Classical capacity

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. Holevo, Schumacher, and Westmoreland proved the following least upper bound on the classical capacity of any quantum channel ${\mathcal {N}}$ :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi(\mathcal{N}) = \max_{\rho^{XA}} I(X;B)_{\mathcal{N}(\rho)} }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho^{XA}} is a classical-quantum state of the following form:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho^{XA} = \sum_x p_X(x) \vert x \rangle \langle x \vert^X \otimes \rho_x^A , }

$p_{X}(x)$ is a probability distribution, and each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_x^A} is a density operator that can be input to the channel Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{N}} .

Achievability using sequential decoding

We briefly review the HSW coding theorem (the statement of the achievability of the Holevo information rate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I(X;B)} for communicating classical data over a quantum channel). We first review the minimal amount of quantum mechanics needed for the theorem. We then cover quantum typicality, and finally we prove the theorem using a recent sequential decoding technique.

Review of quantum mechanics

In order to prove the HSW coding theorem, we really just need a few basic things from quantum mechanics. First, a quantum state is a unit trace, positive operator known as a density operator. Usually, we denote it by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} , etc. The simplest model for a quantum channel is known as a classical-quantum channel:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\mapsto \rho_{x}. }

The meaning of the above notation is that inputting the classical letter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} at the transmitting end leads to a quantum state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_{x}} at the receiving end. It is the task of the receiver to perform a measurement to determine the input of the sender. If it is true that the states Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_{x}} are perfectly distinguishable from one another (i.e., if they have orthogonal supports such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Tr}\,\left\{ \rho_{x}\rho_{x^{\prime}}\right\} =0} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\neq x^{\prime} } ), then the channel is a noiseless channel. We are interested in situations for which this is not the case. If it is true that the states Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_{x}} all commute with one another, then this is effectively identical to the situation for a classical channel, so we are also not interested in these situations. So, the situation in which we are interested is that in which the states Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_{x}} 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 $\left\{\Lambda _{m}\right\}_{m}$ . These operators should satisfy positivity and completeness in order to form a valid POVM:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Lambda_{m} \geq0\ \ \ \ \forall m}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{m}\Lambda_{m} =I. }

The probabilistic interpretation of quantum mechanics states that if someone measures a quantum state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho} using a measurement device corresponding to the POVM $\left\{\Lambda _{m}\right\}$ , then the probability Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\left( m\right) } for obtaining outcome Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} is equal to

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\left( m\right) =\text{Tr}\left\{ \Lambda_{m}\rho\right\} , }

and the post-measurement state is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_{m}^{\prime}=\frac{1}{p\left( m\right) }\sqrt{\Lambda_{m}}\rho \sqrt{\Lambda_{m}}, }

if the person measuring obtains outcome Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} . These rules are sufficient for us to consider classical communication schemes over cq channels.

Quantum typicality

The reader can find a good review of this topic in the article about the typical subspace.

Gentle operator lemma

The following lemma is important for our proofs. It demonstrates that a measurement that succeeds with high probability on average does not disturb the state too much on average:

Lemma: [Winter] Given an ensemble Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\{ p_{X}\left( x\right) ,\rho_{x}\right\} } with expected density operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho\equiv\sum_{x}p_{X}\left( x\right) \rho_{x}} , suppose that an operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Lambda} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I\geq\Lambda\geq0} succeeds with high probability on the state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Tr}\left\{ \Lambda\rho\right\} \geq1-\epsilon. }

Then the subnormalized state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{\Lambda}\rho_{x}\sqrt{\Lambda}} is close in expected trace distance to the original state $\rho _{x}$ :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{E}_{X}\left\{ \left\Vert \sqrt{\Lambda}\rho_{X}\sqrt{\Lambda} -\rho_{X}\right\Vert _{1}\right\} \leq2\sqrt{\epsilon}. }

(Note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\Vert A\right\Vert _{1}} is the nuclear norm of the operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\Vert A\right\Vert _{1}\equiv} TrFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\{ \sqrt{A^{\dagger} A}\right\} } .)

The following inequality is useful for us as well. It holds for any operators Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Lambda} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\leq\rho,\sigma,\Lambda\leq I} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Tr}\left\{ \Lambda\rho\right\} \leq\text{Tr}\left\{ \Lambda \sigma\right\} +\left\Vert \rho-\sigma\right\Vert _{1}. }

(1)

The quantum information-theoretic interpretation of the above inequality is that the probability of obtaining outcome Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Lambda} from a quantum measurement acting on the state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho} is upper bounded by the probability of obtaining outcome Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Lambda} on the state $\sigma$ summed with the distinguishability of the two states Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} .

