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| 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 [[quantum entanglement|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 [[Seth Lloyd|Lloyd]],<ref name="Lloyd1997">{{cite journal | journal = Physical Review A | volume = 55 | issue = 3 | pages = 1613–1622 | year = 1997 | author = Seth Lloyd |title = Capacity of the noisy quantum channel | doi = 10.1103/PhysRevA.55.1613 | arxiv = quant-ph/9604015}}</ref> [[Peter Shor|Shor]],<ref name="Shor2002">{{cite journal | journal = Lecture Notes, MSRI Workshop on Quantum Computation | year = 2002 | author = Peter Shor | title = The quantum channel capacity and coherent information | url = http://www.msri.org/publications/ln/msri/2002/quantumcrypto/shor/1/meta/aux/shor.pdf | format = pdf}}</ref> and Devetak<ref name="Devetak2005">{{cite journal | journal = IEEE Transactions on Information Theory | volume = 51 | pages = 44–55 | year = 2005 | author = Igor Devetak | title = The private classical capacity and quantum capacity of a quantum channel | doi = 10.1109/TIT.2004.839515 | arxiv = quant-ph/0304127}}</ref> who proved it with increasing standards of rigor.
| | e - Shop Word - Press is a excellent cart for your on the web shopping organization. Good luck on continue learning how to make a wordpress website. Step-4 Testing: It is the foremost important of your Plugin development process. Out of the various designs of photography identified these days, sports photography is preferred most, probably for the enjoyment and enjoyment associated with it. It is found that most of the visitors only look for the results that are displayed on the first page of the search engines and so if you get the service from professional service providers then they strive for the first page ranking of your site and improve the online visibility. <br><br>In case you cherished this informative article along with you want to acquire details regarding [http://miniURL.fouiner.info/wordpress_dropbox_backup_15903 wordpress backup] generously stop by the website. Always remember that an effective linkwheel strategy strives to answer all the demands of popular search engines while reacting to the latest marketing number trends. Wordpress have every reason with it which promote wordpress development. It sorts the results of a search according to category, tags and comments. So, if you are looking for some option to build a giant e-commerce website, then e-shopping preferable CMS tools will be helpful for you. Word - Press makes it possible to successfully and manage your website. <br><br>Digital photography is a innovative effort, if you removethe stress to catch every position and viewpoint of a place, you free yourself up to be more innovative and your outcomes will be much better. When a business benefits from its own domain name and a tailor-made blog, the odds of ranking higher in the search engines and being visible to a greater number of people is more likely. Are you considering getting your website redesigned. Our skilled expertise, skillfulness and excellence have been well known all across the world. Websites using this content based strategy are always given top scores by Google. <br><br>You can add keywords but it is best to leave this alone. Russell HR Consulting provides expert knowledge in the practical application of employment law as well as providing employment law training and HR support services. Enterprise, when they plan to hire Word - Press developer resources still PHP, My - SQL and watch with great expertise in codebase. It's now become a great place to sell it thanks to Woo - Commerce. Fortunately, Word - Press Customization Service is available these days, right from custom theme design, to plugin customization and modifying your website, you can take any bespoke service for your Word - Press development project. <br><br>Many developers design websites and give them to the clients, but still the client faces problems to handle the website. If you operate a website that's been built on HTML then you might have to witness traffic losses because such a site isn't competent enough in grabbing the attention of potential consumers. Useful Plugins Uber - Menu Top Megamenu Now it is the time of sticky Top navbar. You should stay away from plugins that are full of flaws and bugs. Article Source: Hostgator discount coupons for your Wordpress site here. |
| | |
| == Hashing Bound for Pauli Channels ==
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| | |
| 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 code]]s and correcting only the likely errors that the channel produces.
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| '''Theorem''' (Hashing Bound). There exists a stabilizer [[quantum error-correcting code]] that
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| achieves the hashing limit <math>R=1-H\left(\mathbf{p}\right)</math> for a Pauli
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| channel of the following form:
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| :<math>
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| \rho \mapsto p_{I}\rho+p_{X}X\rho X+p_{Y}Y\rho Y+p_{Z}Z\rho Z,
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| </math>
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| where <math>\mathbf{p}=\left(p_{I},p_{X},p_{Y},p_{Z}\right)</math> and <math>H\left(\mathbf{p}\right)</math> is the entropy of this probability vector.
