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| In [[quantum information theory]], the '''classical capacity''' of a [[quantum channel]] is the maximum rate at which classical data can be sent over it error-free in the limit of many uses of the channel. [[Alexander Holevo|Holevo]], Schumacher, and Westmoreland proved the following lower bound on the classical capacity of any quantum channel <math>\mathcal{N}</math>:
| | My name: Jay Loewenthal<br>Age: 20<br>Country: Germany<br>Town: Pfaffing <br>Post code: 83537<br>Address: Gubener Str. 83<br><br>Feel free to visit my blog post - [http://www.cheapdomains.com/ Cheap domains] |
| | |
| :<math>
| |
| \chi(\mathcal{N}) = \max_{\rho^{XA}} I(X;B)_{\mathcal{N}(\rho)}
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| </math>
| |
| | |
| where <math>\rho^{XA}</math> is a classical-quantum state of the following form:
| |
| :<math>
| |
| \rho^{XA} = \sum_x p_X(x) \vert x \rangle \langle x \vert^X \otimes \rho_x^A ,
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| </math>
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| <math>p_X(x)</math> is a probability distribution, and each <math>\rho_x^A</math> is a density operator that can be input to the channel <math>\mathcal{N}</math>.
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| | |
| == Achievability using sequential decoding ==
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| | |
| We briefly review the HSW coding theorem (the
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| statement of the achievability of the [[Holevo information]] rate <math>I(X;B)</math> for
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| communicating classical data over a quantum channel). We first review the
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| minimal amount of quantum mechanics needed for the theorem. We then cover
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| quantum typicality, and finally we prove the theorem using a recent sequential
| |
| decoding technique.
| |
| | |
| ==Review of Quantum Mechanics==
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| | |
| In order to prove the HSW coding theorem, we really just need a few basic
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| things from [[quantum mechanics]]. First, a [[quantum state]] is a unit trace,
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| positive operator known as a [[density operator]]. Usually, we denote it
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| by <math>\rho</math>, <math>\sigma</math>, <math>\omega</math>, etc. The simplest model for a [[quantum channel]]
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| is known as a classical-quantum channel:
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| <center><math>
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| x\rightarrow\rho_{x}.
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| </math></center>
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| The meaning of the above notation is that inputting the classical letter <math>x</math>
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| at the transmitting end leads to a quantum state <math>\rho_{x}</math> at the receiving
| |
| end. It is the task of the receiver to perform a measurement to determine the
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| input of the sender. If it is true that the states <math>\rho_{x}</math> are perfectly
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| distinguishable from one another (i.e., if they have orthogonal supports such
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| that Tr<math>\left\{ \rho_{x}\rho_{x^{\prime}}\right\} =0</math> for <math>x\neq x^{\prime}
| |
| </math>), then the channel is a noiseless channel. We are interested in situations
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| for which this is not the case. If it is true that the states <math>\rho_{x}</math> all
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| commute with one another, then this is effectively identical to the situation
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| for a classical channel, so we are also not interested in these situations.
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| So, the situation in which we are interested is that in which the states
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| <math>\rho_{x}</math> have overlapping support and are non-commutative.
| |
| | |
| The most general way to describe a [[quantum measurement]] is with a [[positive
| |
| operator-valued measure]] ([[POVM]]). We usually denote the elements of a POVM as
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| <math>\left\{ \Lambda_{m}\right\} _{m}</math>. These operators should satisfy
| |
| positivity and completeness in order to form a valid POVM:
| |
| :<math>
| |
| \Lambda_{m} \geq0\ \ \ \ \forall m</math>
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| :<math>\sum_{m}\Lambda_{m} =I.
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| </math>
| |
| The probabilistic interpretation of [[quantum mechanics]] states that if someone
| |
| measures a quantum state <math>\rho</math> using a measurement device corresponding to
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| the POVM <math>\left\{ \Lambda_{m}\right\} </math>, then the probability <math>p\left(
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| m\right) </math> for obtaining outcome <math>m</math> is equal to
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| :<math>
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| p\left( m\right) =\text{Tr}\left\{ \Lambda_{m}\rho\right\} ,
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| </math>
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| and the post-measurement state is
| |
| :<math>
| |
| \rho_{m}^{\prime}=\frac{1}{p\left( m\right) }\sqrt{\Lambda_{m}}\rho
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| \sqrt{\Lambda_{m}},
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| </math>
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| if the person measuring obtains outcome <math>m</math>. These rules are sufficient for us
| |
| to consider classical communication schemes over cq channels.
