Given a quantum channel $\mathcal{N}_{A\to B}$ and $\varepsilon\in[0,1]$, the one-shot $\varepsilon$-error entanglement-assisted classical capacity $C_{\operatorname{EA}}^{\varepsilon}(\mathcal{N})$ is
\[
C_{\operatorname{EA}}^{\varepsilon}(\mathcal{N})
:= \sup_{(\mathcal{M},\Psi,\mathcal{E},\mathcal{D})}
\{\log_2 |\mathcal{M}| :
p_{\mathrm{err}}^{*}((\Psi,\mathcal{E},\mathcal{D});\mathcal{N})\le \varepsilon\}.
\]
The supremum ranges over one-shot entanglement-assisted classical communication protocols with message set $\mathcal{M}$, shared state $\Psi_{A'B'}$, encoding channel $\mathcal{E}_{M'A'\to A}$, and decoding channel $\mathcal{D}_{BB'\to\widehat{M}}$, with message and estimate registers indexed by the message set.
Given a quantum channel $\mathcal{N}$, a rate $R\in\mathbb{R}^{+}$ is achievable for entanglement-assisted classical communication over $\mathcal{N}$ if, for all $\varepsilon\in(0,1]$, $\delta>0$, and sufficiently large $n$, there exists an $(n,2^{n(R-\delta)},\varepsilon)$ entanglement-assisted classical communication protocol. The capacity $C_{\operatorname{EA}}(\mathcal{N})$ is
\[
C_{\operatorname{EA}}(\mathcal{N}) :=
\sup\{R : R\text{ is an achievable rate for }\mathcal{N}\}.
\]
For a quantum state $\rho_A$ on a finite-dimensional system $A$, its von Neumann entropy is defined as $H(A)_\rho := -\operatorname{Tr}\!\bigl(\rho_A \log_2 \rho_A\bigr).$ Equivalently, it is the Shannon entropy of the eigenvalues of $\rho_A$.
Let $\rho$ and $\sigma$ be positive semi-definite operators on a finite-dimensional system, with nonzero $\rho$. Define $D(\rho\|\sigma)$, with trace denominator $\operatorname{Tr}\rho$, by
\[
D(\rho\|\sigma):=
\begin{cases}
\dfrac{\operatorname{Tr}\!\left(\rho(\log_2\rho-\log_2\sigma)\right)}{\operatorname{Tr}\rho},&
\operatorname{supp}\rho\subseteq\operatorname{supp}\sigma,\\
+\infty,&\text{otherwise.}
\end{cases}
\]
For a quantum state $\rho$ on a bipartite system $AB$, the conditional von Neumann entropy of $A$ given $B$ is defined as $H(A|B)_\rho := H(AB)_\rho - H(B)_\rho$.
For a quantum state $\rho$ on a tripartite system $ABC$, the conditional mutual information between $A$ and $C$ given $B$ is defined as $I(A : C \mid B)_\rho := H(AB)_\rho + H(BC)_\rho - H(B)_\rho - H(ABC)_\rho$.
Let $\mathcal{N}_{A\to B}$ be a quantum channel. For all $\varepsilon\in(0,1)$ and $\eta\in(0,\varepsilon)$, there exists an $(|\mathcal{M}|,\varepsilon)$ entanglement-assisted classical communication protocol over $\mathcal{N}_{A\to B}$ such that
\[
\log_2 |\mathcal{M}|
= \overline{I}_{H}^{\varepsilon-\eta}(\mathcal{N})
- \log_2\!\left(\frac{4\varepsilon}{\eta^2}\right).
\]
Consequently $C_{\operatorname{EA}}^{\varepsilon}(\mathcal{N})\ge \overline{I}_{H}^{\varepsilon-\eta}(\mathcal{N})-\log_2(4\varepsilon/\eta^2)$. Moreover, for all $\alpha\in(0,1)$, there exists an $(|\mathcal{M}|,\varepsilon)$ protocol satisfying
\[
\log_2 |\mathcal{M}|
\ge \overline{I}_{\alpha}(\mathcal{N})
- \frac{\alpha}{1-\alpha}\log_2\!\left(\frac{1}{\varepsilon-\eta}\right)
- \log_2\!\left(\frac{4\varepsilon}{\eta^2}\right).
\] The hypothesis-testing quantity is denoted inline by $I_H$, and the Renyi parameter appears in $\overline{I}_{\alpha}$.
Let $\rho$ be a quantum state, let $\sigma$ be a positive semi-definite operator, and let $\mathcal{N}$ be a quantum channel. With $D(\rho\|\sigma)$ understood as the quantum relative entropy defined by $D(\rho\|\sigma)=\operatorname{Tr}\!\left[\rho(\log_2 \rho-\log_2 \sigma)\right]$ when the support of $\rho$ is contained in the support of $\sigma$, and as $+\infty$ otherwise, one has $D(\rho\|\sigma) \ge D(\mathcal{N}(\rho)\|\mathcal{N}(\sigma))$.
For any bipartite quantum state $\rho_{AB}$ and local quantum channels $\mathcal{N}_{A \to A'}$ and $\mathcal{N}_{B \to B'}$, let $\omega_{A'B'} := (\mathcal{N}_{A \to A'} \otimes \mathcal{N}_{B \to B'})(\rho_{AB})$. Then, the mutual information satisfies $I(A; B)_\rho \ge I(A'; B')_\omega$.
For any ensemble of bipartite quantum states $\{p(x), \rho_{AB}^x\}$, let $\omega_{AB} := \sum_x p(x)\rho_{AB}^x$. Then, the conditional von Neumann entropy is concave: $H(A \mid B)_\omega \ge \sum_x p(x)H(A \mid B)_{\rho^x}$.
