dorsal/arxiv
View SchemaOn the additivity conjecture for channels with arbitrary constraints
| Authors | M. E. Shirokov |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0308168 |
| URL | https://arxiv.org/abs/quant-ph/0308168 |
Abstract
Recently Shor proved equivalence of several open (sub)additivity problems related to the Holevo capacity and the entanglement of formation [15]. In our previous note [6] equivalence of these to the additivity of the Holevo capacity for channels with arbitrary linear constraints was shown. This note is the development of the previous one in the direction of channels with general constraints. Introducing input constraints provides greater flexibility in the treatment of the additivity conjecture. The Holevo capacity of arbitrarily constrained channel is considered and the characteristic property of an optimal ensemble for such channel is derived, generalizing the maximal distance property of Schumacher and Westmoreland (proposition 1). It is shown that the additivity conjecture for two channels with single linear constraints is equivalent to the similar conjecture for two arbitrarily constrained channels and, hence, to an interesting subadditivity property of the $\chi$-function for the tensor product of these channels (theorem 1). We also propose an alternative way of proving that the additivity conjecture for any two unconstrained channels implies strong superadditivity of the entanglement of formation. The arguments from the convex analysis provide another characterization of channels for which subadditivity of the $\chi $-function holds (theorem 3). This characterization and some modification of Shor's channel extension provide a simple way of proving that global additivity of the minimum output entropy for unconstrained channels implies global subadditivity of the $\chi $-function and strong superadditivity of the entanglement of formation.
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"abstract": "Recently Shor proved equivalence of several open (sub)additivity problems\nrelated to the Holevo capacity and the entanglement of formation [15]. In our\nprevious note [6] equivalence of these to the additivity of the Holevo capacity\nfor channels with arbitrary linear constraints was shown. This note is the\ndevelopment of the previous one in the direction of channels with general\nconstraints. Introducing input constraints provides greater flexibility in the\ntreatment of the additivity conjecture. The Holevo capacity of arbitrarily\nconstrained channel is considered and the characteristic property of an optimal\nensemble for such channel is derived, generalizing the maximal distance\nproperty of Schumacher and Westmoreland (proposition 1). It is shown that the\nadditivity conjecture for two channels with single linear constraints is\nequivalent to the similar conjecture for two arbitrarily constrained channels\nand, hence, to an interesting subadditivity property of the $\\chi$-function for\nthe tensor product of these channels (theorem 1). We also propose an\nalternative way of proving that the additivity conjecture for any two\nunconstrained channels implies strong superadditivity of the entanglement of\nformation. The arguments from the convex analysis provide another\ncharacterization of channels for which subadditivity of the $\\chi $-function\nholds (theorem 3). This characterization and some modification of Shor\u0027s\nchannel extension provide a simple way of proving that global additivity of the\nminimum output entropy for unconstrained channels implies global subadditivity\nof the $\\chi $-function and strong superadditivity of the entanglement of\nformation.",
"arxiv_id": "quant-ph/0308168",
"authors": [
"M. E. Shirokov"
],
"categories": [
"quant-ph"
],
"title": "On the additivity conjecture for channels with arbitrary constraints",
"url": "https://arxiv.org/abs/quant-ph/0308168"
},
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