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| | |
| ==Quantum Typicality==
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| | |
| The reader can find a good review of this topic in the article about the [[typical subspace]].
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| | |
| ==Gentle Operator Lemma==
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| | |
| The following lemma is important for our proofs. It
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| demonstrates that a measurement that succeeds with high probability on average
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| does not disturb the state too much on average:
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| | |
| Lemma: [Winter] Given an
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| ensemble <math>\left\{ p_{X}\left( x\right) ,\rho_{x}\right\} </math> with expected
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| density operator <math>\rho\equiv\sum_{x}p_{X}\left( x\right) \rho_{x}</math>, suppose
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| that an operator <math>\Lambda</math> such that <math>I\geq\Lambda\geq0</math> succeeds with high
| |
| probability on the state <math>\rho</math>:
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| <center><math>
| |
| \text{Tr}\left\{ \Lambda\rho\right\} \geq1-\epsilon.
| |
| </math></center>
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| Then the subnormalized state <math>\sqrt{\Lambda}\rho_{x}\sqrt{\Lambda}</math> is close
| |
| in expected trace distance to the original state <math>\rho_{x}</math>:
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| <center><math>
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| \mathbb{E}_{X}\left\{ \left\Vert \sqrt{\Lambda}\rho_{X}\sqrt{\Lambda}
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| -\rho_{X}\right\Vert _{1}\right\} \leq2\sqrt{\epsilon}.
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| </math></center>
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| (Note that <math>\left\Vert A\right\Vert _{1}</math> is the nuclear norm of the operator
| |
| <math>A</math> so that <math>\left\Vert A\right\Vert _{1}\equiv</math>Tr<math>\left\{ \sqrt{A^{\dagger}
| |
| A}\right\} </math>.)
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| | |
| The following inequality is useful for us as well. It holds for any operators
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| <math>\rho</math>, <math>\sigma</math>, <math>\Lambda</math> such that <math>0\leq\rho,\sigma,\Lambda\leq I</math>:
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| <center><math>
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| \text{Tr}\left\{ \Lambda\rho\right\} \leq\text{Tr}\left\{ \Lambda
| |
| \sigma\right\} +\left\Vert \rho-\sigma\right\Vert _{1}.
| |
| </math></center>
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| The quantum information-theoretic interpretation of the above inequality is
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| that the probability of obtaining outcome <math>\Lambda</math> from a quantum measurement
| |
| acting on the state <math>\rho</math> is upper bounded by the probability of obtaining
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| outcome <math>\Lambda</math> on the state <math>\sigma</math> summed with the distinguishability of
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| the two states <math>\rho</math> and <math>\sigma</math>.
