An Improved Semidefinite Programming Hierarchy for Testing Entanglement

Harrow, Aram W.; Natarajan, Anand; Wu, Xiaodi

doi:10.1007/s00220-017-2859-0

Cited by 32 publications

(28 citation statements)

References 37 publications

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“…The structure of no-signalling codes is also studied in [42]. Semidefinite programming [50] is a subfield of convex optimization and is a powerful tool in quantum information theory with many applications (e.g., [23,28,37,42,[51][52][53][54][55][56]). There are known polynomial-time algorithms for semidefinite programming [57].…”

Section: Preliminariesmentioning

confidence: 99%

Semidefinite Programming Strong Converse Bounds for Classical Capacity

Wang

Xie

Duan

2018

IEEE Trans. Inform. Theory

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We investigate the classical communication over quantum channels when assisted by no-signalling (NS) and PPT-preserving (PPT) codes, for which both the optimal success probability of a given transmission rate and the one-shot -error capacity are formalized as semidefinite programs (SDPs). Based on this, we obtain improved SDP finite blocklength converse bounds of general quantum channels for entanglement-assisted codes and unassisted codes. Furthermore, we derive two SDP strong converse bounds for the classical capacity of general quantum channels: for any code with a rate exceeding either of the two bounds of the channel, the success probability vanishes exponentially fast as the number of channel uses increases. In particular, applying our efficiently computable bounds, we derive an improved upper bound on the classical capacity of the amplitude damping channel. We also establish the strong converse property for the classical and private capacities of a new class of quantum channels. We finally study the zero-error setting and provide efficiently computable upper bounds on the one-shot zero-error capacity of a general quantum channel.

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Section: Preliminariesmentioning

confidence: 99%

Semidefinite Programming Strong Converse Bounds for Classical Capacity

Wang

Xie

Duan

2018

IEEE Trans. Inform. Theory

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“…The no-signalling codes is also studied in [27]. Semidefinite programming [35] is a subfield of convex optimization and is a powerful tool in quantum information theory with many applications (e.g., [15], [22], [27], [28], [36], [37], [38], [39]). In this work, we use CVX [40] and QETLAB [41] to solve the SDPs in this work.…”

Section: Preliminariesmentioning

confidence: 99%

Semidefinite programming converse bounds for classical communication over quantum channels

Wang

Xie

Duan

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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We study the classical communication over quantum channels when assisted by no-signalling (NS) and PPT-preserving (PPT) codes. We first show that both the optimal success probability of a given transmission rate and one-shot-error capacity can be formalized as semidefinite programs (SDPs) when assisted by NS or NS∩PPT codes. Based on this, we derive SDP finite blocklength converse bounds for general quantum channels, which also reduce to the converse bound of Polyanskiy, Poor, and Verdú for classical channels. Furthermore, we derive an SDP strong converse bound for the classical capacity of a general quantum channel: for any code with a rate exceeding this bound, the optimal success probability vanishes exponentially fast as the number of channel uses increases. In particular, applying our efficiently computable bound, we derive improved upper bounds to the classical capacity of the amplitude damping channels and also establish the strong converse property for a new class of quantum channels.

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“…Indeed the only previously known unconditional negative result was in the original DPS paper which showed that the error always remained nonzero for all finite values of k (see also [BS10] showing that this could be amplified). Indeed one can even define an improved version of DPS that removes this limitation and always exactly converges at a finite (but large) value of k [HNW15].…”

Section: Dps Hierarchy For Separability Problem and Integrality Gapsmentioning

confidence: 99%

Limitations of Semidefinite Programs for Separable States and Entangled Games

Harrow

Natarajan

2019

Commun. Math. Phys.

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Semidefinite programs (SDPs) are a framework for exact or approximate optimization with widespread application in quantum information theory. We introduce a new method for using reductions to construct integrality gaps for SDPs, meaning instances where the SDP value is far from the true optimum. These are based on new limitations on the sum-of-squares (SoS) hierarchy in approximating two particularly important sets in quantum information theory, where previously no ω(1)-round integrality gaps were known:

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An Improved Semidefinite Programming Hierarchy for Testing Entanglement

Cited by 32 publications

References 37 publications

Semidefinite Programming Strong Converse Bounds for Classical Capacity

Semidefinite Programming Strong Converse Bounds for Classical Capacity

Semidefinite programming converse bounds for classical communication over quantum channels

Limitations of Semidefinite Programs for Separable States and Entangled Games

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