Definitions for Quantum and private capacities of low-noise channels

QAlgorithms.Papers.Independent.Intakethird100.P50.LowNoiseCapacityBounds

For finite types $A,B,E$ with decidable equality, any complex matrix $V:(A\to B\times E)$, and arbitrary real numbers $\varepsilon,C_q,C_p$, define $\Phi(X)=\operatorname{Tr}_E(VXV^\dagger)$, $\Phi^c(X)=\operatorname{Tr}_B(VXV^\dagger)$, and $e=\operatorname{rank}J$, where $J_{(a,b),(a',b')}=[\Phi(|a\rangle\langle a'|)]_{b,b'}$. Put $S(X)=-\operatorname{Re}\operatorname{tr}[f(X)]$, where $f(x)=x\log_2x$ is applied by functional calculus, and put

$$I_c=\sup_{\rho\ge0,\ \operatorname{tr}\rho=1} \bigl(S(\Phi(\rho))-S(\Phi^c(\rho))\bigr).$$

Let $Q$ be the real supremum of nonnegative rates $r$ admitting a sequence, indexed by every natural $n$, of positive message dimensions $M_n$ and encoding and decoding channels $\mathcal E_n,\mathcal D_n$, with

$$\operatorname{Re}\operatorname{tr}\!\left[ \Omega_{M_n}(\mathrm{id}\otimes(\mathcal D_n\circ\Phi^{\otimes n}\circ\mathcal E_n))(\Omega_{M_n}) \right]\to1, \qquad \forall\delta>0,\quad r-\delta\le\frac{\log_2M_n}{n}\ \text{eventually},$$

where $\Omega_M=M^{-1}\sum_{i,j}|i,i\rangle\langle j,j|$. Let $P$ be the real supremum of nonnegative rates $r$ such that, for every $\epsilon_0>0$ and slack $s>0$, there is a positive $n_0$ such that every $n\ge n_0$ admits $M>0$, density matrices $\rho_m$ on $A^n$, and positive semidefinite measurement matrices $D_j$ on $B^n$ satisfying $\sum_jD_j=I$, for which $r-s\le\log_2M/n$ and $\|\omega-\mathcal K_{M,\sigma}\|_1<\epsilon_0$. In this condition, $V_n(b,e;a)=\prod_{i=0}^{n-1}V(b_i,e_i;a_i)$,

$$\omega_{(m,j,e),(m',j',e')} = \begin{cases} \displaystyle\frac1M\sum_{b,c}D_j(c,b) (V_n\rho_mV_n^\dagger)_{(b,e),(c,e')}, &m=m',\ j=j',\\ 0,&\text{otherwise}, \end{cases} \quad \sigma=\operatorname{Tr}_{m,j}\omega, \quad \mathcal K_{M,\sigma}=\frac1M\sum_m|m,m\rangle\langle m,m|\otimes\sigma.$$

A density matrix is positive semidefinite with trace $1$; a channel has a finite Kraus representation $X\mapsto\sum_jK_jXK_j^\dagger$ with $\sum_jK_j^\dagger K_j=I$; partial traces sum diagonal indices of the discarded registers; $\|X\|_1=\operatorname{Re}\operatorname{tr}\sqrt{X^\dagger X}$; and tensor powers act on matrix units by $\Phi^{\otimes n}(|x\rangle\langle y|)=\bigotimes_i\Phi(|x_i\rangle\langle y_i|)$. With these quantities, the proposition defined here is $Q=I_c$ and $P=I_c$ when $e=1$ or $\varepsilon=0$; otherwise it is the conjunction

$$\begin{aligned} |Q-I_c| &\le -3\varepsilon^{3/2}\log_2\varepsilon +2\left(\frac1{\ln2}+\log_2e+\frac12\log_2(e-1)\right)\varepsilon^{3/2} +C_q\varepsilon^{9/2},\\ |P-I_c| &\le -9\varepsilon^{3/2}\log_2\varepsilon +2\left(\frac3{\ln2}+4\log_2e+\log_2(e-1)\right)\varepsilon^{3/2} +C_p\varepsilon^3. \end{aligned}$$

No isometry condition on $V$, nonemptiness condition on the registers, or sign condition on $\varepsilon,C_q,C_p$ is imposed. Real suprema use their total convention, including $0$ for an empty defining set; $\log_2x=\ln x/\ln2$ uses $\ln0=0$ and the absolute-value convention for negative arguments; fractional powers are total real powers. Zero-length tensor products and division by zero also use their total definitions.