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| | |
| ==Non-Commutative Union Bound==
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| | |
| Lemma: [Sen's bound] The following bound
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| holds for a subnormalized state <math>\sigma</math> such that <math>0\leq\sigma</math> and
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| <math>Tr\left\{ \sigma\right\} \leq1</math> with <math>\Pi_{1}</math>, ... , <math>\Pi_{N}</math> being
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| projectors:
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| <math>
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| \text{Tr}\left\{ \sigma\right\} -\text{Tr}\left\{ \Pi_{N}\cdots\Pi
| |
| _{1}\ \sigma\ \Pi_{1}\cdots\Pi_{N}\right\} \leq2\sqrt{\sum_{i=1}^{N}
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| \text{Tr}\left\{ \left( I-\Pi_{i}\right) \sigma\right\} },
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| </math>
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| | |
| We can think of Sen's bound as a "non-commutative union
| |
| bound" because it is analogous to the following union bound
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| from probability theory:
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| <center><math>
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| \Pr\left\{ \left( A_{1}\cap\cdots\cap A_{N}\right) ^{c}\right\}
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| =\Pr\left\{ A_{1}^{c}\cup\cdots\cup A_{N}^{c}\right\} \leq\sum_{i=1}^{N}
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| \Pr\left\{ A_{i}^{c}\right\} ,
| |
| </math></center>
| |
| where <math>A_{1}</math>, \ldots, <math>A_{N}</math> are events. The analogous bound for projector
| |
| logic would be
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| :<math>
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| \text{Tr}\left\{ \left( I-\Pi_{1}\cdots\Pi_{N}\cdots\Pi_{1}\right)
| |
| \rho\right\} \leq\sum_{i=1}^{N}\text{Tr}\left\{ \left( I-\Pi_{i}\right)
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| \rho\right\} ,
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| </math>
| |
| if we think of <math>\Pi_{1}\cdots\Pi_{N}</math> as a projector onto the intersection of
| |
| subspaces. Though, the above bound only holds if the projectors <math>\Pi_{1}</math>,
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| ..., <math>\Pi_{N}</math> are commuting (choosing <math>\Pi_{1}=\left\vert +\right\rangle
| |
| \left\langle +\right\vert </math>, <math>\Pi_{2}=\left\vert 0\right\rangle \left\langle
| |
| 0\right\vert </math>, and <math>\rho=\left\vert 0\right\rangle \left\langle 0\right\vert
| |
| </math> gives a counterexample). If the projectors are non-commuting, then Sen's
| |
| bound is the next best thing and suffices for our purposes here.
| |
| | |
| ==HSW Theorem with the non-commutative union bound==
| |
| | |
| We now prove the HSW theorem with Sen's non-commutative union bound. We
| |
| divide up the proof into a few parts: codebook generation, POVM construction,
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| and error analysis.
| |
| | |
| '''Codebook Generation.''' We first describe how Alice and Bob agree on a
| |
| random choice of code. They have the channel <math>x\rightarrow\rho_{x}</math> and a
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| distribution <math>p_{X}\left( x\right) </math>. They choose <math>M</math> classical sequences
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| <math>x^{n}</math> according to the IID\ distribution <math>p_{X^{n}}\left( x^{n}\right) </math>.
| |
| After selecting them, they label them with indices as <math>\left\{ x^{n}\left(
| |
| m\right) \right\} _{m\in\left[ M\right] }</math>. This leads to the following
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| quantum codewords:
| |
| <center><math>
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| \rho_{x^{n}\left( m\right) }=\rho_{x_{1}\left( m\right) }\otimes
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| \cdots\otimes\rho_{x_{n}\left( m\right) }.
| |
| </math></center>
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| The quantum codebook is then <math>\left\{ \rho_{x^{n}\left( m\right) }\right\}
| |
| </math>. The average state of the codebook is then
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| <center><math>
| |
| \mathbb{E}_{X^{n}}\left\{ \rho_{X^{n}}\right\} =\sum_{x^{n}}p_{X^{n}}\left(
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| x^{n}\right) \rho_{x^{n}}=\rho^{\otimes n},
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| </math></center>
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| where <math>\rho=\sum_{x}p_{X}\left( x\right) \rho_{x}</math>.
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| | |
| '''POVM Construction''' . Sens' bound from the above lemma
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| suggests a method for Bob to decode a state that Alice transmits. Bob should
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| first ask "Is the received state in the average typical
| |
| subspace?" He can do this operationally by performing a
| |
| typical subspace measurement corresponding to <math>\left\{ \Pi_{\rho,\delta}
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| ^{n},I-\Pi_{\rho,\delta}^{n}\right\} </math>. Next, he asks in sequential order,
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| "Is the received codeword in the <math>m^{\text{th}}</math>
| |
| conditionally typical subspace?" This is in some sense
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| equivalent to the question, "Is the received codeword the
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| <math>m^{\text{th}}</math> transmitted codeword?" He can ask these
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| questions operationally by performing the measurements corresponding to the
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| conditionally typical projectors <math>\left\{ \Pi_{\rho_{x^{n}\left( m\right)
| |
| },\delta},I-\Pi_{\rho_{x^{n}\left( m\right) },\delta}\right\} </math>.
