Enhancement of channel capacities with entanglement

==Introduction== Shared entanglement between a sender and receiver can significantly improve the usefulness of a quantum channel for the communication of either classical or quantum data. Superdense coding and teleportation provide the most well-known examples of this improvement; free entanglement doubles the classical capacity of a noiseless quantum channel and makes it possible for a noiseless classical channel to send quantum data. In fact, the entanglement-assisted classical and quantum capacities of a quantum channel are in many senses simpler and better behaved than their unassisted counterparts~\cite{H98,SW97,D05}. Most importantly, these capacities can be calculated using simple formulas and finite optimization procedures~\cite{BSST99,BSST02}. No such finite procedure is known for either of the unassisted capacities. Moreover, the entanglement-assisted classical and quantum capacities are related by a simple factor of two. The unassisted capacities, in contrast, have completely different formulas. In fact, the simple factor of two generalizes to a statement known as the quantum Reverse Shannon Theorem, which governs the rate at which one quantum channel can simulate another~\cite{QRST}. The answer is given by the ratio of the entanglement-assisted capacities. '''Notation:''' Quantum systems will be denoted by $A$, $B$, and so on as well as their variants such as $A\text{'}$ and $\backslash hat\{A\}$. The choice of letter will generally indicate which party holds a given system, with $A$ reserved for the sender, Alice, and $B$ for the receiver, Bob. Given a quantum system $C$, $C^\{\backslash otimes\; n\}$ will often be written as $C^n$. These symbols will be used to denote both the Hilbert space of the quantum system and the set of density operators on that system. Thus, a quantum channel $\{\backslash mathcal\{N\}\}\; \backslash colon\; A\text{'}\; \backslash rightarrow\; B$ refers to a trace-preserving, completely positive (TPCP) map from the operators on the Hilbert space of $A\text{'}$ to those of $B$. $\{\backslash operatorname\{id\}\}^C$ refers to the identity channel on $C$. The map $\{\backslash mathcal\{N\}\}\; \backslash otimes\; \{\backslash operatorname\{id\}\}^C$ will frequently be abbreviated to $\{\backslash mathcal\{N\}\}$ in order to simplify long expressions. $|\backslash varphi\backslash rangle$ will be abbreviated to $\backslash varphi$. $\backslash pi^C$ will refer to the maximally mixed state on $C$ and $\backslash pi\_d$ to the maximally mixed state on a specified $d$-dimensional quantum system. For a given quantum state $\backslash varphi^\{AB\}$ on the composite system $AB$, $\backslash varphi^A\; =\; \{\backslash operatorname\{tr\}\}\_B\; \backslash varphi^\{AB\}$ and H(A)_\varphi = H(\varphi^A) = - {\operatorname{tr}} ( \varphi^A \log_2 \varphi^A ) is the von Neumann entropy of $\backslash varphi^A$, while H(A|B)_\varphi = -I_c(A\rangle B) = H(AB)_\varphi - H(B)_\varphi is its conditional entropy and $I(A;B)\_\backslash varphi\; =\; H(A)\_\backslash varphi\; +\; H(B)\_\backslash varphi\; -\; H(AB)\_\backslash varphi$ its mutual information. ==Entanglement-assisted classical and quantum capacities== The entanglement-assisted classical capacity of a quantum channel $\{\backslash mathcal\{N\}\}:A\text{'}\backslash rightarrow\; B$ is the optimal rate at which classical information can be communicated through the channel while in addition making use of an unlimited number of maximally entangled states. The formal definition proceeds as follows. Alice and Bob are assumed to share $nS$ ebits in the form of a maximally entangled state $|\backslash phi\backslash rangle^\{\backslash tilde\{A\}\backslash tilde\{B\}\}$ of Schmidt rank $2^\{nS\}$. Conditioned on her message $m\; \backslash in\; \backslash \{1,2,\backslash