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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Quantum capacity ： ウィキペディア英語版 Quantum capacity In the theory of quantum communication, the quantum capacity is the highest rate at which quantum information can be communicated over many independent uses of a noisy quantum channel from a sender to a receiver. It is also equal to the highest rate at which entanglement can be generated over the channel, and forward classical communication cannot improve it. The quantum capacity theorem is important for the theory of quantum error correction, and more broadly for the theory of quantum computation. The theorem giving a lower bound on the quantum capacity of any channel is colloquially known as the LSD theorem, after the authors Lloyd, Shor, and Devetak who proved it with increasing standards of rigor. == Hashing Bound for Pauli Channels == The LSD theorem states that the coherent information of a quantum channel is an achievable rate for reliable quantum communication. For a Pauli channel, the coherent information has a simple form and the proof that it is achievable is particularly simple as well. We prove the theorem for this special case by exploiting random stabilizer codes and correcting only the likely errors that the channel produces. Theorem (Hashing Bound). There exists a stabilizer quantum error-correcting code that achieves the hashing limit $R=1-H\backslash left(\backslash mathbf\backslash right)$ for a Pauli channel of the following form: :$$ \rho \mapsto p_\rho+p_X\rho X+p_Y\rho Y+p_Z\rho Z, where $\backslash mathbf=\backslash left(p\_,p\_,p\_,p\_\backslash right)$ and $H\backslash left(\backslash mathbf\backslash right)$ is the entropy of this probability vector. Proof. We consider correcting only the typical errors. That is, consider defining the typical set of errors as follows: :$$ T_^}\equiv\left\ \log_\left( \Pr\left\ \right) -H\left( \mathbf\right) \right\vert \leq\delta\right\} , where $a^$ is some sequence consisting of the letters $\backslash left\backslash$ and $\backslash Pr\backslash left\backslash$ is the probability that an IID Pauli channel issues some tensor-product error $E\_\}\backslash otimes\backslash cdots\backslash otimes\; E\_\backslash in\; T\_^\}\}\backslash Pr\backslash left\backslash \; \backslash geq\; 1-\backslash epsilon,$ for all $\backslash epsilon>0$ and sufficiently large $n$. The error-correcting conditions〔.〕 for a stabilizer code $\backslash mathcal$ in this case are that $\backslash \backslash in\; T\_^\}\backslash \}$ is a correctable set of errors if :$$ E_E_\right) \backslash \mathcal, for all error pairs $E\_\}$ such that $a^,b^\backslash in$ T_^} where $N(\backslash mathcal)$ is the normalizer of $\backslash mathcal$. Also, we consider the expectation of the error probability under a random choice of a stabilizer code. We proceed as follows: :$$ \begin \mathbb_\right\} &= \mathbb_} \Pr \left\ \mathcal\left(E_\mathcal\right) \right\} \\ &\leq \mathbb_ \in T_^}} \Pr\left\ \mathcal\left(E_\mathcal\right) \right\} + \epsilon \\ &= \sum_^}} \Pr\left\ \mathbb_\left(E_\mathcal\right) \right\} + \epsilon \\ &= \sum_^}} \Pr\left\ \Pr_}\text\mathcal\right\} + \epsilon. \end The first equality follows by definition—$\backslash mathcal$ is an indicator function equal to one if $E\_$ and equal to zero otherwise. The first inequality follows, since we correct only the typical errors because the atypical error set has negligible probability mass. The second equality follows by exchanging the expectation and the sum. The third equality follows because the expectation of an indicator function is the probability that the event it selects occurs. Continuing, we have :$$ =\sum_^}}\Pr\left\ \Pr_}:b^\in T_^},\ b^\neq a^,\ E_E_ \right) \backslash\mathcal\right\} :$\backslash leq\backslash sum\_^\backslash Pr\backslash left\backslash$ \Pr_}:b^\in T_^},\ b^\neq a^,\ E_E_ \right) \right\} :$=\backslash sum\_^\}\}\backslash Pr\backslash left\backslash$ \Pr_\in T_^ },\ b^\neq a^}E_E_\right) \right\} :$\backslash leq\backslash sum\_\backslash in\; T\_^\},\backslash \; b^\backslash neq\; a^\}$ \Pr\left\ \Pr_}^E_\right) \right\} :$\backslash leq\backslash sum\_\backslash in\; T\_^\},\backslash \; b^\backslash neq\; a^\}$ \Pr\left\ 2^ :$\backslash leq2^2^2^$ :$=2^.$ The first equality follows from the error-correcting conditions for a quantum stabilizer code, where $N\backslash left(\; \backslash mathcal\backslash right)$ is the normalizer of $\backslash mathcal$. The first inequality follows by ignoring any potential degeneracy in the code—we consider an error uncorrectable if it lies in the normalizer $N\backslash left(\; \backslash mathcal\backslash right)$ and the probability can only be larger because $N\backslash left(\; \backslash mathcal\backslash right)\; \backslash backslash\backslash mathcal\backslash in\; N\backslash left($ \mathcal\right) . The second equality follows by realizing that the probabilities for the existence criterion and the union of events are equivalent. The second inequality follows by applying the union bound. The third inequality follows from the fact that the probability for a fixed operator $E\_E\_\}\backslash left\backslash E\_$ \right) \right\} =\frac\leq2^. The reasoning here is that the random choice of a stabilizer code is equivalent to fixing operators $Z\_$, ..., $Z\_$ and performing a uniformly random Clifford unitary. The probability that a fixed operator commutes with $\backslash overline\_$, ..., $\backslash overline\_$ is then just the number of non-identity operators in the normalizer ($2^-1$) divided by the total number of non-identity operators ($2^-1$). After applying the above bound, we then exploit the following typicality bounds: :$$ \forall a^ \in T_^}:\Pr\left\\right\} \leq2^, :$\backslash left\backslash vert\; T\_^\}\backslash right\backslash vert\; \backslash leq2^.$ We conclude that as long as the rate $k/n=1-H\backslash left(\; \backslash mathbf\backslash right)$ -4\delta, the expectation of the error probability becomes arbitrarily small, so that there exists at least one choice of a stabilizer code with the same bound on the error probability. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Quantum capacity」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.