For bipartite quantum states $\rho$ and $\sigma$ on a finite-dimensional system $AB$ with $\frac{1}{2}\|\rho - \sigma\|_1 \le \varepsilon \le 1$, the conditional von Neumann entropy satisfies $|H(A \mid B)_\rho - H(A \mid B)_\sigma| \le 2\varepsilon \log |A| + (1+\varepsilon)h\left(\frac{\varepsilon}{1+\varepsilon}\right)$.
Let $\rho$ be a quantum state, $\sigma$ a positive semidefinite operator, and $\mathcal{N}$ a quantum channel. For $\frac{1}{2} \le \alpha < 1$ or $\alpha > 1$, the sandwiched Renyi relative entropy satisfies $\widetilde{D}_\alpha(\rho \| \sigma) \ge \widetilde{D}_\alpha(\mathcal{N}(\rho) \| \mathcal{N}(\sigma))$.
Let $\rho_{ABC}$ be a pure subnormalized state and let $0 \le \varepsilon < \sqrt{\operatorname{Tr}\rho}$. Then $H_{\max}^{\varepsilon}(A|B)_{\rho} = -H_{\min}^{\varepsilon}(A|C)_{\rho}$.
Let $\rho_{AB}$ be a bipartite state on finite-dimensional systems $A$ and $B$, let $0<\varepsilon<1$, let $\eta$ be the finite-AEP parameter associated to $\rho_{AB}$, and let $n\in\mathbb{N}$ be such that $\rho_{AB}^{\otimes n}$ is the i.i.d. state on $A^nB^n$. If $n\ge \frac85\log(2/\varepsilon^2)$, then the conditional entropy quantity $H$ satisfies\[\frac1n H_{\min}^{\varepsilon}(A^n|B^n)_{\rho^{\otimes n}} \ge H(A|B)_{\rho}-\frac{\delta(\varepsilon,\eta)}{\sqrt n},\]where $\delta(\varepsilon,\eta):=4\log\eta\sqrt{\log(2/\varepsilon^2)}$.
Let $\rho_{AB}$ be a bipartite state on finite-dimensional systems $A$ and $B$, and let $\varepsilon>0$ be the smoothing parameter. For each $n\in\mathbb{N}$, write $\rho_{AB}^{\otimes n}$ for the i.i.d. state on $A^nB^n$. Then, for the entropy quantities $H_{\min}^{\varepsilon}$, $H_{\max}^{\varepsilon}$, and $H$,\[\lim_{\varepsilon\to0}\lim_{n\to\infty}\frac1n H_{\min}^{\varepsilon}(A^n|B^n)_{\rho^{\otimes n}}=H(A|B)_\rho,\qquad\lim_{\varepsilon\to0}\lim_{n\to\infty}\frac1n H_{\max}^{\varepsilon}(A^n|B^n)_{\rho^{\otimes n}}=H(A|B)_\rho.\]
Let $\mathcal{N}_{A\to B}$ be a quantum channel and let $\varepsilon\in[0,1)$. For every $(|\mathcal{M}|,\varepsilon)$ entanglement-assisted classical communication protocol over $\mathcal{N}$,
\[
\log_2 |\mathcal{M}| \le
\frac{1}{1-\varepsilon}\bigl(I(\mathcal{N})+h_2(\varepsilon)\bigr),
\]
and, for all $\alpha>1$,
\[
\log_2 |\mathcal{M}| \le
\widetilde{I}_{\alpha}(\mathcal{N})
+ \frac{\alpha}{\alpha-1}\log_2\!\left(\frac{1}{1-\varepsilon}\right).
\]
Consequently the same two upper bounds hold for $C_{\operatorname{EA}}^{\varepsilon}(\mathcal{N})$. The two information quantities are $I(\mathcal{N})$ and $\widetilde{I}_{\alpha}(\mathcal{N})$, and the binary entropy term is $h_2(\varepsilon)$.
Let $\mathcal{N}$ be a quantum channel. Let $I$ denote the channel mutual information functional and let $C$ denote entanglement-assisted classical capacity. The channel mutual information $I(\mathcal{N})$ is an achievable entanglement-assisted classical communication rate. Consequently,
\[
C_{\operatorname{EA}}(\mathcal{N}) \ge I(\mathcal{N}).
\]
Let $\phi^{AA'}$ be a pure state whose marginal $\phi^A$ has full rank $\dim A$, and let $W^{A'\to BE}$ be a Stinespring isometry for a channel $N^{A'\to B}$ with environment $E$. For a subspace $R\subseteq A$ with projection $P$, choose a Haar-random unitary $U$ on $A$, set $V=\sqrt{\dim A/\dim R}\,PU$, and let $\psi^{RBE}$ be the resulting non-normalized state after applying $V$ and then $W$. Writing $\phi^{AE}=\operatorname{Tr}_B(W\phi^{AA'}W^*)$ and $\pi^R$ for the maximally mixed state on $R$, \[\mathbb{E}_U\bigl\|\psi^{RE}-\pi^R\otimes\phi^E\bigr\|_1 \le \sqrt{\dim R\,\dim E\,\operatorname{Tr}[(\phi^{AE})^2]},\] where $\|\cdot\|_1$ is the trace norm.
Let $\mathcal{N}$ be a quantum channel. For all $\varepsilon\in[0,1)$, $n\in\mathbb{N}$, and every $(n,|\mathcal{M}|,\varepsilon)$ entanglement-assisted classical communication protocol over $n$ uses of $\mathcal{N}$,
\[
\frac{1}{n}\log_2 |\mathcal{M}|
\le \frac{1}{1-\varepsilon}\left(\frac{1}{n}I(\mathcal{N}^{\otimes n})+\frac{1}{n}h_2(\varepsilon)\right),
\]
and, for all $\alpha>1$,
\[
\frac{1}{n}\log_2 |\mathcal{M}|
\le \frac{1}{n}\widetilde{I}_{\alpha}(\mathcal{N}^{\otimes n})
+ \frac{\alpha}{n(\alpha-1)}\log_2\!\left(\frac{1}{1-\varepsilon}\right).