ldots,2^\{nR\}\backslash \}$, Alice will apply an encoding operation $\{\backslash mathcal\{E\}\}\_m\; \backslash colon\; \backslash tilde\{A\}\; \backslash rightarrow\; A\text{'}^n$. Bob's decoding is given by a POVM $\backslash \{\backslash Lambda\_m\; \backslash \}\_\{m=1\}^\{2^\{nR\}\}$ on the composite system $\backslash tilde\{B\}B^n$. The procedure is said to have maximum probability of error $\backslash epsilon$ if \label{eqn:errorProb} \max_m\ {\operatorname{tr}}\Big[ \Lambda_m ({\mathcal{N}}^{\otimes n} \circ {\mathcal{E}}_m)(\phi) \Big] \geq 1 - \epsilon. These elements, illustrated in Figure~\ref{fig:C_E}, consisting of the shared entanglement, as well as the encoding and decoding operations meeting the criterion of Eq.~(\ref{eqn:errorProb}), are called a $(2^\{nR\},2^\{nS\},n,\backslash epsilon)$ entanglement-assisted classical code for the channel $\{\backslash mathcal\{N\}\}$. A rate $R$ is said to be achievable if there exists a choice of $S\backslash geq\; 0$ and a sequence of entanglement-assisted classical codes $(2^\{nR\},2^\{nS\},n,\backslash epsilon\_n)$ with $\backslash epsilon\_n\; \backslash rightarrow\; 0$. The entanglement-assisted classical capacity $C\_E(\{\backslash mathcal\{N\}\})$ of $\{\backslash mathcal\{N\}\}$ is defined to be the supremum over all achievable rates. Theorem [BSST~\cite{BSST99,BSST02}] \label{thm:C_E} The entanglement-assisted classical capacity $C\_E$ of a quantum channel $\{\backslash mathcal\{N\}\}\; \backslash colon\; A\text{'}\; \backslash rightarrow\; B$ is given by \label{eqn:C_E} C_E({\mathcal{N}}) = \max_\sigma I(A;B)_\sigma, where the maximization is over states $\backslash sigma^\{AB\}\; =\; \{\backslash mathcal\{N\}\}(\backslash varphi^\{AA\text{'}\})$ arising from the channel by acting on the $A\text{'}$ half of any pure state $|\backslash varphi\backslash rangle^\{AA\text{'}\}$. \end{theorem} The theorem bears a strong formal resemblance to Shannon's noisy coding theorem for the classical capacity of a classical noisy channel. There the capacity formula is also given by an optimization of the mutual information, but over joint distributions between the input and output alphabets arising from the action of the channel. Such a joint distribution cannot exist in general for a quantum channel because the no-cloning theorem excludes the possibility of the input and output existing simultaneously. Eq.~(\ref{eqn:C_E}) instead refers to a natural substitute for the joint input-output distribution: a quantum state arising from the quantum channel acting on half of an entangled pure state. Another point worth stressing is that, unlike the known formulas for the unassisted classical and quantum capacities of a quantum channel, Eq.~(\ref{eqn:C_E}) refers to only a single use of $\{\backslash mathcal\{N\}\}$ instead of the limit of many uses, $\{\backslash mathcal\{N\}\}^\{\backslash otimes\; n\}$. The formula can therefore readily be used to evaluate $C\_E$ for any channel of interest. Consider, for example, the $d$-dimensional depolarizing channel {\mathcal{D}}_p(\rho) = (1-p)\rho + p \, \pi_d that with probability $p$ completely randomizes the input but otherwise leaves the input invariant. For such channels, the maximum is achieved by choosing a maximally entangled state for $|\backslash varphi>^\{AA\text{'}\}$, yielding C_E({\mathcal{D}}_p) = 2 \log_2 d - h_{d^2}\Big( 1 - p \frac{d^2-1}{d^2} \Big), where for any $0\; \backslash leq\; q\; \backslash leq\; 1$ and integer $r\; \backslash geq\; 1$, h_r(q) = -q \log_2 q - (1-q) \log_2 \Big(\frac{1-q}{r-1}\Big) is the Shannon entropy of the distribution $\backslash big(q,(1-q)/(r-1),\backslash ldots,(1-q)/(r-1)\backslash big)$. Entanglement-assistance also simplifies the relationship between the classical and quantum capacities of a channel. Proceeding as before to formally define