Non-commutative union bound

Lemma: [Sen's bound] The following bound holds for a subnormalized state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\leq\sigma} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Tr\left\{ \sigma\right\} \leq1} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Pi_{1}} , ... , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Pi_{N}} being projectors: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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} \text{Tr}\left\{ \left( I-\Pi_{i}\right) \sigma\right\} }, }

We can think of Sen's bound as a "non-commutative union bound" because it is analogous to the following union bound from probability theory:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Pr\left\{ \left( A_{1}\cap\cdots\cap A_{N}\right) ^{c}\right\} =\Pr\left\{ A_{1}^{c}\cup\cdots\cup A_{N}^{c}\right\} \leq\sum_{i=1}^{N} \Pr\left\{ A_{i}^{c}\right\} , }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_{1}, \ldots, A_{N}} are events. The analogous bound for projector logic would be

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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) \rho\right\} , }

if we think of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Pi_{1}\cdots\Pi_{N}} as a projector onto the intersection of subspaces. Though, the above bound only holds if the projectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Pi_{1}} , ..., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Pi_{N}} are commuting (choosing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Pi_{1}=\left\vert +\right\rangle \left\langle +\right\vert } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Pi_{2}=\left\vert 0\right\rangle \left\langle 0\right\vert } , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho=\left\vert 0\right\rangle \left\langle 0\right\vert } 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, and error analysis.

Codebook Generation. We first describe how Alice and Bob agree on a random choice of code. They have the channel Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\rightarrow\rho_{x}} and a distribution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_{X}\left( x\right) } . They choose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} classical sequences Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^{n}} according to the IID\ distribution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_{X^{n}}\left( x^{n}\right) } . After selecting them, they label them with indices as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\{ x^{n}\left( m\right) \right\} _{m\in\left[ M\right] }} . This leads to the following quantum codewords:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_{x^{n}\left( m\right) }=\rho_{x_{1}\left( m\right) }\otimes \cdots\otimes\rho_{x_{n}\left( m\right) }. }

The quantum codebook is then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\{ \rho_{x^{n}\left( m\right) }\right\} } . The average state of the codebook is then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{E}_{X^{n}}\left\{ \rho_{X^{n}}\right\} =\sum_{x^{n}}p_{X^{n}}\left( x^{n}\right) \rho_{x^{n}}=\rho^{\otimes n}, }

(2)

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho=\sum_{x}p_{X}\left( x\right) \rho_{x}} .

POVM Construction . Sens' bound from the above lemma suggests a method for Bob to decode a state that Alice transmits. Bob should first ask "Is the received state in the average typical subspace?" He can do this operationally by performing a typical subspace measurement corresponding to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\{ \Pi_{\rho,\delta} ^{n},I-\Pi_{\rho,\delta}^{n}\right\} } . Next, he asks in sequential order, "Is the received codeword in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m^{\text{th}}} conditionally typical subspace?" This is in some sense equivalent to the question, "Is the received codeword the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m^{\text{th}}} transmitted codeword?" He can ask these questions operationally by performing the measurements corresponding to the conditionally typical projectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\{ \Pi_{\rho_{x^{n}\left( m\right) },\delta},I-\Pi_{\rho_{x^{n}\left( m\right) },\delta}\right\} } .

Why should this sequential decoding scheme work well? The reason is that the transmitted codeword lies in the typical subspace on average:

\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\}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\text{Tr}\left\{ \Pi_{\rho,\delta}\ \rho^{\otimes n}\right\} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \geq1-\epsilon,}

where the inequality follows from (\ref{eq:1st-typ-prop}). Also, the projectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Pi_{\rho_{x^{n}\left( m\right) },\delta}} are "good detectors" for the states Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_{x^{n}\left( m\right) }} (on average) because the following condition holds from conditional quantum typicality:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{E}_{X^{n}}\left\{ \text{Tr}\left\{ \Pi_{\rho_{X^{n}},\delta} \ \rho_{X^{n}}\right\} \right\} \geq1-\epsilon. }

Error Analysis. The probability of detecting the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m^{\text{th}}} codeword correctly under our sequential decoding scheme is equal to

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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\} , }

where we make the abbreviation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\Pi}\equiv I-\Pi} . (Observe that we project into the average typical subspace just once.) Thus, the probability of an incorrect detection for the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m^{\text{th}}} codeword is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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\} , }

and the average error probability of this scheme is equal to

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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\} . }

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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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\} . }

(3)