\]
Using additivity, these bounds imply that $I(\mathcal{N})$ is a strong converse rate and $\widetilde{C}_{\operatorname{EA}}(\mathcal{N})\le I(\mathcal{N})$. The single-letter strong-converse quantity is $\widetilde{I}_{\alpha}(\mathcal{N})$, and the entropy correction is $h_2(\varepsilon)$.
For every quantum channel $\mathcal{N}$, its entanglement-assisted classical capacity $C_{\operatorname{EA}}(\mathcal{N})$ and its strong converse entanglement-assisted classical capacity $\widetilde{C}_{\operatorname{EA}}(\mathcal{N})$ are both equal to the mutual information $I(\mathcal{N})$:
\[
C_{\operatorname{EA}}(\mathcal{N})
= \widetilde{C}_{\operatorname{EA}}(\mathcal{N})
= I(\mathcal{N}).
\]
Here $I(\mathcal{N}) := \sup_{\psi_{RA}} I(R;B)_{\omega}$ with $\omega_{RB}=\mathcal{N}_{A\to B}(\psi_{RA})$. This is the final capacity identity for the information quantity $I(\mathcal{N})$.
Let $\rho$ and $\sigma$ be quantum states on a finite-dimensional system. Denote by $P_e(\rho^{\otimes n}, \sigma^{\otimes n})$ the minimum error probability for distinguishing their $n$-fold tensor powers under equal priors. Then
\[
\lim_{n\to\infty} -\frac{1}{n} \log P_e(\rho^{\otimes n}, \sigma^{\otimes n}) = -\log \min_{0 \le s \le 1} \operatorname{Tr}(\rho^s \sigma^{1-s}).
\]
For finite-dimensional positive semidefinite operators $A$ and $B$ and any real parameter $s$ with $0\le s\le1$, one has $\operatorname{Tr}(A^sB^{1-s})\ge\frac12\operatorname{Tr}(A+B-|A-B|)$, where $|A-B|$ denotes the operator absolute value.
Let $x,b\in\{0,1\}$, where $b=0$ denotes the rectilinear/computational basis and $b=1$ denotes the diagonal/Hadamard basis. Let $\rho_{x,b}$ be the BB84 qubit state prepared by encoding bit $x$ in basis $b$. For the projective measurement in the same basis $b$, write $p(x\mid \rho_{x,b},b)$ for the Born-rule probability assigned to the matching outcome $x$. Then
\[
p(x\mid \rho_{x,b},b)=1.
\]
For states $\rho$ and $\sigma$, let $\psi_\rho$ and $\psi_\sigma$ be the canonical purifications of $\rho$ and $\sigma$, respectively. There exists a reference unitary $U$ such that $F(\rho,\sigma)^2 = |\langle \psi_\rho | U | \psi_\sigma \rangle|^2$, and for every reference unitary $V$, $|\langle \psi_\rho | V | \psi_\sigma \rangle|^2 \le F(\rho,\sigma)^2$.
For every CHSH behavior $p$ with binary outcome signs $a$ and $b$, if $p$ admits a local hidden-variable decomposition, then the CHSH functional $S(p)=\langle a_0b_0\rangle_p+\langle a_0b_1\rangle_p+\langle a_1b_0\rangle_p-\langle a_1b_1\rangle_p$ satisfies $S(p)\le2$.
For a bipartite behavior $p(a,b \mid x,y)$ with inputs $x,y \in \{0,1\}$ and outcomes $a,b \in \{-1,1\}$, the CHSH value is
\[
S(p) = \langle a_0 b_0 \rangle + \langle a_0 b_1 \rangle + \langle a_1 b_0 \rangle - \langle a_1 b_1 \rangle,
\]
where $\langle a_x b_y \rangle := \sum_{a,b} ab \, p(a,b \mid x,y)$ is the correlator for settings $x$ and $y$.
A real rate $R$ is achievable for classical communication over a quantum channel $N$ if, for every $\varepsilon>0$ and every slack $\delta>0$, all sufficiently large blocklengths admit classical message codes over $N^{\otimes n}$ with rate at least $R-\delta$ and maximal error at most $\varepsilon$. The classical capacity is the supremum over all achievable real rates:$$
C(N) := \sup R.
$$
For every finite-dimensional pure tripartite source $\psi_{ABR}$, the fully quantum Slepian-Wolf communication rate $\frac{1}{2} I(A;R)_\psi$ is directly achievable: for every slack $\delta>0$ and error tolerance $\varepsilon>0$, all sufficiently large i.i.d. blocklengths admit FQSW operational surfaces with communication rate at most $\frac{1}{2} I(A;R)_\psi+\delta$, entanglement yield at least $\frac{1}{2} I(A;B)_\psi-\delta$, and normalized trace-distance error at most $\varepsilon$.
Let $A$ and $B$ be positive semidefinite operators with $A \le 1$. The Hayashi-Nagaoka operator inequality (parameter 1) gives $1 - (A+B)^{-1/2}\, A\, (A+B)^{-1/2} \;\le\; 2\,(1 - A) + 4\,B$.
For quantum states $\rho$ and $\sigma$ with equal prior probabilities, the minimum average error probability over binary hypothesis tests is $P^*(\rho,\sigma)=\frac12(1-D(\rho,\sigma))$, where $D(\rho,\sigma)=\frac12\|\rho-\sigma\|_1$ is the normalized trace distance.
