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Previous work on the classical information capacities of bosonic channels has established the capacity of the single-user pure-loss channel, bounded the capacity of the single-user thermal-noise channel, and bounded the capacity region of the multiple-access channel.
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V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, J. H. Shapiro, B. J. Yen, and H. P. Yuen, “Classical capacity of free-space optical communication,” in O. Hirota, ed., Quantum Information, Statistics, Probability
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𝒜 n {\cal A}^{n} , ℬ n {\cal B}^{n} , and 𝒞 n {\cal C}^{n} are the n n channel use alphabets of Alice, Bob, and Charlie, with respective sizes | 𝒜 n | |{\cal A}^{n}| , | ℬ n | |{\cal B}^{n}| , and | 𝒞 n | |{\cal C}^{n}|
Cited in the paper.
When | 𝒯 | |{\cal T}| and | 𝒜 | |{\cal A}| are finite, and we are using coherent states, there will be a finite number of possible transmitted states, which leads to a finite number of possible states received by Bob and Charlie. Suppose we limit the auxiliary-input alphabet ( T T )—and hence the input ( A A ) and the output alphabets ( B B and C C )—to truncated coherent states within the finite-dimensional Hilbert space spanned by the Fock states { | 0 ⟩ , | 1 ⟩ , … , | K ⟩ } \left\{|0\rangle,|1\rangle,\ldots,|K\rangle\right\} , where K ≫ N ¯ K\gg{\bar{N}} . Applying the theorem from Yard et al
Cited in the paper.
Holevo’s bound [ 14 ] : Let X X be the input alphabet for a channel, { p i , ρ ^ i } \left\{p_{i},{\hat{\rho}}_{i}\right\} be the priors and modulating states, { Π j } \left\{\Pi_{j}\right\} be a POVM, and Y Y the resulting output (classical) alphabet. The Shannon mutual information I ( X , Y ) I(X;Y) cannot exceed the Holevo information χ ( p i , ρ ^ i ) \chi(p_{i},{\hat{\rho}}_{i})
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2006
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M. de Gosson, Symplectic Geometry and Quantum Mechanics
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