| |
| | |
| Why should this sequential decoding scheme work well? The reason is that the
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| transmitted codeword lies in the typical subspace on average:
| |
| :<math> | |
| \mathbb{E}_{X^{n}}\left\{ \text{Tr}\left\{ \Pi_{\rho,\delta}\ \rho_{X^{n}
| |
| }\right\} \right\} =\text{Tr}\left\{ \Pi_{\rho,\delta}\ \mathbb{E}
| |
| _{X^{n}}\left\{ \rho_{X^{n}}\right\} \right\} </math>
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| :<math> =\text{Tr}\left\{ \Pi_{\rho,\delta}\ \rho^{\otimes n}\right\} </math> | |
| :<math> \geq1-\epsilon,</math> | |
| where the inequality follows from (\ref{eq:1st-typ-prop}). Also, the
| |
| projectors <math>\Pi_{\rho_{x^{n}\left( m\right) },\delta}</math>
| |
| are "good detectors" for the states <math>\rho_{x^{n}\left( m\right)
| |
| }</math> (on average) because the following condition holds from conditional quantum
| |
| typicality:
| |
| <center><math>
| |
| \mathbb{E}_{X^{n}}\left\{ \text{Tr}\left\{ \Pi_{\rho_{X^{n}},\delta}
| |
| \ \rho_{X^{n}}\right\} \right\} \geq1-\epsilon.
| |
| </math></center>
| |
| | |
| '''Error Analysis''' . The probability of detecting the <math>m^{\text{th}}</math>
| |
| codeword correctly under our sequential decoding scheme is equal to
| |
| <center><math>
| |
| \text{Tr}\left\{ \Pi_{\rho_{X^{n}\left( m\right) },\delta}\hat{\Pi}
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| _{\rho_{X^{n}\left( m-1\right) },\delta}\cdots\hat{\Pi}_{\rho_{X^{n}\left(
| |
| 1\right) },\delta}\ \Pi_{\rho,\delta}^{n}\ \rho_{x^{n}\left( m\right)
| |
| }\ \Pi_{\rho,\delta}^{n}\ \hat{\Pi}_{\rho_{X^{n}\left( 1\right) },\delta
| |
| }\cdots\hat{\Pi}_{\rho_{X^{n}\left( m-1\right) },\delta}\Pi_{\rho
| |
| _{X^{n}\left( m\right) },\delta}\right\} ,
| |
| </math></center>
| |
| where we make the abbreviation <math>\hat{\Pi}\equiv I-\Pi</math>. (Observe that we
| |
| project into the average typical subspace just once.) Thus, the probability of
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| an incorrect detection for the <math>m^{\text{th}}</math> codeword is given by
| |
| <center><math>
| |
| 1-\text{Tr}\left\{ \Pi_{\rho_{X^{n}\left( m\right) },\delta}\hat{\Pi}
| |
| _{\rho_{X^{n}\left( m-1\right) },\delta}\cdots\hat{\Pi}_{\rho_{X^{n}\left(
| |
| 1\right) },\delta}\ \Pi_{\rho,\delta}^{n}\ \rho_{x^{n}\left( m\right)
| |
| }\ \Pi_{\rho,\delta}^{n}\ \hat{\Pi}_{\rho_{X^{n}\left( 1\right) },\delta
| |
| }\cdots\hat{\Pi}_{\rho_{X^{n}\left( m-1\right) },\delta}\Pi_{\rho
| |
| _{X^{n}\left( m\right) },\delta}\right\} ,
| |
| </math></center>
| |
| and the average error probability of this scheme is equal to
| |
| <center><math>
| |
| 1-\frac{1}{M}\sum_{m}\text{Tr}\left\{ \Pi_{\rho_{X^{n}\left( m\right)
| |
| },\delta}\hat{\Pi}_{\rho_{X^{n}\left( m-1\right) },\delta}\cdots\hat{\Pi
| |
| }_{\rho_{X^{n}\left( 1\right) },\delta}\ \Pi_{\rho,\delta}^{n}\ \rho
| |
| _{x^{n}\left( m\right) }\ \Pi_{\rho,\delta}^{n}\ \hat{\Pi}_{\rho
| |
| _{X^{n}\left( 1\right) },\delta}\cdots\hat{\Pi}_{\rho_{X^{n}\left(
| |
| m-1\right) },\delta}\Pi_{\rho_{X^{n}\left( m\right) },\delta}\right\} .
