Computing Classical-Quantum Channel Capacity Using Blahut–Arimoto Type Algorithm: A Theoretical and Numerical Analysis
Based on Arimoto’s work in 1972, we propose an iterative algorithm for computing the capacity of a discrete memoryless classical-quantum channel with a finite input alphabet and a finite dimensional output, which we call the Blahut−Arimoto algorithm for classical-quantum channel,...
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2020-02-01
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| Online Access: | https://www.mdpi.com/1099-4300/22/2/222 |
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doaj-f28333603bba4753b2c7387f40dbbc8c2020-11-25T01:45:08ZengMDPI AGEntropy1099-43002020-02-0122222210.3390/e22020222e22020222Computing Classical-Quantum Channel Capacity Using Blahut–Arimoto Type Algorithm: A Theoretical and Numerical AnalysisHaobo Li0Ning Cai1School of Information Science and Technology, ShanghaiTech University, Shanghai 201210, ChinaSchool of Information Science and Technology, ShanghaiTech University, Shanghai 201210, ChinaBased on Arimoto’s work in 1972, we propose an iterative algorithm for computing the capacity of a discrete memoryless classical-quantum channel with a finite input alphabet and a finite dimensional output, which we call the Blahut−Arimoto algorithm for classical-quantum channel, and an input cost constraint is considered. We show that, to reach <inline-formula> <math display="inline"> <semantics> <mi>ε</mi> </semantics> </math> </inline-formula> accuracy, the iteration complexity of the algorithm is upper bounded by <inline-formula> <math display="inline"> <semantics> <mfrac> <mrow> <mo form="prefix">log</mo> <mi>n</mi> <mo form="prefix">log</mo> <mi>ε</mi> </mrow> <mi>ε</mi> </mfrac> </semantics> </math> </inline-formula> where <i>n</i> is the size of the input alphabet. In particular, when the output state <inline-formula> <math display="inline"> <semantics> <msub> <mrow> <mo>{</mo> <msub> <mi>ρ</mi> <mi>x</mi> </msub> <mo>}</mo> </mrow> <mrow> <mi>x</mi> <mo>∈</mo> <mi mathvariant="script">X</mi> </mrow> </msub> </semantics> </math> </inline-formula> is linearly independent in complex matrix space, the algorithm has a geometric convergence. We also show that the algorithm reaches an <inline-formula> <math display="inline"> <semantics> <mi>ε</mi> </semantics> </math> </inline-formula> accurate solution with a complexity of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>O</mi> <mo>(</mo> <mfrac> <mrow> <msup> <mi>m</mi> <mn>3</mn> </msup> <mo form="prefix">log</mo> <mi>n</mi> <mo form="prefix">log</mo> <mi>ε</mi> </mrow> <mi>ε</mi> </mfrac> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>O</mi> <mo>(</mo> <msup> <mi>m</mi> <mn>3</mn> </msup> <mo form="prefix">log</mo> <mi>ε</mi> <msub> <mo form="prefix">log</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>−</mo> <mi>δ</mi> <mo>)</mo> </mrow> </msub> <mfrac> <mi>ε</mi> <mrow> <mi>D</mi> <mo>(</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>|</mo> <mo>|</mo> <msup> <mi>p</mi> <msub> <mi>N</mi> <mn>0</mn> </msub> </msup> <mo>)</mo> </mrow> </mfrac> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> in the special case, where <i>m</i> is the output dimension, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>D</mi> <mo>(</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>|</mo> <mo>|</mo> <msup> <mi>p</mi> <msub> <mi>N</mi> <mn>0</mn> </msub> </msup> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> is the relative entropy of two distributions, and <inline-formula> <math display="inline"> <semantics> <mi>δ</mi> </semantics> </math> </inline-formula> is a positive number. Numerical experiments were performed and an approximate solution for the binary two-dimensional case was analysed.https://www.mdpi.com/1099-4300/22/2/222capacityclassical-quantum channelblahut–arimoto type algorithmconvergence speed |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Haobo Li Ning Cai |
| spellingShingle |
Haobo Li Ning Cai Computing Classical-Quantum Channel Capacity Using Blahut–Arimoto Type Algorithm: A Theoretical and Numerical Analysis Entropy capacity classical-quantum channel blahut–arimoto type algorithm convergence speed |