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| '''Proof'''. We consider correcting only the typical errors. That is, consider defining the
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| [[typical set]] of errors as follows:
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| :<math>
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| T_{\delta}^{\mathbf{p}^{n}}\equiv\left\{ a^{n}:\left\vert -\frac{1}{n}
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| \log_{2}\left( \Pr\left\{ E_{a^{n}}\right\} \right) -H\left(
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| \mathbf{p}\right) \right\vert \leq\delta\right\} ,
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| </math>
| |
| where <math>a^{n}</math> is some sequence consisting of the letters <math>\left\{I,X,Y,Z\right\}</math> and <math>\Pr\left\{E_{a^{n}}\right\}</math> is the probability
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| that an IID Pauli channel issues some tensor-product error <math>E_{a^{n}}\equiv
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| E_{a_{1}}\otimes\cdots\otimes E_{a_{n}}</math>. This typical set consists of the
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| likely errors in the sense that
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| :<math>
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| \sum_{a^{n}\in T_{\delta}^{\mathbf{p}^{n}}}\Pr\left\{E_{a^{n}}\right\} \geq 1-\epsilon,
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| </math>
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| for all <math>\epsilon>0</math> and sufficiently large <math>n</math>. The error-correcting | |
| conditions<ref name="NC">{{citation | last1 = Nielsen | first1 = Michael A. | author1-link = Michael Nielsen | last2 = Chuang | first2 = Isaac L. | author2-link = Isaac Chuang | title = Quantum Computation and Quantum Information | publisher = [[Cambridge University Press]] | date = 2000 | isbn = 9780521635035}}.</ref> for a stabilizer code <math>\mathcal{S}</math> in this case are that <math>\{E_{a^{n}}:a^{n}\in T_{\delta}^{\mathbf{p}^{n}}\}</math> is a correctable set of errors if
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| :<math>
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| E_{a^{n}}^{\dagger}E_{b^{n}}\notin N\left(\mathcal{S}\right) \backslash \mathcal{S},
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| </math>
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| for all error pairs <math>E_{a^{n}}</math> and <math>E_{b^{n}}</math> such that <math>a^{n},b^{n}\in
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| T_{\delta}^{\mathbf{p}^{n}}</math> where <math>N(\mathcal{S})</math> is the [[normalizer]] of <math>\mathcal{S}</math>. Also, we consider the expectation of the error
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| probability under a random choice of a stabilizer code.
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| We proceed as follows:
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| :<math>
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| \begin{align}
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| \mathbb{E}_{\mathcal{S}}\left\{p_{e}\right\}
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| &= \mathbb{E}_{\mathcal{S}}
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| \left\{
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| \sum_{a^{n}} \Pr \left\{ E_{a^{n}}\right\}
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| \mathcal{I}\left(E_{a^{n}}\text{ is uncorrectable under }\mathcal{S}\right)
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| \right\} \\
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| &\leq \mathbb{E}_{\mathcal{S}}
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| \left\{
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| \sum_{a^{n} \in T_{\delta}^{\mathbf{p}^{n}}}
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| \Pr\left\{E_{a^{n}}\right\}
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| \mathcal{I}\left(E_{a^{n}}\text{ is uncorrectable under }\mathcal{S}\right)
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| \right\} + \epsilon \\
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| &= \sum_{a^{n} \in T_{\delta}^{\mathbf{p}^{n}}}
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| \Pr\left\{E_{a^{n}}\right\} \mathbb{E}_{\mathcal{S}}
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| \left\{
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| \mathcal{I}\left(E_{a^{n}}\text{ is uncorrectable under }\mathcal{S}\right)
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| \right\} + \epsilon \\
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| &= \sum_{a^{n} \in T_{\delta}^{\mathbf{p}^{n}}}
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| \Pr\left\{E_{a^{n}}\right\}
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| \Pr_{\mathcal{S}} \left\{E_{a^{n}}\text{ is uncorrectable under }\mathcal{S}\right\} + \epsilon.