QAlgorithms.Papers.Independent.Intakethird100.P50.PolynomialNoise

For three real polynomials $P_0,P_1,P_2$, this predicate means that $P_i(0)=0$ for every $i\in\{0,1,2\}$, and that there exists a real $\delta>0$ such that, for every real $p\in[0,\delta]$, all four entries of

$$w(p)=\left(1-\sum_{i=0}^{2}P_i(p),\ P_0(p),\ P_1(p),\ P_2(p)\right)$$

are nonnegative and $\sum_{k=0}^{3}w_k(p)=1$. Both endpoints of the interval are included; no condition is required outside that interval.

QAlgorithms.Papers.Independent.Intakethird100.P50.Probabilities

For a finite type $I$ and a function $w:I\to\mathbb R$, this predicate means that $w_i\ge 0$ for every $i\in I$ and $\sum_{i\in I}w_i=1$. If $I$ is empty, the sum is $0$, so the predicate cannot hold.

QAlgorithms.Papers.Independent.Intakethird100.P50.channelDifference

For arbitrary types $A,B$ and arbitrary maps $\Phi,\Psi$ from complex $A\times A$ matrices to complex $B\times B$ matrices, their defined difference is the map

$$X\longmapsto\Phi(X)-\Psi(X).$$

There are no assumptions of linearity, positivity, normalization, finiteness, or nonemptiness.

QAlgorithms.Papers.Independent.Intakethird100.P50.composedDilation

For types $A,B,E,F$, with $B$ finite so that the following sum is defined, and arbitrary complex matrices $V$ with rows $B\times E$ and columns $A$, and $W$ with rows $A\times F$ and columns $B$, this defines the complex matrix $Z$ with rows $A\times(E\times F)$ and columns $A$ by

$$Z_{(a_{\mathrm{out}},(e,f)),a_{\mathrm{in}}} =\sum_{b\in B}W_{(a_{\mathrm{out}},f),b}\,V_{(b,e),a_{\mathrm{in}}}.$$

Neither input matrix is assumed to be an isometry. Empty index types are allowed; in particular, if $B$ is empty, every entry of $Z$ is zero.

QAlgorithms.Papers.Independent.Intakethird100.P50.degradability

For finite types $A,B,E$ with decidable equality and any complex matrix $V:(A\to B\times E)$, put $\Phi_V(X)=\operatorname{Tr}_E(VXV^\dagger)$ and $\Phi_V^c(X)=\operatorname{Tr}_B(VXV^\dagger)$. This defines

$$\inf\left\{\eta\in[0,2]: \exists\mathcal D,\quad \mathcal D(X)=\sum_jK_jXK_j^\dagger,\quad \sum_jK_j^\dagger K_j=I_B,\quad \|\Phi_V^c-\mathcal D\circ\Phi_V\|_\diamond\le\eta \right\},$$

where the Kraus family is finite, each $K_j$ has rows indexed by $E$ and columns indexed by $B$, and the particular norm used is

$$\|\Lambda\|_\diamond = \sup_{\substack{X\text{ on }A\times A\\\|X\|_1\le1}} \|(\mathrm{id}_A\otimes\Lambda)(X)\|_1, \qquad \|X\|_1=\operatorname{Re}\operatorname{tr}\sqrt{X^\dagger X}.$$

The square root is defined by functional calculus, and $\mathrm{id}_A\otimes\Lambda$ applies $\Lambda$ to each reference-indexed matrix block. The two partial traces sum the corresponding diagonal register indices. No isometry condition on $V$ is assumed, and the set of admissible $\eta$ need not be nonempty; an empty defining set gives value $0$ under the real infimum convention.

