dorsal/arxiv
View SchemaProperties of Conjugate Channels with Applications to Additivity and Multiplicativity
| Authors | Christopher King, Keiji Matsumoto, Michael Nathanson, Mary Beth Ruskai |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0509126 |
| URL | https://arxiv.org/abs/quant-ph/0509126 |
| Journal | Markov Process and Related Fields 13, 391-423 (2007) |
Abstract
Quantum channels can be described via a unitary coupling of system and environment, followed by a trace over the environment state space. Taking the trace instead over the system state space produces a different mapping which we call the conjugate channel. We explore the properties of conjugate channels and describe several different methods of construction. In general, conjugate channels map M_d to M_d' with d < d', and different constructions may differ by conjugation with a partial isometry. We show that a channel and its conjugate have the same minimal output entropy and maximal output p-norm. It then follows that the additivity and multiplicativity conjectures for these measures of optimal output purity hold for a product of channels if and only if they also hold for the product of their conjugates. This allows us to reduce these conjectures to the special case of maps taking M_d to M_d' with a minimal representation of dimension at most d. We find explicit expressions for the conjugates for a number of well-known examples, including entanglement-breaking channels, unital qubit channels, the depolarizing channel, and a subclass of random unitary channels. For the entanglement-breaking channels, channels this yields a new class of channels for which additivity and multiplicativity of optimal output purity can be established. For random unitary channels using the generalized Pauli matrices, we obtain a new formulation of the multiplicativity conjecture. The conjugate of the completely noisy channel plays a special role and suggests a mechanism for using noise to transmit information.
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"abstract": "Quantum channels can be described via a unitary coupling of system and\nenvironment, followed by a trace over the environment state space. Taking the\ntrace instead over the system state space produces a different mapping which we\ncall the conjugate channel. We explore the properties of conjugate channels and\ndescribe several different methods of construction. In general, conjugate\nchannels map M_d to M_d\u0027 with d \u003c d\u0027, and different constructions may differ by\nconjugation with a partial isometry.\n We show that a channel and its conjugate have the same minimal output entropy\nand maximal output p-norm. It then follows that the additivity and\nmultiplicativity conjectures for these measures of optimal output purity hold\nfor a product of channels if and only if they also hold for the product of\ntheir conjugates. This allows us to reduce these conjectures to the special\ncase of maps taking M_d to M_d\u0027 with a minimal representation of dimension at\nmost d.\n We find explicit expressions for the conjugates for a number of well-known\nexamples, including entanglement-breaking channels, unital qubit channels, the\ndepolarizing channel, and a subclass of random unitary channels. For the\nentanglement-breaking channels, channels this yields a new class of channels\nfor which additivity and multiplicativity of optimal output purity can be\nestablished. For random unitary channels using the generalized Pauli matrices,\nwe obtain a new formulation of the multiplicativity conjecture. The conjugate\nof the completely noisy channel plays a special role and suggests a mechanism\nfor using noise to transmit information.",
"arxiv_id": "quant-ph/0509126",
"authors": [
"Christopher King",
"Keiji Matsumoto",
"Michael Nathanson",
"Mary Beth Ruskai"
],
"categories": [
"quant-ph"
],
"journal_ref": "Markov Process and Related Fields 13, 391-423 (2007)",
"title": "Properties of Conjugate Channels with Applications to Additivity and Multiplicativity",
"url": "https://arxiv.org/abs/quant-ph/0509126"
},
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