| |
| </math></center>
| |
| Instead of analyzing the average error probability, we analyze the expectation
| |
| of the average error probability, where the expectation is with respect to the
| |
| random choice of code:
| |
| <center><math>
| |
| 1-\mathbb{E}_{X^{n}}\left\{ \frac{1}{M}\sum_{m}\text{Tr}\left\{ \Pi
| |
| _{\rho_{X^{n}\left( m\right) },\delta}\hat{\Pi}_{\rho_{X^{n}\left(
| |
| m-1\right) },\delta}\cdots\hat{\Pi}_{\rho_{X^{n}\left( 1\right) },\delta
| |
| }\ \Pi_{\rho,\delta}^{n}\ \rho_{X^{n}\left( m\right) }\ \Pi_{\rho,\delta
| |
| }^{n}\ \hat{\Pi}_{\rho_{X^{n}\left( 1\right) },\delta}\cdots\hat{\Pi}
| |
| _{\rho_{X^{n}\left( m-1\right) },\delta}\Pi_{\rho_{X^{n}\left( m\right)
| |
| },\delta}\right\} \right\} .
| |
| </math></center>
| |
| | |
| Our first step is to apply Sen's bound to the above quantity. But before doing
| |
| so, we should rewrite the above expression just slightly, by observing that
| |
| :<math>
| |
| 1 =\mathbb{E}_{X^{n}}\left\{ \frac{1}{M}\sum_{m}\text{Tr}\left\{
| |
| \rho_{X^{n}\left( m\right) }\right\} \right\} </math>
| |
| :<math> =\mathbb{E}_{X^{n}}\left\{ \frac{1}{M}\sum_{m}\text{Tr}\left\{ \Pi
| |
| _{\rho,\delta}^{n}\rho_{X^{n}\left( m\right) }\right\} +\text{Tr}\left\{
| |
| \hat{\Pi}_{\rho,\delta}^{n}\rho_{X^{n}\left( m\right) }\right\} \right\}
| |
| </math> | |
| :<math> =\mathbb{E}_{X^{n}}\left\{ \frac{1}{M}\sum_{m}\text{Tr}\left\{ \Pi | |
| _{\rho,\delta}^{n}\rho_{X^{n}\left( m\right) }\Pi_{\rho,\delta}^{n}\right\}
| |
| \right\} +\frac{1}{M}\sum_{m}\text{Tr}\left\{ \hat{\Pi}_{\rho,\delta}
| |
| ^{n}\mathbb{E}_{X^{n}}\left\{ \rho_{X^{n}\left( m\right) }\right\}
| |
| \right\} </math>
| |
| :<math> =\mathbb{E}_{X^{n}}\left\{ \frac{1}{M}\sum_{m}\text{Tr}\left\{ \Pi
| |
| _{\rho,\delta}^{n}\rho_{X^{n}\left( m\right) }\Pi_{\rho,\delta}^{n}\right\}
| |
| \right\} +\text{Tr}\left\{ \hat{\Pi}_{\rho,\delta}^{n}\rho^{\otimes
| |
| n}\right\} </math>
| |
| :<math> \leq\mathbb{E}_{X^{n}}\left\{ \frac{1}{M}\sum_{m}\text{Tr}\left\{
| |
| \Pi_{\rho,\delta}^{n}\rho_{X^{n}\left( m\right) }\Pi_{\rho,\delta}
| |
| ^{n}\right\} \right\} +\epsilon
| |
| </math>
| |
| Substituting into (\ref{eq:error-term}) (and forgetting about the small
| |
| <math>\epsilon</math> term for now) gives an upper bound of
| |
| :<math>
| |
| \mathbb{E}_{X^{n}}\left\{ \frac{1}{M}\sum_{m}\text{Tr}\left\{ \Pi
| |
| _{\rho,\delta}^{n}\rho_{X^{n}\left( m\right) }\Pi_{\rho,\delta}^{n}\right\}
| |
| \right\} </math>
| |
| :<math>
| |
| -\mathbb{E}_{X^{n}}\left\{ \frac{1}{M}\sum_{m}\text{Tr}\left\{ \Pi
| |
| _{\rho_{X^{n}\left( m\right) },\delta}\hat{\Pi}_{\rho_{X^{n}\left(
| |
| m-1\right) },\delta}\cdots\hat{\Pi}_{\rho_{X^{n}\left( 1\right) },\delta
| |
| }\ \Pi_{\rho,\delta}^{n}\ \rho_{X^{n}\left( m\right) }\ \Pi_{\rho,\delta
| |
| }^{n}\ \hat{\Pi}_{\rho_{X^{n}\left( 1\right) },\delta}\cdots\hat{\Pi}
| |
| _{\rho_{X^{n}\left( m-1\right) },\delta}\Pi_{\rho_{X^{n}\left( m\right)
| |
| },\delta}\right\} \right\} .