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 =\mathbb{E}_{X^{n}}\left\{ \frac{1}{M}\sum_{m}\text{Tr}\left\{ \rho_{X^{n}\left( m\right) }\right\} \right\} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\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\} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\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\} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\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\} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 }

Substituting into (3) (and forgetting about the small Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon} term for now) gives an upper bound of

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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\} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\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\} . }

We then apply Sen's bound to this expression with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma=\Pi_{\rho,\delta }^{n}\rho_{X^{n}\left( m\right) }\Pi_{\rho,\delta}^{n}} and the sequential projectors as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Pi_{\rho_{X^{n}\left( m\right) },\delta}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\Pi} _{\rho_{X^{n}\left( m-1\right) },\delta}} , ..., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\Pi}_{\rho _{X^{n}\left( 1\right) },\delta}} . This gives the upper bound Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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\} . } Due to concavity of the square root, we can bound this expression from above by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}, }

where the second bound follows by summing over all of the codewords not equal to the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m^{\text{th}}} 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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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\} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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\} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \leq\epsilon+2\sqrt{\epsilon}. }

where the first inequality follows from (1) 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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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\} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\sum_{i\neq m}\text{Tr}\left\{ \mathbb{E}_{X^{n}}\left\{ \Pi _{\rho_{X^{n}\left( i\right) },\delta}\right\} \ \Pi_{\rho,\delta} ^{n}\ \mathbb{E}_{X^{n}}\left\{ \rho_{X^{n}\left( m\right) }\right\} \ \Pi_{\rho,\delta}^{n}\right\} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\sum_{i\neq m}\text{Tr}\left\{ \mathbb{E}_{X^{n}}\left\{ \Pi _{\rho_{X^{n}\left( i\right) },\delta}\right\} \ \Pi_{\rho,\delta} ^{n}\ \rho^{\otimes n}\ \Pi_{\rho,\delta}^{n}\right\} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \leq\sum_{i\neq m}2^{-n\left[ H\left( B\right) -\delta\right] }\ \text{Tr}\left\{ \mathbb{E}_{X^{n}}\left\{ \Pi_{\rho_{X^{n}\left( i\right) },\delta}\right\} \ \Pi_{\rho,\delta}^{n}\right\} }

The first equality follows because the codewords Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X^{n}\left( m\right) } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X^{n}\left( i\right) } are independent since they are different. The second equality follows from (2). The first inequality follows from (\ref{eq:3rd-typ-prop}). Continuing, we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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\} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \leq\sum_{i\neq m}2^{-n\left[ H\left( B\right) -\delta\right] }\ 2^{n\left[ H\left( B|X\right) +\delta\right] }}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\sum_{i\neq m}2^{-n\left[ I\left( X;B\right) -2\delta\right] }}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \leq M\ 2^{-n\left[ I\left( X;B\right) -2\delta\right] }. }

The first inequality follows from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Pi_{\rho,\delta}^{n}\leq I} and exchanging the trace with the expectation. The second inequality follows from (\ref{eq:2nd-cond-typ}). The next two are straightforward.

Putting everything together, we get our final bound on the expectation of the average error probability:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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\} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}. }

Thus, as long as we choose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M=2^{n\left[ I\left( X;B\right) -3\delta \right] }} , there exists a code with vanishing error probability.

References

Holevo, Alexander S. (1998), "The Capacity of Quantum Channel with General Signal States", IEEE Transactions on Information Theory, 44 (1): 269–273, arXiv:quant-ph/9611023, doi:10.1109/18.651037.
Schumacher, Benjamin; Westmoreland, Michael (1997), "Sending classical information via noisy quantum channels", Phys. Rev. A, 56 (1): 131–138, Bibcode:1997PhRvA..56..131S, doi:10.1103/PhysRevA.56.131.
Wilde, Mark M. (2017), Quantum Information Theory, Cambridge University Press, arXiv:1106.1445, Bibcode:2011arXiv1106.1445W, doi:10.1017/9781316809976.001, S2CID 2515538
Sen, Pranab (2012), "Achieving the Han-Kobayashi inner bound for the quantum interference channel by sequential decoding", IEEE International Symposium on Information Theory Proceedings (ISIT 2012), pp. 736–740, arXiv:1109.0802, doi:10.1109/ISIT.2012.6284656, S2CID 15119225.
Guha, Saikat; Tan, Si-Hui; Wilde, Mark M. (2012), "Explicit capacity-achieving receivers for optical communication and quantum reading", IEEE International Symposium on Information Theory Proceedings (ISIT 2012), pp. 551–555, arXiv:1202.0518, doi:10.1109/ISIT.2012.6284251, ISBN 978-1-4673-2579-0, S2CID 8786400.