the quantum capacity, Alice and Bob are again assumed to share a maximally entangled state $|\backslash phi>^\{\backslash tilde\{A\}\backslash tilde\{B\}\}$ of Schmidt rank $2^\{nS\}$. Alice's encoding operation will be a TPCP map $\{\backslash mathcal\{E\}\}\backslash colon\; \backslash hat\{A\}\; \backslash tilde\{A\}\; \backslash rightarrow\; A\text{'}^n$ acting on an input system $\backslash hat\{A\}$ and her half of the shared entanglement, $\backslash tilde\{A\}$. Bob's decoding will likewise be a TPCP map $\{\backslash mathcal\{D\}\}\; \backslash colon\; \backslash tilde\{B\}B^n\; \backslash rightarrow\; \backslash hat\{B\}$ acting on the output of the channel, $B^n$, and his half of the shared entanglement, $\backslash tilde\{B\}$. $\backslash hat\{A\}$ and $\backslash hat\{B\}$ are assumed to be isomorphic quantum systems of some fixed dimension $2^\{nQ\}$. The procedure is said to have subspace fidelity $1-\backslash epsilon$ if \label{eqn:fidelity} \,^{\hat{B}}<\varphi | ({\mathcal{D}} \circ {\mathcal{N}}^{\otimes n} \circ {\mathcal{E}})(\phi^{\tilde{A}\tilde{B}} \otimes \varphi^{\hat{A}}) |\varphi\rangle^{\hat{B}} \geq 1 - \epsilon for all $|\backslash varphi>^\{\backslash hat\{A\}\}\; \backslash in\; \backslash hat\{A\}$. These elements, illustrated in Figure~\ref{fig:Q_E}, are together called a $(2^\{nQ\},2^\{nS\},n,\backslash epsilon)$ entanglement-assisted quantum code for the channel {\mathcal{N}}. A rate $Q$ is said to be achievable if there exists a choice of $S\backslash geq\; 0$ and a sequence of entanglement-assisted quantum codes $(2^\{nR\},2^\{nS\},n,\backslash epsilon\_n)$ with $\backslash epsilon\_n\; \backslash rightarrow\; 0$. The entanglement-assisted quantum capacity $Q\_E(\{\backslash mathcal\{N\}\})$ of $\{\backslash mathcal\{N\}\}$ is defined to be the supremum over all achievable rates. There is considerable freedom in the definition of the entanglement-assisted quantum capacity. It could, for example, be defined as the largest amount of maximal entanglement that can be generated using the channel, minus the entanglement consumed during the protocol itself. Alternatively, the fidelity criterion Eq.~(\ref{eqn:fidelity}) could be strengthened to require that $\{\backslash mathcal\{D\}\}\; \backslash circ\; \{\backslash mathcal\{N\}\}^\{\backslash otimes\; n\}\; \backslash circ\; \{\backslash mathcal\{E\}\}$ preserve not only pure states on $\backslash hat\{A\}$ but any entanglement between $\backslash hat\{A\}$ and a reference system. All of these variants yield the same capacity formula: Q_E({\mathcal{N}}) = \frac{1}{2} C_E({\mathcal{N}}). This equivalence is a direct consequence of the existence of the [[Teleportation Protocol|teleportation]] and [[Superdense Coding Protocol|superdense coding]] protocols. When maximal entanglement is available, teleportation converts the ability to send classical data into the ability to send quantum data at half the classical rate. Conversely, by consuming maximal entanglement, superdense coding converts the ability to send quantum data into the ability to send classical data at double the quantum rate. ==Sketch of proof== The proof of a capacity theorem can usually be broken into two parts, achievability and optimality. The achievability part demonstrates the existence of a sequence of codes reaching the prescribed rate while the optimality part shows that it is impossible to do better. The main idea in the achievability proof can be understood by studying the special case where $\backslash varphi^\{A\text{'}\}\; =\; \backslash pi^\{A\text{'}\}$. Let $d\_\{A\text{'}\}\; =\; \backslash dim\; A\text{'}$ and $\backslash \{U\_j\backslash \}\_\{j=1\}^\{d\_\{A\text{'}\}^\{2n\}\}$ be a set of Weyl operators for $A\text{'}^n$. The relevant property of these operators is that averaging over them implements the constant map: for