| |
| </math> | |
| We then apply Sen's bound to this expression with <math>\sigma=\Pi_{\rho,\delta
| |
| }^{n}\rho_{X^{n}\left( m\right) }\Pi_{\rho,\delta}^{n}</math> and the sequential
| |
| projectors as <math>\Pi_{\rho_{X^{n}\left( m\right) },\delta}</math>, <math>\hat{\Pi}
| |
| _{\rho_{X^{n}\left( m-1\right) },\delta}</math>, ..., <math>\hat{\Pi}_{\rho
| |
| _{X^{n}\left( 1\right) },\delta}</math>. This gives the upper bound
| |
| <math>
| |
| \mathbb{E}_{X^{n}}\left\{ \frac{1}{M}\sum_{m}2\left[ \text{Tr}\left\{
| |
| \left( I-\Pi_{\rho_{X^{n}\left( m\right) },\delta}\right) \Pi_{\rho
| |
| ,\delta}^{n}\rho_{X^{n}\left( m\right) }\Pi_{\rho,\delta}^{n}\right\}
| |
| +\sum_{i=1}^{m-1}\text{Tr}\left\{ \Pi_{\rho_{X^{n}\left( i\right) },\delta
| |
| }\Pi_{\rho,\delta}^{n}\rho_{X^{n}\left( m\right) }\Pi_{\rho,\delta}
| |
| ^{n}\right\} \right] ^{1/2}\right\} .
| |
| </math>
| |
| Due to concavity of the square root, we can bound this expression from above
| |
| by
| |
| :<math>
| |
| 2\left[ \mathbb{E}_{X^{n}}\left\{ \frac{1}{M}\sum_{m}\text{Tr}\left\{
| |
| \left( I-\Pi_{\rho_{X^{n}\left( m\right) },\delta}\right) \Pi_{\rho
| |
| ,\delta}^{n}\rho_{X^{n}\left( m\right) }\Pi_{\rho,\delta}^{n}\right\}
| |
| +\sum_{i=1}^{m-1}\text{Tr}\left\{ \Pi_{\rho_{X^{n}\left( i\right) },\delta
| |
| }\Pi_{\rho,\delta}^{n}\rho_{X^{n}\left( m\right) }\Pi_{\rho,\delta}
| |
| ^{n}\right\} \right\} \right] ^{1/2}</math>
| |
| :<math> \leq2\left[ \mathbb{E}_{X^{n}}\left\{ \frac{1}{M}\sum_{m}\text{Tr}\left\{
| |
| \left( I-\Pi_{\rho_{X^{n}\left( m\right) },\delta}\right) \Pi_{\rho
| |
| ,\delta}^{n}\rho_{X^{n}\left( m\right) }\Pi_{\rho,\delta}^{n}\right\}
| |
| +\sum_{i\neq m}\text{Tr}\left\{ \Pi_{\rho_{X^{n}\left( i\right) },\delta
| |
| }\Pi_{\rho,\delta}^{n}\rho_{X^{n}\left( m\right) }\Pi_{\rho,\delta}
| |
| ^{n}\right\} \right\} \right] ^{1/2},
| |
| </math>
| |
| where the second bound follows by summing over all of the codewords not equal
| |
| to the <math>m^{\text{th}}</math> codeword (this sum can only be larger).
