”

Section: Introductionmentioning

confidence: 99%

Formulas for Generalized Two-Qubit Separability Probabilities

Slater

2018

Advances in Mathematical Physics

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To begin, we find certain formulas ( , ) = 1 ( ) 2 ( ), for = −1, 0, 1, . . . , 9. These yield that part of the total separability probability, ( , ), for generalized (real, complex, quaternionic, etc.) two-qubit states endowed with random induced measure, for which the determinantal inequality | PT | > | | holds. Here denotes a 4×4 density matrix, obtained by tracing over the pure states in 4×(4+ )-dimensions, and PT denotes its partial transpose. Further, is a Dyson-index-like parameter with = 1 for the standard (15-dimensional) convex set of (complex) two-qubit states. For = 0, we obtain the previously reported Hilbert-Schmidt formulas, with (0, 1/2) = 29/128 (the real case), (0, 1) = 4/33 (the standard complex case), and (0, 2) = 13/323 (the quaternionic case), the three simply equalling (0, )/2. The factors 2 ( ) are sums of polynomial-weighted generalized hypergeometric functions −1 , ≥ 7, all with argument = 27/64 = (3/4) 3 . We find number-theoretic-based formulas for the upper ( ) and lower ( ) parameter sets of these functions and, then, equivalently express 2 ( ) in terms of first-order difference equations. Applications of Zeilberger's algorithm yield "concise" forms of (−1, ), (1, ), and (3, ), parallel to the one obtained previously (Slater 2013) for (0, ) = 2 (0, ). For nonnegative half-integer and integer values of , ( , ) (as well as ( , )) has descending roots starting at = − − 1. Then, we (Dunkl and I) construct a remarkably compact (hypergeometric) form for ( , ) itself. The possibility of an analogous "master" formula for ( , ) is, then, investigated, and a number of interesting results are found.

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Entanglement Thresholds for Random Induced States

Cited by 61 publications

References 58 publications

Random matrix techniques in quantum information theory

Random matrix techniques in quantum information theory

Lp Geominimal Surface Areas and their Inequalities

Formulas for Generalized Two-Qubit Separability Probabilities

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