For a quantum channel $\mathcal{N}$, the Holevo information $\chi(\mathcal{N})$ is the supremum of the Holevo quantity over all finite input ensembles:$$
\chi(\mathcal{N}) := \sup_{\{p_X(x),\rho_x\}}\left[H\!\left(\sum_x p_X(x)\,\mathcal{N}(\rho_x)\right) - \sum_x p_X(x)\,H\!(\mathcal{N}(\rho_x))\right]\!.
$$
For any ensemble $\mathcal{E}$ of states on a finite-dimensional quantum system $B$, the Holevo information is bounded by
\[
\chi(\mathcal{E}) \le \log \dim B.
\]
For every quantum channel $\mathcal{N}$, its classical capacity is given by the regularized Holevo-information limit$$
C(\mathcal{N}) = \lim_{n\to\infty} \frac{1}{n}\,\chi(\mathcal{N}^{\otimes n}).
$$
Let $p$ be a finite Bell behavior and let $\psi$ be a bipartite target state. Suppose that a chosen finite tensor-product quantum realization reproduces the probability table of $p$ and has shared state $\rho$. If local isometries $V_A$ and $V_B$ satisfy
\[
(V_A\otimes V_B)\rho(V_A\otimes V_B)^\dagger = \psi,
\]
then the chosen realization supplies a target-state realization witness for $p$ and $\psi$.
The statement is existential in the chosen realization; it does not assert that every realization of $p$ extracts $\psi$.
Let $\{p(x),\sigma_x\}_{x\in S}$ be a finite ensemble with average state $\sigma=\sum_x p(x)\sigma_x$, let $P$ be a code subspace projector, let $\{P_x\}_{x\in S}$ be codeword subspace projectors, and let $\varepsilon,d,D$ be real parameters with $D>0$ and $\varepsilon\ge0$. Suppose that for every $x\in S$ the four packing conditions hold: $\operatorname{Tr}\{P\sigma_x\}\ge 1-\varepsilon$, $\operatorname{Tr}\{P_x\sigma_x\}\ge 1-\varepsilon$, $\operatorname{Tr}\{P_x\}\le d$, and $P\sigma P\le D^{-1}P$. Then for any nonempty finite message set $M$, there exist a code choosing one codeword from $S$ for each message and a square-root-measurement decoder POVM whose uniform-message average error is at most $2(\varepsilon + 2\sqrt{\varepsilon}) + 4(|M|-1)\, d/D$.
Let $\rho_{ZE}$ be a subnormalized classical--quantum state on $ZE$, classical on $Z$ with side information $E$, and let $0<\varepsilon<1$. For any $0<\delta<\varepsilon$, set $\varepsilon'=(\varepsilon-\delta)/2$, $\varepsilon''=\sqrt{2\varepsilon-\varepsilon^2}$, and
\[
L := H_{\min}^{\varepsilon'}(Z|E)_{\rho}-2\log_2\frac{1}{\delta}.
\]
In the strict finite-alphabet convention, where output alphabet sizes are integers, the extractable-randomness length satisfies
\[
\log_2 \max\{1,\lfloor 2^L\rfloor\}
\le \log_2 \ell^{\varepsilon}(Z|E)_{\rho}
\le H_{\min}^{\varepsilon''}(Z|E)_{\rho}.
\]
Let $\rho$ be a density operator on a finite-dimensional Hilbert space with von Neumann entropy $H(\rho)$. If $R$ is any achievable Schumacher compression rate for the i.i.d. source $\rho$ under the joint (purification) trace-distance error criterion, then $H(\rho) \le R$.
Let $\rho$ be a density operator on a finite-dimensional Hilbert space with von Neumann entropy $H(\rho) = -\operatorname{Tr}(\rho \log \rho)$. For every $\delta > 0$ and $\varepsilon > 0$ there exists $N \in \mathbb{N}$ such that for all block lengths $n \ge N$, the i.i.d. source $\rho^{\otimes n}$ admits a Schumacher compression code of rate at most $H(\rho) + \delta$ with joint (purification) trace-distance error at most $\varepsilon$. Hence $H(\rho)$ is a directly achievable Schumacher compression rate for $\rho$.
Let $\rho$ be a density operator on a finite-dimensional Hilbert space with von Neumann entropy $H(\rho)$. The optimal asymptotic Schumacher compression rate of the i.i.d. quantum source $\rho$, defined as the infimum over all achievable compression rates $R$, equals $H(\rho)$.
Let $\rho$ be a density operator on a finite-dimensional Hilbert space with von Neumann entropy $H(\rho) = -\operatorname{Tr}(\rho \log \rho)$. If a typical-compression witness family supplies, for every $\delta, \varepsilon > 0$, an $N \in \mathbb{N}$ such that every blocklength $n \ge N$ has a Schumacher compression code for $\rho^{\otimes n}$ with rate at most $H(\rho)+\delta$ and error at most $\varepsilon$, then $H(\rho)$ is a directly achievable Schumacher compression rate for $\rho$. This bridge theorem assumes the typical-compression witness family; unconditional direct achievability is tracked separately.
For every finite-dimensional tensor-product quantum realization of the two-input, two-output CHSH scenario, let $S$ be its CHSH value computed from the four signed outcome-probability correlators. Then
\[
S \le 2\sqrt{2}.
\]
No matching results.
Entanglement-assisted classical communication / Definition
One-Shot Entanglement-Assisted Classical Capacity
Given a quantum channel $\mathcal{N}_{A\to B}$ and $\varepsilon\in[0,1]$, the one-shot $\varepsilon$-error entanglement-assisted classical capacity $C_{\operatorname{EA}}^{\varepsilon}(\mathcal{N})$ is
\[
C_{\operatorname{EA}}^{\varepsilon}(\mathcal{N})
:= \sup_{(\mathcal{M},\Psi,\mathcal{E},\mathcal{D})}
\{\log_2 |\mathcal{M}| :
p_{\mathrm{err}}^{*}((\Psi,\mathcal{E},\mathcal{D});\mathcal{N})\le \varepsilon\}.