| |
| | |
| We now focus exclusively on showing that the term inside the square root can
| |
| be made small. Consider the first term:
| |
| :<math>
| |
| \mathbb{E}_{X^{n}}\left\{ \frac{1}{M}\sum_{m}\text{Tr}\left\{ \left(
| |
| I-\Pi_{\rho_{X^{n}\left( m\right) },\delta}\right) \Pi_{\rho,\delta}
| |
| ^{n}\rho_{X^{n}\left( m\right) }\Pi_{\rho,\delta}^{n}\right\} \right\} </math>
| |
| :<math> \leq\mathbb{E}_{X^{n}}\left\{ \frac{1}{M}\sum_{m}\text{Tr}\left\{ \left(
| |
| I-\Pi_{\rho_{X^{n}\left( m\right) },\delta}\right) \rho_{X^{n}\left(
| |
| m\right) }\right\} +\left\Vert \rho_{X^{n}\left( m\right) }-\Pi
| |
| _{\rho,\delta}^{n}\rho_{X^{n}\left( m\right) }\Pi_{\rho,\delta}
| |
| ^{n}\right\Vert _{1}\right\} </math>
| |
| :<math> \leq\epsilon+2\sqrt{\epsilon}.
| |
| </math>
| |
| where the first inequality follows from (\ref{eq:trace-inequality}) and the
| |
| second inequality follows from the Gentle Operator Lemma and the
| |
| properties of unconditional and conditional typicality. Consider now the
| |
| second term and the following chain of inequalities:
| |
| :<math>
| |
| \sum_{i\neq m}\mathbb{E}_{X^{n}}\left\{ \text{Tr}\left\{ \Pi_{\rho
| |
| _{X^{n}\left( i\right) },\delta}\ \Pi_{\rho,\delta}^{n}\ \rho_{X^{n}\left(
| |
| m\right) }\ \Pi_{\rho,\delta}^{n}\right\} \right\} </math>
| |
| :<math> =\sum_{i\neq m}\text{Tr}\left\{ \mathbb{E}_{X^{n}}\left\{ \Pi
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| _{\rho_{X^{n}\left( i\right) },\delta}\right\} \ \Pi_{\rho,\delta}
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| ^{n}\ \mathbb{E}_{X^{n}}\left\{ \rho_{X^{n}\left( m\right) }\right\}
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| \ \Pi_{\rho,\delta}^{n}\right\} </math>
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| :<math> =\sum_{i\neq m}\text{Tr}\left\{ \mathbb{E}_{X^{n}}\left\{ \Pi
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| _{\rho_{X^{n}\left( i\right) },\delta}\right\} \ \Pi_{\rho,\delta}
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| ^{n}\ \rho^{\otimes n}\ \Pi_{\rho,\delta}^{n}\right\} </math>
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| :<math> \leq\sum_{i\neq m}2^{-n\left[ H\left( B\right) -\delta\right]
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| }\ \text{Tr}\left\{ \mathbb{E}_{X^{n}}\left\{ \Pi_{\rho_{X^{n}\left(
| |
| i\right) },\delta}\right\} \ \Pi_{\rho,\delta}^{n}\right\}
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| </math>
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| The first equality follows because the codewords <math>X^{n}\left( m\right) </math> and
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| <math>X^{n}\left( i\right) </math> are independent since they are different. The second
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| equality follows from (\ref{eq:avg-state}). The first inequality follows from
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| (\ref{eq:3rd-typ-prop}). Continuing, we have
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| :<math>
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| \leq\sum_{i\neq m}2^{-n\left[ H\left( B\right) -\delta\right]
| |
| }\ \mathbb{E}_{X^{n}}\left\{ \text{Tr}\left\{ \Pi_{\rho_{X^{n}\left(
| |
| i\right) },\delta}\right\} \right\} </math>
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| :<math> \leq\sum_{i\neq m}2^{-n\left[ H\left( B\right) -\delta\right]
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| }\ 2^{n\left[ H\left( B|X\right) +\delta\right] }</math>
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| :<math> =\sum_{i\neq m}2^{-n\left[ I\left( X;B\right) -2\delta\right] }</math>
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| :<math> \leq M\ 2^{-n\left[ I\left( X;B\right) -2\delta\right] }.