all density operators $\backslash rho$, \frac{1}{d_{A'}^{2n}} \sum_{j=1}^{d_{A'}^{2n}} U_j \, \rho \, U_j^\dagger = \pi^{A'^n}. Consider the state $\backslash sigma\_j$ that arises if Alice acts with $U\_j$ on the $A\text{'}^n$ half of a rank $d\_\{A\text{'}\}^n$ maximally entangled state $|\backslash varphi\; >^\{AA\text{'}^n\}$ and then sends the $A\text{'}^n$ half of the resulting state through $\{\backslash mathcal\{N\}\}$. (Note that here $A\text{'}^n$ also plays the role of $\backslash tilde\{A\}$.) The entropy of the resulting state is H(\sigma_j)= H\big( {\mathcal{N}}( ( U_j \otimes I_{\tilde{B}} ) \varphi ( U_j^\dagger \otimes I_{\tilde{B}} ) ) \big) \\ = H\big( {\mathcal{N}}( \varphi ) \big) since $U\_j$ does not change the local density operator on $A\text{'}^n$. On the other hand, if Alice selects a value of $j$ from the uniform distribution, then the resulting average input state to the channel will be \pi^{A'^n} \otimes \pi^{A} = \varphi^{A'^n} \otimes \varphi^{A} and the corresponding average output state will be $\{\backslash mathcal\{N\}\}(\backslash varphi^\{A\text{'}^n\})\; \backslash otimes\; \backslash varphi^A$, which has entropy H( {\mathcal{N}}(\varphi^{A'^n}) ) + H( \varphi^{A} ). Therefore, the Holevo quantity of the ensemble of output states, defined as the entropy of the average state minus the average of the entropies of the individual output states, will be equal to H(\varphi^{A}) + H({\mathcal{N}}(\varphi^{A'^n})) - H\big( {\mathcal{N}}( \varphi^{AA'^n} ) \big). This is precisely the quantity $I(A;B)\_\backslash sigma$ for the state $\{\backslash mathcal\{N\}\}(\backslash varphi^\{AA\text{'}^n\})$ since the channel $\{\backslash mathcal\{N\}\}$ transforms the $A\text{'}^n$ system into $B$. Moreover, if Bob is given the $A$ part of the maximally entangled state, then this is the Holevo quantity of an ensemble of states that can be produced by Alice acting on half of a shared entangled state and then sending her half through the channel. Invoking the HSW theorem for the classical capacity~\cite{H98,SW97} therefore completes the proof; using coding, the Holevo quantity is an achievable communication rate. The proof that Eq.~(\ref{eqn:C_E}) is optimal involves a series of entropy manipulations similar to the optimality proofs for the unassisted classical and quantum capacities. From the point of view of quantum information, the truly unusual part of the proof is the demonstration that it is unnecessary to consider multiple copies of $\{\backslash mathcal\{N\}\}$~\cite{CA97}. Specifically, let f({\mathcal{N}}) = \max_\sigma I(A;B)_\sigma, where the maximization is defined as in Theorem~\ref{thm:C_E}. Techniques analogous to those used for the unassisted capacities yield the upper bound C_E({\mathcal{N}}) \leq \lim_{n\rightarrow\infty} \frac{1}{n} f({\mathcal{N}}^{\otimes n}). Unlike the unassisted case, however, a relatively easy argument shows that \label{eqn:additivity} f({\mathcal{N}}_1 \otimes {\mathcal{N}}_2) = f({\mathcal{N}}_1) + f({\mathcal{N}}_2). (The analogous statement is an important conjecture for the classical capacity and is known to be false for the quantum capacity~\cite{DSS98}.) As a result, $C\_E(\{\backslash mathcal\{N\}\})\; \backslash leq\; f(\{\backslash mathcal\{N\}\})$, which is the optimality part of Theorem~\ref{thm:C_E}. To see the origin of Eq.~(\ref{eqn:additivity}), it will be helpful to invoke Stinespring's theorem to write $\{\backslash mathcal\{N\}\}\_j\; =\; \{\backslash operatorname\{tr\}\}\_\{E\_j\}\; \{\backslash mathcal\{U\}\}\_j^\{B\_j\; E\_j\}$, where $\{\backslash mathcal\{U\}\}\_j\; \backslash colon\; A\text{'}\_j\; \backslash rightarrow\; B\_j\; E\_j$ is an isometry. Fix a state $|\backslash varphi\; >^\{AA\_1\text{'}A\_2\text{'}\}$ and let $\backslash sigma\; =\; (\{\backslash mathcal\{U\}\}\_1\; \backslash otimes\; \{\backslash mathcal\{U\}\}\_2)(\backslash varphi)$. Eq.