\]
The supremum ranges over one-shot entanglement-assisted classical communication protocols with message set $\mathcal{M}$, shared state $\Psi_{A'B'}$, encoding channel $\mathcal{E}_{M'A'\to A}$, and decoding channel $\mathcal{D}_{BB'\to\widehat{M}}$, with message and estimate registers indexed by the message set.
Entanglement-assisted classical communication / Definition
Entanglement-Assisted Classical Capacity of a Quantum Channel
Given a quantum channel $\mathcal{N}$, a rate $R\in\mathbb{R}^{+}$ is achievable for entanglement-assisted classical communication over $\mathcal{N}$ if, for all $\varepsilon\in(0,1]$, $\delta>0$, and sufficiently large $n$, there exists an $(n,2^{n(R-\delta)},\varepsilon)$ entanglement-assisted classical communication protocol. The capacity $C_{\operatorname{EA}}(\mathcal{N})$ is
\[
C_{\operatorname{EA}}(\mathcal{N}) :=
\sup\{R : R\text{ is an achievable rate for }\mathcal{N}\}.
\]
For a quantum state $\rho_A$ on a finite-dimensional system $A$, its von Neumann entropy is defined as $H(A)_\rho := -\operatorname{Tr}\!\bigl(\rho_A \log_2 \rho_A\bigr).$ Equivalently, it is the Shannon entropy of the eigenvalues of $\rho_A$.
Let $\rho$ and $\sigma$ be positive semi-definite operators on a finite-dimensional system, with nonzero $\rho$. Define $D(\rho\|\sigma)$, with trace denominator $\operatorname{Tr}\rho$, by
\[
D(\rho\|\sigma):=
\begin{cases}
\dfrac{\operatorname{Tr}\!\left(\rho(\log_2\rho-\log_2\sigma)\right)}{\operatorname{Tr}\rho},&
\operatorname{supp}\rho\subseteq\operatorname{supp}\sigma,\\
+\infty,&\text{otherwise.}
\end{cases}
\]
For a quantum state $\rho$ on a bipartite system $AB$, the conditional von Neumann entropy of $A$ given $B$ is defined as $H(A|B)_\rho := H(AB)_\rho - H(B)_\rho$.
For a quantum state $\rho$ on a tripartite system $ABC$, the conditional mutual information between $A$ and $C$ given $B$ is defined as $I(A : C \mid B)_\rho := H(AB)_\rho + H(BC)_\rho - H(B)_\rho - H(ABC)_\rho$.
Entanglement-assisted classical communication / Theorem
One-Shot Lower Bounds for Entanglement-Assisted Classical Communication
Let $\mathcal{N}_{A\to B}$ be a quantum channel. For all $\varepsilon\in(0,1)$ and $\eta\in(0,\varepsilon)$, there exists an $(|\mathcal{M}|,\varepsilon)$ entanglement-assisted classical communication protocol over $\mathcal{N}_{A\to B}$ such that
\[
\log_2 |\mathcal{M}|
= \overline{I}_{H}^{\varepsilon-\eta}(\mathcal{N})
- \log_2\!\left(\frac{4\varepsilon}{\eta^2}\right).
\]
Consequently $C_{\operatorname{EA}}^{\varepsilon}(\mathcal{N})\ge \overline{I}_{H}^{\varepsilon-\eta}(\mathcal{N})-\log_2(4\varepsilon/\eta^2)$. Moreover, for all $\alpha\in(0,1)$, there exists an $(|\mathcal{M}|,\varepsilon)$ protocol satisfying
\[
\log_2 |\mathcal{M}|
\ge \overline{I}_{\alpha}(\mathcal{N})
- \frac{\alpha}{1-\alpha}\log_2\!\left(\frac{1}{\varepsilon-\eta}\right)
- \log_2\!\left(\frac{4\varepsilon}{\eta^2}\right).
\] The hypothesis-testing quantity is denoted inline by $I_H$, and the Renyi parameter appears in $\overline{I}_{\alpha}$.
Let $\rho$ be a quantum state, let $\sigma$ be a positive semi-definite operator, and let $\mathcal{N}$ be a quantum channel. With $D(\rho\|\sigma)$ understood as the quantum relative entropy defined by $D(\rho\|\sigma)=\operatorname{Tr}\!\left[\rho(\log_2 \rho-\log_2 \sigma)\right]$ when the support of $\rho$ is contained in the support of $\sigma$, and as $+\infty$ otherwise, one has $D(\rho\|\sigma) \ge D(\mathcal{N}(\rho)\|\mathcal{N}(\sigma))$.
$sigma$
Positive semi-definite reference operator in the second argument of quantum relative entropy.
For any bipartite quantum state $\rho_{AB}$ and local quantum channels $\mathcal{N}_{A \to A'}$ and $\mathcal{N}_{B \to B'}$, let $\omega_{A'B'} := (\mathcal{N}_{A \to A'} \otimes \mathcal{N}_{B \to B'})(\rho_{AB})$. Then, the mutual information satisfies $I(A; B)_\rho \ge I(A'; B')_\omega$.
For any ensemble of bipartite quantum states $\{p(x), \rho_{AB}^x\}$, let $\omega_{AB} := \sum_x p(x)\rho_{AB}^x$. Then, the conditional von Neumann entropy is concave: $H(A \mid B)_\omega \ge \sum_x p(x)H(A \mid B)_{\rho^x}$.