| |
| </math>
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| The first inequality follows from <math>\Pi_{\rho,\delta}^{n}\leq I</math> and exchanging
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| the trace with the expectation. The second inequality follows from
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| (\ref{eq:2nd-cond-typ}). The next two are straightforward.
| |
| | |
| Putting everything together, we get our final bound on the expectation of the
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| average error probability:
| |
| :<math>
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| 1-\mathbb{E}_{X^{n}}\left\{ \frac{1}{M}\sum_{m}\text{Tr}\left\{ \Pi
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| _{\rho_{X^{n}\left( m\right) },\delta}\hat{\Pi}_{\rho_{X^{n}\left(
| |
| m-1\right) },\delta}\cdots\hat{\Pi}_{\rho_{X^{n}\left( 1\right) },\delta
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| }\ \Pi_{\rho,\delta}^{n}\ \rho_{X^{n}\left( m\right) }\ \Pi_{\rho,\delta
| |
| }^{n}\ \hat{\Pi}_{\rho_{X^{n}\left( 1\right) },\delta}\cdots\hat{\Pi}
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| _{\rho_{X^{n}\left( m-1\right) },\delta}\Pi_{\rho_{X^{n}\left( m\right)
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| },\delta}\right\} \right\} </math>
| |
| :<math>\leq\epsilon+2\left[ \left( \epsilon+2\sqrt{\epsilon}\right) | |
| +M\ 2^{-n\left[ I\left( X;B\right) -2\delta\right] }\right] ^{1/2}.
| |
| </math>
| |
| Thus, as long as we choose <math>M=2^{n\left[ I\left( X;B\right) -3\delta
| |
| \right] }</math>, there exists a code with vanishing error probability.
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| | |
| == See also ==
| |
| | |
| * [[Quantum capacity]]
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| * [[Entanglement-assisted classical capacity]]
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| * [[Typical subspace]]
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| * [[Quantum information theory]]
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| | |
| == References ==
| |
| *{{citation|first=Alexander S.|last=Holevo|authorlink=Alexander Holevo|arxiv=quant-ph/9611023|title=The Capacity of Quantum Channel with General Signal States|year=1998|doi=10.1109/18.651037|journal=IEEE Transactions on Information Theory|volume=44|issue=1|pages=269–273}}.
| |
| *{{citation|first1=Benjamin|last1=Schumacher|first2=Michael|last2=Westmoreland|doi=10.1103/PhysRevA.56.131|title=Sending classical information via noisy quantum channels|journal=Phys. Rev. A|volume=56|pages=131–138|year=1997}}.
| |
| *{{citation|first=Mark M.|last=Wilde|arxiv=1106.1445|title=Quantum Information Theory|year=2013|publisher=Cambridge University Press}}
| |
| *{{citation|first=Pranab|last=Sen|arxiv=1109.0802|contribution=Achieving the Han-Kobayashi inner bound for the quantum interference channel by sequential decoding|year=2012|title=IEEE International Symposium on Information Theory Proceedings (ISIT 2012)|pages=736–740|doi=10.1109/ISIT.2012.6284656}}.
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| *{{citation|first1=Saikat|last1=Guha|first2=Si-Hui|last2=Tan|first3=Mark M.|last3=Wilde|arxiv=1202.0518|contribution=Explicit capacity-achieving receivers for optical communication and quantum reading|year=2012|title=IEEE International Symposium on Information Theory Proceedings (ISIT 2012)|pages=551–555|doi=10.1109/ISIT.2012.6284251}}.
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| {{Quantum computing}}
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| [[Category:Quantum information theory]]
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