~(\ref{eqn:additivity}) follows from the fact that \label{eqn:infoInequality} I(A;B_1B_2)_\sigma \leq I(AB_2E_2;B_1)_\sigma + I(AB_1E_1;B_2)_\sigma. Simply redefining $A$ to be $AB\_2E\_2$ shows that the first term of the righthand side is upper bounded by $f(\{\backslash mathcal\{N\}\}\_1)$. The second term, likewise, is upper bounded by $f(\{\backslash mathcal\{N\}\}\_2)$. Eq.~(\ref{eqn:infoInequality}) is itself equivalent to the inequality \begin{multline} H(B_1 B_2 | E_1 E_2)_\sigma + H(B_1 B_2)_\sigma \\ \;\;\leq H(B_1|E_1)_\sigma + H(B_2|E_2)_\sigma + H(B_1)_\sigma + H(B_2)_\sigma. \end{multline} The inequality $H(B\_1B\_2)\_\backslash sigma\; \backslash leq\; H(B\_1)\_\backslash sigma+H(B\_2)\_\backslash sigma$ holds by the subadditivity of the von Neumann entropy. Repeated applications of the \emph{strong} subadditivity inequality, moreover, lead to the inequality H(B_1 B_2 | E_1 E_2)_\sigma \leq H(B_1|E_1)_\sigma + H(B_2|E_2)_\sigma. Together, they prove Eq.~(\ref{eqn:infoInequality}) and, thence, Eq.~(\ref{eqn:additivity}). The intuitive meaning of this ``single-letterization'' is unclear, but regardless, it is interesting to note that the proof involved invoking a pair of purifying environment systems, $E\_1$ and $E\_2$, and studying the entropy relationships between the true outputs of the channel and the environment's share. ==The quantum Reverse Shannon Theorem== A strong argument can be made that the entanglement-assisted capacity of a quantum channel is the most important capacity of that channel and that all the other capacities are, in some sense, of less significance. The fact that it is unnecessary to distinguish between the classical and quantum entanglement-assisted capacities because they are related by a factor of two is a hint in that direction, as is the simple, single-letter formula for $C\_E(\{\backslash mathcal\{N\}\})$. A more general argument can be made by considering the problem of having one channel simulate another. Indeed, the quantum capacity of a quantum channel is simply the optimal rate at which that channel can simulate the noiseless channel $\{\backslash operatorname\{id\}\}\_2$ on a single qubit. Likewise, the classical capacity of a quantum channel is its optimal rate for simulation of a qubit dephasing channel \rho \mapsto |0\rangle\langle 0| \, \rho \, |0\rangle\langle 0| + |1\rangle\langle 1| \, \rho \, |1\rangle\langle 1|. In this spirit, the fact that $C\_E(\{\backslash mathcal\{N\}\})\; =\; 2\; Q\_E(\{\backslash mathcal\{N\}\})$ can be re-expressed in the form Q_E({\mathcal{N}}) = \frac{C_E({\mathcal{N}})}{C_E({\operatorname{id}}_2)}. Equivalently, when entanglement is free, the optimal rate at which $\{\backslash mathcal\{N\}\}$ can simulate a noiseless qubit channel is given by the ratio between the entanglement-assisted classical capacities of $\{\backslash mathcal\{N\}\}$ and $\{\backslash operatorname\{id\}\}\_2$. The quantum Reverse Shannon Theorem generalizes this statement to the simulation of arbitrary channels in the presence of free entanglement. Suppose that Alice and Bob would like to use $\{\backslash mathcal\{N\}\}\_1\backslash colon\; A\text{'}\backslash rightarrow\; B$ to simulate another channel $\{\backslash mathcal\{N\}\}\_2\; \backslash colon\; A\text{'}\; \backslash rightarrow\; B$. Fix an input state $\backslash varphi^\{A\text{'}\}$ and let $|\backslash varphi\; >^\{AA\text{'}^n\}$ be a purification of $(\backslash varphi^\{A\text{'}\})^\{\backslash otimes\; n\}$. As always, assume that Alice and Bob share a maximally entangled state $|\backslash phi\; >^\{\backslash tilde\{A\}\backslash tilde\{B\}\}$ of Schmidt rank $2^\{nS\}$. Alice's encoding operation will be a TPCP map $\{\backslash mathcal\{E\}\}\backslash colon\; \backslash tilde\{A\}A\text{'}^n\; \backslash