Entanglement Distillation from Greenberger–Horne–Zeilinger Shares

Vrana, Péter; Christandl, Matthias

doi:10.1007/s00220-017-2861-6

Cited by 19 publications

(17 citation statements)

References 8 publications

(11 reference statements)

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“…Recall that we defined g(G) to be the size of a minimum cut of G in Example 1.1.21. The following theorem is a special case of the result proved in [VC16].…”

Section: Preliminariesmentioning

confidence: 82%

“…For k = 3 this question was affirmatively answered in [Str91] (see Theorem 1.1.32 above), and for λ = (k − 1, 1) and any k ≥ 3 this question was affirmatively answered in [VC16] (see Theorem 1.1.35 above). Theorem 1.2.4 extends both results and covers more cases, including λ = (2, 2), see Corollary 1.2.5.…”

Section: Questionsmentioning

confidence: 98%

“…We now introduce the second family of tensors, D λ . SGDM03,VC16]). Let k be a positive integer and let λ = (λ 1 , .…”

Section: Preliminariesmentioning

confidence: 99%

“…In quantum information theory, if φ and ψ are pure quantum states, then φ ≥ ψ means precisely that ψ can be obtained from φ by means of stochastic local operations and classical communication (SLOCC). We refer to [VC15] and [VC16] for general properties of ω(φ, ψ).…”

Section: Preliminariesmentioning

confidence: 99%

“…Our third question is about the (monomial) subexponent of Dicke tensors. In [VC16] the upper bound q(D λ ) ≤ H(λ/k) was proven for any λ k, and the following problem was posed.…”

Section: Questionsmentioning

confidence: 99%

See 4 more Smart Citations

Asymptotic tensor rank of graph tensors: beyond matrix multiplication

2018

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We present an upper bound on the exponent of the asymptotic behaviour of the tensor rank of a family of tensors defined by the complete graph on k vertices. For k ≥ 4, we show that the exponent per edge is at most 0.77, outperforming the best known upper bound on the exponent per edge for matrix multiplication (k = 3), which is approximately 0.79. We raise the question whether for some k the exponent per edge can be below 2/3, i.e. can outperform matrix multiplication even if the matrix multiplication exponent equals 2. In order to obtain our results, we generalise to higher order tensors a result by Strassen on the asymptotic subrank of tight tensors and a result by Coppersmith and Winograd on the asymptotic rank of matrix multiplication. Our results have applications in entanglement theory and communication complexity.

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“…Recall that we defined g(G) to be the size of a minimum cut of G in Example 1.1.21. The following theorem is a special case of the result proved in [VC16].…”

Section: Preliminariesmentioning

confidence: 82%

Section: Questionsmentioning

confidence: 98%

“…We now introduce the second family of tensors, D λ . SGDM03,VC16]). Let k be a positive integer and let λ = (λ 1 , .…”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

“…Our third question is about the (monomial) subexponent of Dicke tensors. In [VC16] the upper bound q(D λ ) ≤ H(λ/k) was proven for any λ k, and the following problem was posed.…”

Section: Questionsmentioning

confidence: 99%

See 3 more Smart Citations

Asymptotic tensor rank of graph tensors: beyond matrix multiplication

2018

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Multipartite Entanglement

Walter¹,

Gross²,

Eisert³

2016

Quantum Information

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Interpolating Between Rényi Entanglement Entropies for Arbitrary Bipartitions via Operator Geometric Means

Bugár,

Vrana

2024

Ann. Henri Poincaré

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The asymptotic restriction problem for tensors can be reduced to finding all parameters that are normalized, monotone under restrictions, additive under direct sums and multiplicative under tensor products, the simplest of which are the flattening ranks. Over the complex numbers, a refinement of this problem, originating in the theory of quantum entanglement, is to find the optimal rate of entanglement transformations as a function of the error exponent. This trade-off can also be characterized in terms of the set of normalized, additive, multiplicative functionals that are monotone in a suitable sense, which includes the restriction-monotones as well. For example, the flattening ranks generalize to the (exponentiated) Rényi entanglement entropies of order $$\alpha \in [0,1]$$ α ∈ [ 0 , 1 ] . More complicated parameters of this type are known, which interpolate between the flattening ranks or Rényi entropies for special bipartitions, with one of the parts being a single tensor factor. We introduce a new construction of subadditive and submultiplicative monotones in terms of a regularized Rényi divergence between many copies of the pure state represented by the tensor and a suitable sequence of positive operators. We give explicit families of operators that correspond to the flattening-based functionals, and show that they can be combined in a nontrivial way using weighted operator geometric means. This leads to a new characterization of the previously known additive and multiplicative monotones, and gives new submultiplicative and subadditive monotones that interpolate between the Rényi entropies for all bipartitions. We show that for each such monotone there exist pointwise smaller multiplicative and additive ones as well. In addition, we find lower bounds on the new functionals that are superadditive and supermultiplicative.

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Entanglement Distillation from Greenberger–Horne–Zeilinger Shares

Cited by 19 publications

References 8 publications

Asymptotic tensor rank of graph tensors: beyond matrix multiplication

Asymptotic tensor rank of graph tensors: beyond matrix multiplication

Multipartite Entanglement

Interpolating Between Rényi Entanglement Entropies for Arbitrary Bipartitions via Operator Geometric Means

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