For bipartite quantum states $\rho$ and $\sigma$ on a finite-dimensional system $AB$ with $\frac{1}{2}\|\rho - \sigma\|_1 \le \varepsilon \le 1$, the conditional von Neumann entropy satisfies $|H(A \mid B)_\rho - H(A \mid B)_\sigma| \le 2\varepsilon \log |A| + (1+\varepsilon)h\left(\frac{\varepsilon}{1+\varepsilon}\right)$.
Let $\rho$ be a quantum state, $\sigma$ a positive semidefinite operator, and $\mathcal{N}$ a quantum channel. For $\frac{1}{2} \le \alpha < 1$ or $\alpha > 1$, the sandwiched Renyi relative entropy satisfies $\widetilde{D}_\alpha(\rho \| \sigma) \ge \widetilde{D}_\alpha(\mathcal{N}(\rho) \| \mathcal{N}(\sigma))$.
Let $\rho_{ABC}$ be a pure subnormalized state and let $0 \le \varepsilon < \sqrt{\operatorname{Tr}\rho}$. Then $H_{\max}^{\varepsilon}(A|B)_{\rho} = -H_{\min}^{\varepsilon}(A|C)_{\rho}$.
Let $\rho_{AB}$ be a bipartite state on finite-dimensional systems $A$ and $B$, let $0<\varepsilon<1$, let $\eta$ be the finite-AEP parameter associated to $\rho_{AB}$, and let $n\in\mathbb{N}$ be such that $\rho_{AB}^{\otimes n}$ is the i.i.d. state on $A^nB^n$. If $n\ge \frac85\log(2/\varepsilon^2)$, then the conditional entropy quantity $H$ satisfies\[\frac1n H_{\min}^{\varepsilon}(A^n|B^n)_{\rho^{\otimes n}} \ge H(A|B)_{\rho}-\frac{\delta(\varepsilon,\eta)}{\sqrt n},\]where $\delta(\varepsilon,\eta):=4\log\eta\sqrt{\log(2/\varepsilon^2)}$.
Let $\rho_{AB}$ be a bipartite state on finite-dimensional systems $A$ and $B$, and let $\varepsilon>0$ be the smoothing parameter. For each $n\in\mathbb{N}$, write $\rho_{AB}^{\otimes n}$ for the i.i.d. state on $A^nB^n$. Then, for the entropy quantities $H_{\min}^{\varepsilon}$, $H_{\max}^{\varepsilon}$, and $H$,\[\lim_{\varepsilon\to0}\lim_{n\to\infty}\frac1n H_{\min}^{\varepsilon}(A^n|B^n)_{\rho^{\otimes n}}=H(A|B)_\rho,\qquad\lim_{\varepsilon\to0}\lim_{n\to\infty}\frac1n H_{\max}^{\varepsilon}(A^n|B^n)_{\rho^{\otimes n}}=H(A|B)_\rho.\]
Entanglement-assisted classical communication / Bound
One-Shot Upper Bounds for Entanglement-Assisted Classical Communication
Let $\mathcal{N}_{A\to B}$ be a quantum channel and let $\varepsilon\in[0,1)$. For every $(|\mathcal{M}|,\varepsilon)$ entanglement-assisted classical communication protocol over $\mathcal{N}$,
\[
\log_2 |\mathcal{M}| \le
\frac{1}{1-\varepsilon}\bigl(I(\mathcal{N})+h_2(\varepsilon)\bigr),
\]
and, for all $\alpha>1$,
\[
\log_2 |\mathcal{M}| \le
\widetilde{I}_{\alpha}(\mathcal{N})
+ \frac{\alpha}{\alpha-1}\log_2\!\left(\frac{1}{1-\varepsilon}\right).
\]
Consequently the same two upper bounds hold for $C_{\operatorname{EA}}^{\varepsilon}(\mathcal{N})$. The two information quantities are $I(\mathcal{N})$ and $\widetilde{I}_{\alpha}(\mathcal{N})$, and the binary entropy term is $h_2(\varepsilon)$.
Entanglement-assisted classical communication / Theorem
Lower Bound for Entanglement-Assisted Classical Communication in Asymptotic Setting
Let $\mathcal{N}$ be a quantum channel. Let $I$ denote the channel mutual information functional and let $C$ denote entanglement-assisted classical capacity. The channel mutual information $I(\mathcal{N})$ is an achievable entanglement-assisted classical communication rate. Consequently,
\[
C_{\operatorname{EA}}(\mathcal{N}) \ge I(\mathcal{N}).
\]
Let $\phi^{AA'}$ be a pure state whose marginal $\phi^A$ has full rank $\dim A$, and let $W^{A'\to BE}$ be a Stinespring isometry for a channel $N^{A'\to B}$ with environment $E$. For a subspace $R\subseteq A$ with projection $P$, choose a Haar-random unitary $U$ on $A$, set $V=\sqrt{\dim A/\dim R}\,PU$, and let $\psi^{RBE}$ be the resulting non-normalized state after applying $V$ and then $W$. Writing $\phi^{AE}=\operatorname{Tr}_B(W\phi^{AA'}W^*)$ and $\pi^R$ for the maximally mixed state on $R$, \[\mathbb{E}_U\bigl\|\psi^{RE}-\pi^R\otimes\phi^E\bigr\|_1 \le \sqrt{\dim R\,\dim E\,\operatorname{Tr}[(\phi^{AE})^2]},\] where $\|\cdot\|_1$ is the trace norm.
