On Nonuniqueness of Quantum Channel for Fixed Input-Output States: Case of Decoherence Channel
For a fixed pair of input and output states in the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>A</mi></msub></mrow></semantics>...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-01-01
|
| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/14/2/214 |
| _version_ | 1797476479647350784 |
|---|---|
| author | Congjie Ou Sumiyoshi Abe |
| author_facet | Congjie Ou Sumiyoshi Abe |
| author_sort | Congjie Ou |
| collection | DOAJ |
| description | For a fixed pair of input and output states in the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>A</mi></msub></mrow></semantics></math></inline-formula> of a system <i>A</i>, a quantum channel, i.e., a linear, completely positive and trace-preserving map, between them is not unique, in general. Here, this point is discussed specifically for a decoherence channel, which maps from a pure input state to a completely decoherent state like the thermal state. In particular, decoherence channels of two different types are analyzed: one is unital and the other is not, and both of them can be constructed through reduction of <i>B</i> in the total extended space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>A</mi></msub><mo>⊗</mo><msub><mi>H</mi><mi>B</mi></msub></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>B</mi></msub></mrow></semantics></math></inline-formula> is the space of an ancillary system <i>B</i> that is a replica of <i>A</i>. The nonuniqueness is seen to have its origin in the unitary symmetry in the extended space. It is shown in an example of a two-qubit system how such symmetry is broken in the objective subspace <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>A</mi></msub></mrow></semantics></math></inline-formula> due to entanglement between <i>A</i> and <i>B</i>. A comment is made on possible relevance of the present work to nanothermodynamics in view of quantum Darwinism. |
| first_indexed | 2024-03-09T20:58:24Z |
| format | Article |
| id | doaj.art-758d32f09345437f81436e984d239494 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T20:58:24Z |
| publishDate | 2022-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-758d32f09345437f81436e984d2394942023-11-23T22:15:08ZengMDPI AGSymmetry2073-89942022-01-0114221410.3390/sym14020214On Nonuniqueness of Quantum Channel for Fixed Input-Output States: Case of Decoherence ChannelCongjie Ou0Sumiyoshi Abe1Department of Physics, College of Information Science and Engineering, Huaqiao University, Xiamen 361021, ChinaDepartment of Physics, College of Information Science and Engineering, Huaqiao University, Xiamen 361021, ChinaFor a fixed pair of input and output states in the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>A</mi></msub></mrow></semantics></math></inline-formula> of a system <i>A</i>, a quantum channel, i.e., a linear, completely positive and trace-preserving map, between them is not unique, in general. Here, this point is discussed specifically for a decoherence channel, which maps from a pure input state to a completely decoherent state like the thermal state. In particular, decoherence channels of two different types are analyzed: one is unital and the other is not, and both of them can be constructed through reduction of <i>B</i> in the total extended space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>A</mi></msub><mo>⊗</mo><msub><mi>H</mi><mi>B</mi></msub></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>B</mi></msub></mrow></semantics></math></inline-formula> is the space of an ancillary system <i>B</i> that is a replica of <i>A</i>. The nonuniqueness is seen to have its origin in the unitary symmetry in the extended space. It is shown in an example of a two-qubit system how such symmetry is broken in the objective subspace <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>A</mi></msub></mrow></semantics></math></inline-formula> due to entanglement between <i>A</i> and <i>B</i>. A comment is made on possible relevance of the present work to nanothermodynamics in view of quantum Darwinism.https://www.mdpi.com/2073-8994/14/2/214channel nonuniquenessdecoherence channelunitary symmetrytwo-qubit |
| spellingShingle | Congjie Ou Sumiyoshi Abe On Nonuniqueness of Quantum Channel for Fixed Input-Output States: Case of Decoherence Channel Symmetry channel nonuniqueness decoherence channel unitary symmetry two-qubit |
| title | On Nonuniqueness of Quantum Channel for Fixed Input-Output States: Case of Decoherence Channel |
| title_full | On Nonuniqueness of Quantum Channel for Fixed Input-Output States: Case of Decoherence Channel |
| title_fullStr | On Nonuniqueness of Quantum Channel for Fixed Input-Output States: Case of Decoherence Channel |
| title_full_unstemmed | On Nonuniqueness of Quantum Channel for Fixed Input-Output States: Case of Decoherence Channel |
| title_short | On Nonuniqueness of Quantum Channel for Fixed Input-Output States: Case of Decoherence Channel |
| title_sort | on nonuniqueness of quantum channel for fixed input output states case of decoherence channel |
| topic | channel nonuniqueness decoherence channel unitary symmetry two-qubit |
| url | https://www.mdpi.com/2073-8994/14/2/214 |
| work_keys_str_mv | AT congjieou onnonuniquenessofquantumchannelforfixedinputoutputstatescaseofdecoherencechannel AT sumiyoshiabe onnonuniquenessofquantumchannelforfixedinputoutputstatescaseofdecoherencechannel |