rightarrow\; A\text{'}^m$ acting on $n$ copies of the input system $A\text{'}$ and her half of the shared entanglement, $\backslash tilde\{A\}$. Bob's decoding will likewise be a TPCP map $\{\backslash mathcal\{D\}\}\; \backslash colon\; B^m\; \backslash tilde\{B\}\; \backslash rightarrow\; B^n$ acting on $m$ copies of the output of the channel, and his half of the shared entanglement, $\backslash tilde\{B\}$. This procedure is said to $\backslash epsilon$-simulate $\{\backslash mathcal\{N\}\}\_2^\{\backslash otimes\; n\}$ on $(\backslash varphi^\{A\text{'}\})^\{\backslash otimes\; n\}$ if F\big({\mathcal{N}}_2^{\otimes n}(\varphi^{AA'^n}),({\mathcal{D}}\circ{\mathcal{N}}_1^{\otimes m}\circ {\mathcal{E}})(\phi^{\tilde{A}\tilde{B}} \otimes \varphi^{AA'^n}) \big) \geq 1 - \epsilon, where $F$ is the mixed state fidelity $F(\backslash rho,\backslash sigma)\; =\; \backslash big(\; \{\backslash operatorname\{tr\}\}\backslash sqrt\{\backslash rho^\{1/2\}\backslash sigma\backslash rho^\{1/2\}\}\backslash big)^2$. The entire procedure, illustrated in Figure~\ref{fig:QRST}, is said to be a $(2^\{nS\},m,n,\backslash epsilon)$ entanglement-assisted simulation of $\{\backslash mathcal\{N\}\}\_2$ by $\{\backslash mathcal\{N\}\}\_1$. A rate $R$, measured in copies of $\{\backslash mathcal\{N\}\}\_2$ per copy of $\{\backslash mathcal\{N\}\}\_1$, is said to be achievable for $\backslash varphi^\{A\text{'}\}$ if there exists a choice of $S\backslash geq\; 0$ and a sequence of $(2^\{nS\},m\_n,n,\backslash epsilon\_n)$ entanglement-assisted simulations with $n/m\_n\; \backslash rightarrow\; R$ while $\backslash epsilon\_n\; \backslash rightarrow\; 0$. The quantum Reverse Shannon Theorem states that the entanglement-assisted capacity completely governs the achievable simulation rates. \begin{theorem}[BDHSW~\cite{W02,QRST}] \label{thm:QRST} Given two channels $\{\backslash mathcal\{N\}\}\_1\; \backslash colon\; A\text{'}\; \backslash rightarrow\; B$ and $\{\backslash mathcal\{N\}\}\_2\backslash colon\; A\text{'}\; \backslash rightarrow\; B$, $R$ is an achievable simulation rate for $\{\backslash mathcal\{N\}\}\_2$ by $\{\backslash mathcal\{N\}\}\_1$ and all input states $\backslash varphi^\{A\text{'}\}$ if and only if \label{eqn:simulationRatio} R \leq \frac{C_E({\mathcal{N}}_1)}{C_E({\mathcal{N}}_2)}. \end{theorem} Note that the form of Eq.~(\ref{eqn:simulationRatio}) ensures that the simulation is asymptotically reversible: if a channel $\{\backslash mathcal\{N\}\}\_1$ is used to simulate $\{\backslash mathcal\{N\}\}\_2$ and the simulation is then used to simulate $\{\backslash mathcal\{N\}\}\_1$ again, then the overall rate becomes \frac{C_E({\mathcal{N}}_1)}{C_E({\mathcal{N}}_2)}\frac{C_E({\mathcal{N}}_2)}{C_E({\mathcal{N}}_1)} = 1. Thus, in the presence of free entanglement and for a known input density operator of the form $(\backslash varphi^\{A\text{'}\})^\{\backslash otimes\; n\}$, a single parameter, the entanglement-assisted classical capacity, suffices to completely characterize the asymptotic properties of a quantum channel. Moreover, since two channels that are asymptotically equivalent without free entanglement will surely remain equivalent if free entanglement is permitted, Eq.~(\ref{eqn:simulationRatio}) gives essentially the only possible non-trivial, single-parameter asymptotic characterization of quantum channels. This is the sense in which the entanglement-assisted capacity should be regarded as the most important capacity of a quantum channel. The proof of the quantum Reverse Shannon Theorem is quite involved, but some of its features can be understood without much work. First, note that by the optimality statement of the entanglement-assisted classical capacity, the desired simulation can exist only if Eq.