Entanglement-assisted classical communication / Bound
Upper Bounds for Entanglement-Assisted Classical Communication in Asymptotic Setting
Let $\mathcal{N}$ be a quantum channel. For all $\varepsilon\in[0,1)$, $n\in\mathbb{N}$, and every $(n,|\mathcal{M}|,\varepsilon)$ entanglement-assisted classical communication protocol over $n$ uses of $\mathcal{N}$,
\[
\frac{1}{n}\log_2 |\mathcal{M}|
\le \frac{1}{1-\varepsilon}\left(\frac{1}{n}I(\mathcal{N}^{\otimes n})+\frac{1}{n}h_2(\varepsilon)\right),
\]
and, for all $\alpha>1$,
\[
\frac{1}{n}\log_2 |\mathcal{M}|
\le \frac{1}{n}\widetilde{I}_{\alpha}(\mathcal{N}^{\otimes n})
+ \frac{\alpha}{n(\alpha-1)}\log_2\!\left(\frac{1}{1-\varepsilon}\right).
\]
Using additivity, these bounds imply that $I(\mathcal{N})$ is a strong converse rate and $\widetilde{C}_{\operatorname{EA}}(\mathcal{N})\le I(\mathcal{N})$. The single-letter strong-converse quantity is $\widetilde{I}_{\alpha}(\mathcal{N})$, and the entropy correction is $h_2(\varepsilon)$.
Entanglement-assisted classical communication / Theorem
Entanglement-Assisted Classical Capacity
For every quantum channel $\mathcal{N}$, its entanglement-assisted classical capacity $C_{\operatorname{EA}}(\mathcal{N})$ and its strong converse entanglement-assisted classical capacity $\widetilde{C}_{\operatorname{EA}}(\mathcal{N})$ are both equal to the mutual information $I(\mathcal{N})$:
\[
C_{\operatorname{EA}}(\mathcal{N})
= \widetilde{C}_{\operatorname{EA}}(\mathcal{N})
= I(\mathcal{N}).
\]
Here $I(\mathcal{N}) := \sup_{\psi_{RA}} I(R;B)_{\omega}$ with $\omega_{RB}=\mathcal{N}_{A\to B}(\psi_{RA})$. This is the final capacity identity for the information quantity $I(\mathcal{N})$.
Let $\rho$ and $\sigma$ be quantum states on a finite-dimensional system. Denote by $P_e(\rho^{\otimes n}, \sigma^{\otimes n})$ the minimum error probability for distinguishing their $n$-fold tensor powers under equal priors. Then
\[
\lim_{n\to\infty} -\frac{1}{n} \log P_e(\rho^{\otimes n}, \sigma^{\otimes n}) = -\log \min_{0 \le s \le 1} \operatorname{Tr}(\rho^s \sigma^{1-s}).
\]
For finite-dimensional positive semidefinite operators $A$ and $B$ and any real parameter $s$ with $0\le s\le1$, one has $\operatorname{Tr}(A^sB^{1-s})\ge\frac12\operatorname{Tr}(A+B-|A-B|)$, where $|A-B|$ denotes the operator absolute value.
Let $x,b\in\{0,1\}$, where $b=0$ denotes the rectilinear/computational basis and $b=1$ denotes the diagonal/Hadamard basis. Let $\rho_{x,b}$ be the BB84 qubit state prepared by encoding bit $x$ in basis $b$. For the projective measurement in the same basis $b$, write $p(x\mid \rho_{x,b},b)$ for the Born-rule probability assigned to the matching outcome $x$. Then
\[
p(x\mid \rho_{x,b},b)=1.
\]
For states $\rho$ and $\sigma$, let $\psi_\rho$ and $\psi_\sigma$ be the canonical purifications of $\rho$ and $\sigma$, respectively. There exists a reference unitary $U$ such that $F(\rho,\sigma)^2 = |\langle \psi_\rho | U | \psi_\sigma \rangle|^2$, and for every reference unitary $V$, $|\langle \psi_\rho | V | \psi_\sigma \rangle|^2 \le F(\rho,\sigma)^2$.
For every CHSH behavior $p$ with binary outcome signs $a$ and $b$, if $p$ admits a local hidden-variable decomposition, then the CHSH functional $S(p)=\langle a_0b_0\rangle_p+\langle a_0b_1\rangle_p+\langle a_1b_0\rangle_p-\langle a_1b_1\rangle_p$ satisfies $S(p)\le2$.
For a bipartite behavior $p(a,b \mid x,y)$ with inputs $x,y \in \{0,1\}$ and outcomes $a,b \in \{-1,1\}$, the CHSH value is
\[
S(p) = \langle a_0 b_0 \rangle + \langle a_0 b_1 \rangle + \langle a_1 b_0 \rangle - \langle a_1 b_1 \rangle,
\]
where $\langle a_x b_y \rangle := \sum_{a,b} ab \, p(a,b \mid x,y)$ is the correlator for settings $x$ and $y$.
A real rate $R$ is achievable for classical communication over a quantum channel $N$ if, for every $\varepsilon>0$ and every slack $\delta>0$, all sufficiently large blocklengths admit classical message codes over $N^{\otimes n}$ with rate at least $R-\delta$ and maximal error at most $\varepsilon$. The classical capacity is the supremum over all achievable real rates:$$
C(N) := \sup R.
$$
For every finite-dimensional pure tripartite source $\psi_{ABR}$, the fully quantum Slepian-Wolf communication rate $\frac{1}{2} I(A;R)_\psi$ is directly achievable: for every slack $\delta>0$ and error tolerance $\varepsilon>0$, all sufficiently large i.i.d. blocklengths admit FQSW operational surfaces with communication rate at most $\frac{1}{2} I(A;R)_\psi+\delta$, entanglement yield at least $\frac{1}{2} I(A;B)_\psi-\delta$, and normalized trace-distance error at most $\varepsilon$.
Let $A$ and $B$ be positive semidefinite operators with $A \le 1$. The Hayashi-Nagaoka operator inequality (parameter 1) gives $1 - (A+B)^{-1/2}\, A\, (A+B)^{-1/2} \;\le\; 2\,(1 - A) + 4\,B$.