| author_facet |
Haobo Li Ning Cai |
| author_sort |
Haobo Li |
| title |
Computing Classical-Quantum Channel Capacity Using Blahut–Arimoto Type Algorithm: A Theoretical and Numerical Analysis |
| title_short |
Computing Classical-Quantum Channel Capacity Using Blahut–Arimoto Type Algorithm: A Theoretical and Numerical Analysis |
| title_full |
Computing Classical-Quantum Channel Capacity Using Blahut–Arimoto Type Algorithm: A Theoretical and Numerical Analysis |
| title_fullStr |
Computing Classical-Quantum Channel Capacity Using Blahut–Arimoto Type Algorithm: A Theoretical and Numerical Analysis |
| title_full_unstemmed |
Computing Classical-Quantum Channel Capacity Using Blahut–Arimoto Type Algorithm: A Theoretical and Numerical Analysis |
| title_sort |
computing classical-quantum channel capacity using blahut–arimoto type algorithm: a theoretical and numerical analysis |
| publisher |
MDPI AG |
| series |
Entropy |
| issn |
1099-4300 |
| publishDate |
2020-02-01 |
| description |
Based on Arimoto’s work in 1972, we propose an iterative algorithm for computing the capacity of a discrete memoryless classical-quantum channel with a finite input alphabet and a finite dimensional output, which we call the Blahut−Arimoto algorithm for classical-quantum channel, and an input cost constraint is considered. We show that, to reach <inline-formula> <math display="inline"> <semantics> <mi>ε</mi> </semantics> </math> </inline-formula> accuracy, the iteration complexity of the algorithm is upper bounded by <inline-formula> <math display="inline"> <semantics> <mfrac> <mrow> <mo form="prefix">log</mo> <mi>n</mi> <mo form="prefix">log</mo> <mi>ε</mi> </mrow> <mi>ε</mi> </mfrac> </semantics> </math> </inline-formula> where <i>n</i> is the size of the input alphabet. In particular, when the output state <inline-formula> <math display="inline"> <semantics> <msub> <mrow> <mo>{</mo> <msub> <mi>ρ</mi> <mi>x</mi> </msub> <mo>}</mo> </mrow> <mrow> <mi>x</mi> <mo>∈</mo> <mi mathvariant="script">X</mi> </mrow> </msub> </semantics> </math> </inline-formula> is linearly independent in complex matrix space, the algorithm has a geometric convergence. We also show that the algorithm reaches an <inline-formula> <math display="inline"> <semantics> <mi>ε</mi> </semantics> </math> </inline-formula> accurate solution with a complexity of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>O</mi> <mo>(</mo> <mfrac> <mrow> <msup> <mi>m</mi> <mn>3</mn> </msup> <mo form="prefix">log</mo> <mi>n</mi> <mo form="prefix">log</mo> <mi>ε</mi> </mrow> <mi>ε</mi> </mfrac> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>O</mi> <mo>(</mo> <msup> <mi>m</mi> <mn>3</mn> </msup> <mo form="prefix">log</mo> <mi>ε</mi> <msub> <mo form="prefix">log</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>−</mo> <mi>δ</mi> <mo>)</mo> </mrow> </msub> <mfrac> <mi>ε</mi> <mrow> <mi>D</mi> <mo>(</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>|</mo> <mo>|</mo> <msup> <mi>p</mi> <msub> <mi>N</mi> <mn>0</mn> </msub> </msup> <mo>)</mo> </mrow> </mfrac> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> in the special case, where <i>m</i> is the output dimension, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>D</mi> <mo>(</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>|</mo> <mo>|</mo> <msup> <mi>p</mi> <msub> <mi>N</mi> <mn>0</mn> </msub> </msup> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> is the relative entropy of two distributions, and <inline-formula> <math display="inline"> <semantics> <mi>δ</mi> </semantics> </math> </inline-formula> is a positive number. Numerical experiments were performed and an approximate solution for the binary two-dimensional case was analysed. |
| topic |
capacity classical-quantum channel blahut–arimoto type algorithm convergence speed |
| url |
https://www.mdpi.com/1099-4300/22/2/222 |
| work_keys_str_mv |
AT haoboli computingclassicalquantumchannelcapacityusingblahutarimototypealgorithmatheoreticalandnumericalanalysis AT ningcai computingclassicalquantumchannelcapacityusingblahutarimototypealgorithmatheoreticalandnumericalanalysis |
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