~(\ref{eqn:simulationRatio}) holds. Otherwise, composing the simulation of $\{\backslash mathcal\{N\}\}\_2$ by $\{\backslash mathcal\{N\}\}\_1$ with a sequence of codes achieving $C\_E(\{\backslash mathcal\{N\}\}\_2)$ would result in a sequence of codes beating the capacity formula for $\{\backslash mathcal\{N\}\}\_1$. Similarly, note that one method to simulate a channel $\{\backslash mathcal\{N\}\}\_1$ using $\{\backslash mathcal\{N\}\}\_2$ is to first use $\{\backslash mathcal\{N\}\}\_2$ to simulate the noiseless channel and then use the simulated noiseless channel to simulate $\{\backslash mathcal\{N\}\}\_1$. Since the achievable rates for the first step are characterized by the entanglement-assisted capacity theorem, proving the achievability part of Theorem \ref{thm:QRST} reduces to finding protocols for simulating a general noisy quantum channel $\{\backslash mathcal\{N\}\}\_2$ by a noiseless one. That perhaps sounds like a strange goal, but it is the difficult part of the quantum Reverse Shannon Theorem nonetheless. It is likely that the quantum Reverse Shannon Theorem can be extended to cover other types of inputs than the known tensor power states $(\backslash varphi^\{A\; \text{'}\})^\{\backslash otimes\; n\}$. The most desirable form of the theorem would be one valid for all possible input density operators on $A\text{'}^\{\backslash otimes\; n\}$, providing a single simulation procedure dependent only on the channels and not the input state. It is known that without modifying the form of the free entanglement, this most ambitious form of the theorem fails, but it is conjectured that the full-strength theorem does hold provided very large amounts of entanglement are supplied in the form of so-called embezzling states~\cite{DH03}. ==Relationships between protocols== There is another sense in which the entanglement-assisted capacity can be viewed as the fundamental capacity of a quantum channel: an efficient protocol for achieving the entanglement-assisted capacity can be converted into protocols achieving the unassisted quantum and classical capacities, or at least very close variants thereof. An efficient protocol in this case refers to one that doesn't waste entanglement. Suppose that $\{\backslash mathcal\{N\}\}\; \backslash colon\; A\text{'}\; \backslash rightarrow\; B$ can be written $\{\backslash operatorname\{tr\}\}\_E\; \{\backslash mathcal\{U\}\}^\{BE\}$ for some isometry $\{\backslash mathcal\{U\}\}^\{BE\}$. Let $|\backslash varphi\; \backslash rangle^\{AA\text{'}\}$ be a pure state and $|\backslash sigma\; \backslash rangle^\{ABE\}\; =\; \{\backslash mathcal\{U\}\}^\{BE\}|\backslash varphi\; \backslash rangle^\{AA\text{'}\}$ the corresponding purified channel output state. Careful analysis of the entanglement-assisted classical communication protocol achieving the rate $I(A;B)\_\backslash sigma$ leads to an entanglement-assisted quantum communication protocol consuming entanglement at the rate $\backslash frac\{1\}\{2\}I(A;E)\_\backslash sigma$ ebits per use of $\{\backslash mathcal\{N\}\}$ and yielding communication at the rate of $\backslash frac\{1\}\{2\}I(A;B)\_\backslash sigma$ qubits per use $\{\backslash mathcal\{N\}\}$. The protocol achieving this goal is known as the ``father''~\cite{DHW04}. If the entanglement consumed in the father were actually supplied by quantum communication from Alice to Bob, then the net rate of quantum communication produced by the resulting protocol would be $\backslash frac\{1\}\{2\}I(A;B)\_\backslash sigma-\backslash frac\{1\}\{2\}I(A;E)\_\backslash sigma$ qubits from Alice to Bob, that is, the total produced minus the total consumed. This quantity, how much more information $B$ has about $A$ than $E$ does, can be simplified using an interesting identity. Since $|\backslash sigma\; >^\{ABE\}$ is pure, I(A;E)_\sigma = H(A)_\sigma + H(E)_\sigma - H(AE)_\sigma \\ = H(A)_\sigma + H(AB)_\sigma - H(B)_\sigma. Expanding $I(A;B)\_\backslash sigma$ and cancelling terms then reveals that \frac{1}{2}I(A;B)-\frac{1}{2}I(A;E) = -H(A|B)_\sigma = I_c(A\rangle B)_\sigma, where the function $I\_c$ is known as the coherent information. After optimizing over input states and multiple channel uses, this is precisely the formula for the unassisted quantum capacity of a quantum channel~\cite{D05}. Thus, the net rate of qubit communication for the protocol derived from the father exactly matches the rates necessary to achieve the unassisted quantum capacity. The only caveat is that the protocol derived from the father uses quantum communication catalytically, meaning that some communication needs to be invested in order to get a gain of $I\_c(A\backslash rangle\; B)$. For the unassisted quantum capacity, no investment is necessary. Nonetheless, detailed analysis of the situation reveals that the amount of catalytic communication required can be reduced to an amount sublinear in the number of channel uses, meaning the rate of required investment can be made arbitrarily small. In this sense, the father protocol essentially generates the optimal protocols for the unassisted quantum capacity. Protocols achieving the unassisted classical capacity can be constructed in a similar way. In this case, one starts from an ensemble $\{\backslash mathcal\{E\}\}\; =\; \backslash \{p\_j,\{\backslash mathcal\{N\}\}(\backslash psi\_j^\{A\text{'}\})\backslash \}$ of states generated by the channel. Achievability of the unassisted classical capacity formula follows from achievability of rates of the form \chi({\mathcal{E}}) = H\big(\sum_j p_j {\mathcal{N}}(\psi_j^{A'})\big) - \sum_j p_j H\big( {\mathcal{N}}(\psi_j^{A'})\big) for arbitrary ensembles of output states. Consider the channel \tilde{{\mathcal{N}}}(\rho) = \sum_j \langle j| \rho |j\rangle \cdot {\mathcal{N}}(\psi_j) and input state $|\backslash varphi\; \backslash rangle^\{AA\text{'}\}\; =\; \backslash sum\_j\; \backslash sqrt\{p\_j\}\; |j\backslash rangle^\{A\}\; |j\backslash rangle^\{A\text{'}\}$. If $\backslash sigma\; =\; \backslash tilde\{\{\backslash mathcal\{N\}\}\}(\backslash varphi)$, then $I(A;B)\_\backslash sigma$ is equal to $\backslash chi(\{\backslash mathcal\{E\}\})$. Thus, there are protocols consuming entanglement that achieve the classical communications rate $\backslash chi(\{\backslash mathcal\{E\}\})$ for the modified channel $\backslash tilde\{\{\backslash mathcal\{N\}\}\}$. Because the channel $\backslash tilde\{\{\backslash mathcal\{N\}\}\}$ includes an orthonormal measurement which destroys all entanglement between $A$ and $B$, however, it can be argued that any entanglement used in such a protocol could be replaced by shared randomness, which could then in turn be eliminated by a standard derandomization argument. The net result is a procedure for choosing rate $\backslash chi(\{\backslash mathcal\{E\}\})$ codes for the channel $\{\backslash mathcal\{N\}\}$ consisting of states of the form $\backslash psi\_\{j\_1\}\; \backslash otimes\; \backslash cdots\; \backslash otimes\; \backslash psi\_\{j\_n\}$, which is the essence of the achievability proof for the unassisted classical capacity. This may seem like an unnecessarily cumbersome and even circular approach to the unassisted classical capacity given that the proof sketched above for the entanglement-assisted classical capacity itself invokes the unassisted result in the form of the HSW theorem. The approach becomes more satisfying when one learns that simple and direct proofs of the father protocol exist that completely bypass the HSW theorem~\cite{ADHW05}. Thus, the entanglement-assisted communication protocols can be easily transformed into their unassisted analogs, confirming the central place of entanglement-assisted communication in quantum information theory. [[Category:Quantum Communication]]