For quantum states $\rho$ and $\sigma$ with equal prior probabilities, the minimum average error probability over binary hypothesis tests is $P^*(\rho,\sigma)=\frac12(1-D(\rho,\sigma))$, where $D(\rho,\sigma)=\frac12\|\rho-\sigma\|_1$ is the normalized trace distance.
For a quantum channel $\mathcal{N}$, the Holevo information $\chi(\mathcal{N})$ is the supremum of the Holevo quantity over all finite input ensembles:$$
\chi(\mathcal{N}) := \sup_{\{p_X(x),\rho_x\}}\left[H\!\left(\sum_x p_X(x)\,\mathcal{N}(\rho_x)\right) - \sum_x p_X(x)\,H\!(\mathcal{N}(\rho_x))\right]\!.
$$
For any ensemble $\mathcal{E}$ of states on a finite-dimensional quantum system $B$, the Holevo information is bounded by
\[
\chi(\mathcal{E}) \le \log \dim B.
\]
For every quantum channel $\mathcal{N}$, its classical capacity is given by the regularized Holevo-information limit$$
C(\mathcal{N}) = \lim_{n\to\infty} \frac{1}{n}\,\chi(\mathcal{N}^{\otimes n}).
$$
Let $p$ be a finite Bell behavior and let $\psi$ be a bipartite target state. Suppose that a chosen finite tensor-product quantum realization reproduces the probability table of $p$ and has shared state $\rho$. If local isometries $V_A$ and $V_B$ satisfy
\[
(V_A\otimes V_B)\rho(V_A\otimes V_B)^\dagger = \psi,
\]
then the chosen realization supplies a target-state realization witness for $p$ and $\psi$.
The statement is existential in the chosen realization; it does not assert that every realization of $p$ extracts $\psi$.
Let $\{p(x),\sigma_x\}_{x\in S}$ be a finite ensemble with average state $\sigma=\sum_x p(x)\sigma_x$, let $P$ be a code subspace projector, let $\{P_x\}_{x\in S}$ be codeword subspace projectors, and let $\varepsilon,d,D$ be real parameters with $D>0$ and $\varepsilon\ge0$. Suppose that for every $x\in S$ the four packing conditions hold: $\operatorname{Tr}\{P\sigma_x\}\ge 1-\varepsilon$, $\operatorname{Tr}\{P_x\sigma_x\}\ge 1-\varepsilon$, $\operatorname{Tr}\{P_x\}\le d$, and $P\sigma P\le D^{-1}P$. Then for any nonempty finite message set $M$, there exist a code choosing one codeword from $S$ for each message and a square-root-measurement decoder POVM whose uniform-message average error is at most $2(\varepsilon + 2\sqrt{\varepsilon}) + 4(|M|-1)\, d/D$.
Let $\rho_{ZE}$ be a subnormalized classical--quantum state on $ZE$, classical on $Z$ with side information $E$, and let $0<\varepsilon<1$. For any $0<\delta<\varepsilon$, set $\varepsilon'=(\varepsilon-\delta)/2$, $\varepsilon''=\sqrt{2\varepsilon-\varepsilon^2}$, and
\[
L := H_{\min}^{\varepsilon'}(Z|E)_{\rho}-2\log_2\frac{1}{\delta}.
\]
In the strict finite-alphabet convention, where output alphabet sizes are integers, the extractable-randomness length satisfies
\[
\log_2 \max\{1,\lfloor 2^L\rfloor\}
\le \log_2 \ell^{\varepsilon}(Z|E)_{\rho}
\le H_{\min}^{\varepsilon''}(Z|E)_{\rho}.
\]
Let $\rho$ be a density operator on a finite-dimensional Hilbert space with von Neumann entropy $H(\rho)$. If $R$ is any achievable Schumacher compression rate for the i.i.d. source $\rho$ under the joint (purification) trace-distance error criterion, then $H(\rho) \le R$.
Let $\rho$ be a density operator on a finite-dimensional Hilbert space with von Neumann entropy $H(\rho) = -\operatorname{Tr}(\rho \log \rho)$. For every $\delta > 0$ and $\varepsilon > 0$ there exists $N \in \mathbb{N}$ such that for all block lengths $n \ge N$, the i.i.d. source $\rho^{\otimes n}$ admits a Schumacher compression code of rate at most $H(\rho) + \delta$ with joint (purification) trace-distance error at most $\varepsilon$. Hence $H(\rho)$ is a directly achievable Schumacher compression rate for $\rho$.
Schumacher quantum data-compression limit equality
Let $\rho$ be a density operator on a finite-dimensional Hilbert space with von Neumann entropy $H(\rho)$. The optimal asymptotic Schumacher compression rate of the i.i.d. quantum source $\rho$, defined as the infimum over all achievable compression rates $R$, equals $H(\rho)$.
Let $\rho$ be a density operator on a finite-dimensional Hilbert space with von Neumann entropy $H(\rho) = -\operatorname{Tr}(\rho \log \rho)$. If a typical-compression witness family supplies, for every $\delta, \varepsilon > 0$, an $N \in \mathbb{N}$ such that every blocklength $n \ge N$ has a Schumacher compression code for $\rho^{\otimes n}$ with rate at most $H(\rho)+\delta$ and error at most $\varepsilon$, then $H(\rho)$ is a directly achievable Schumacher compression rate for $\rho$. This bridge theorem assumes the typical-compression witness family; unconditional direct achievability is tracked separately.
Tsirelson upper bound for CHSH quantum realizations
For every finite-dimensional tensor-product quantum realization of the two-input, two-output CHSH scenario, let $S$ be its CHSH value computed from the four signed outcome-probability correlators. Then
\[
S \le 2\sqrt{2}.
\]