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| \end{align}
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| </math> | |
| The first equality follows by definition—<math>\mathcal{I}</math> is an indicator function equal to one if <math>E_{a^{n}}</math> is uncorrectable under <math>\mathcal{S}</math> 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
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| :<math>
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| =\sum_{a^{n}\in T_{\delta}^{\mathbf{p}^{n}}}\Pr\left\{ E_{a^{n}}\right\}
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| \Pr_{\mathcal{S}}\left\{ \exists E_{b^{n}}:b^{n}\in T_{\delta}^{\mathbf{p}
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| ^{n}},\ b^{n}\neq a^{n},\ E_{a^{n}}^{\dagger}E_{b^{n}}\in N\left( \mathcal{S}
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| \right) \backslash\mathcal{S}\right\} </math>
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| :<math> \leq\sum_{a^{n}\in T_{\delta}^{A^{n}}}\Pr\left\{ E_{a^{n}}\right\}
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| \Pr_{\mathcal{S}}\left\{ \exists E_{b^{n}}:b^{n}\in T_{\delta}^{\mathbf{p}
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| ^{n}},\ b^{n}\neq a^{n},\ E_{a^{n}}^{\dagger}E_{b^{n}}\in N\left( \mathcal{S}
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| \right) \right\} </math>
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| :<math> =\sum_{a^{n}\in T_{\delta}^{\mathbf{p}^{n}}}\Pr\left\{ E_{a^{n}}\right\}
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| \Pr_{\mathcal{S}}\left\{ \bigcup\limits_{b^{n}\in T_{\delta}^{\mathbf{p}^{n}
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| },\ b^{n}\neq a^{n}}E_{a^{n}}^{\dagger}E_{b^{n}}\in N\left( \mathcal{S}\right)
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| \right\} </math>
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| :<math> \leq\sum_{a^{n},b^{n}\in T_{\delta}^{\mathbf{p}^{n}},\ b^{n}\neq a^{n}}
| |
| \Pr\left\{ E_{a^{n}}\right\} \Pr_{\mathcal{S}}\left\{ E_{a^{n}}^{\dagger
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| }E_{b^{n}}\in N\left( \mathcal{S}\right) \right\} </math>
| |
| :<math> \leq\sum_{a^{n},b^{n}\in T_{\delta}^{\mathbf{p}^{n}},\ b^{n}\neq a^{n}}
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| \Pr\left\{ E_{a^{n}}\right\} 2^{-\left( n-k\right) }</math>
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| :<math> \leq2^{2n\left[ H\left( \mathbf{p}\right) +\delta\right] }2^{-n\left[
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| H\left( \mathbf{p}\right) +\delta\right] }2^{-\left( n-k\right) }</math>
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| :<math> =2^{-n\left[ 1-H\left( \mathbf{p}\right) -k/n-3\delta\right] }.
| |
| </math>
| |
| The first equality follows from the error-correcting conditions for a quantum
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| stabilizer code, where <math>N\left( \mathcal{S}\right) </math> is the normalizer of
| |
| <math>\mathcal{S}</math>. The first inequality follows by ignoring any potential
| |
| degeneracy in the code—we consider an error uncorrectable if it lies in the
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| normalizer <math>N\left( \mathcal{S}\right) </math> and the probability can only be
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| larger because <math>N\left( \mathcal{S}\right) \backslash\mathcal{S}\in N\left(
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| \mathcal{S}\right) </math>. The second equality follows by realizing that the
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| probabilities for the existence criterion and the union of events are
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| equivalent. The second inequality follows by applying the union bound. The
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| third inequality follows from the fact that the probability for a fixed
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| operator <math>E_{a^{n}}^{\dagger}E_{b^{n}}</math> not equal to the identity commuting with
| |
| the stabilizer operators of a random stabilizer can be upper bounded as | |
| follows:
| |
| <center><math>
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| \Pr_{\mathcal{S}}\left\{ E_{a^{n}}^{\dagger}E_{b^{n}}\in N\left( \mathcal{S}
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| \right) \right\} =\frac{2^{n+k}-1}{2^{2n}-1}\leq2^{-\left( n-k\right) }.
| |
| </math></center>
| |
| The reasoning here is that the random choice of a stabilizer code is equivalent to
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| fixing operators <math>Z_{1}</math>, ..., <math>Z_{n-k}</math> and performing a uniformly random
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| Clifford unitary. The probability that a fixed operator commutes with
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| <math>\overline{Z}_{1}</math>, ..., <math>\overline{Z}_{n-k}</math> is then just the number of
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| non-identity operators in the normalizer (<math>2^{n+k}-1</math>) divided by the total
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| number of non-identity operators (<math>2^{2n}-1</math>). After applying the above bound,
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| we then exploit the following typicality bounds:
| |
| :<math>
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| \forall a^{n} \in T_{\delta}^{\mathbf{p}^{n}}:\Pr\left\{ E_{a^{n}
| |
| }\right\} \leq2^{-n\left[ H\left( \mathbf{p}\right) +\delta\right] },</math>
| |
| :<math>\left\vert T_{\delta}^{\mathbf{p}^{n}}\right\vert \leq2^{n\left[ H\left(
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| \mathbf{p}\right) +\delta\right] }.
| |
| </math>
| |
| We conclude that as long as the rate <math>k/n=1-H\left( \mathbf{p}\right)
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| -4\delta</math>, 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.
| |
| | |
| == References ==
| |
| | |
| <references/>
| |
| | |
| == Further reading ==
| |
| | |
| * Mark M. Wilde, [http://arxiv.org/abs/1106.1445 "From Classical to Quantum Shannon Theory", arXiv:1106.1445].
| |
| | |
| | |
| {{quantum computing}}
| |
| | |
| [[Category:Quantum information science]]
| |
| [[Category:Quantum information theory]]
| |
| [[Category:Models of computation]]
| |
| [[Category:Quantum cryptography]]
| |
| [[Category:Theoretical computer science]]
| |
| [[Category:Classes of computers]]
| |
| [[Category:Information theory]]
| |
| [[Category:Computational complexity theory]]
| |
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