QAlgorithms.Papers.Independent.Intakethird100.P50.depolarizingDilation

For every real $p$, this defines a complex matrix with entries

$$V_{(b,k),a}=\sqrt{w_k}\,(P_k)_{b,a}, \qquad (w_0,w_1,w_2,w_3)=\left(1-p,\frac p3,\frac p3,\frac p3\right),$$

where $a,b\in\{0,1\}$, $k\in\{0,1,2,3\}$, and

$$P_0=\begin{pmatrix}1&0\\0&1\end{pmatrix},\quad P_1=\begin{pmatrix}0&1\\1&0\end{pmatrix},\quad P_2=\begin{pmatrix}0&-\mathrm i\\\mathrm i&0\end{pmatrix},\quad P_3=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$

There is no restriction $0\le p\le1$; any negative weight contributes a zero square-root factor.

QAlgorithms.Papers.Independent.Intakethird100.P50.depolarizingHashing

For every real $p$, this defines the real number

$$1-\frac{-p\log p-(1-p)\log(1-p)}{\log2} -p\,\frac{\log3}{\log2}.$$

Here $\log$ is the total real logarithm used by the code: $\log0=0$, and for negative arguments it equals the logarithm of the absolute value. Thus the definition includes $p=0$, $p=1$, and values outside $[0,1]$.

QAlgorithms.Papers.Independent.Intakethird100.P50.dilationChoiRank

For finite types $A,B,E$, with decidable equality on $A$, and any complex matrix $V:(A\to B\times E)$, this defines the matrix rank of $J_V$, indexed by $A\times B$, whose entries are

$$(J_V)_{(a,b),(a',b')} = \left[\operatorname{Tr}_E\!\left(V|a\rangle\langle a'|V^\dagger\right)\right]_{b,b'}.$$

Here $|a\rangle\langle a'|$ is the matrix with its only nonzero entry, equal to $1$, at $(a,a')$, and $(\operatorname{Tr}_EY)_{b,b'}=\sum_eY_{(b,e),(b',e)}$. There is no normalization factor in $J_V$, and no isometry or nonemptiness hypothesis is imposed on $V$ or the registers; empty indexing cases therefore include rank $0$.

QAlgorithms.Papers.Independent.Intakethird100.P50.dilationQuantumCapacity

For finite types $A,B,E$, with decidable equality on $A$ and $B$, and any complex matrix $V$ with rows indexed by $B\times E$ and columns indexed by $A$, define $\Phi_V(X)=\operatorname{Tr}_E(VXV^\dagger)$, where $(\operatorname{Tr}_E Y)_{b,b'}=\sum_eY_{(b,e),(b',e)}$. The value defined here is the real supremum of all $r\ge0$ for which there is, for every natural number $n$, a positive integer $M_n$, an encoding channel $\mathcal E_n$ from $M_n$-dimensional matrices to matrices on $A^n$, and a decoding channel $\mathcal D_n$ from matrices on $B^n$ to $M_n$-dimensional matrices, such that

$$\operatorname{Re}\operatorname{tr}\!\left[ \Omega_{M_n}\, (\mathrm{id}\otimes(\mathcal D_n\circ\Phi_V^{\otimes n}\circ\mathcal E_n))(\Omega_{M_n}) \right]\longrightarrow1, \qquad \forall\delta>0,\quad r-\delta\le\frac{\log_2M_n}{n}\ \text{eventually}.$$

Here $A^n$ means the functions $\{0,\ldots,n-1\}\to A$, $\Omega_M=M^{-1}\sum_{i,j=0}^{M-1}|i,i\rangle\langle j,j|$, and a channel means a map $X\mapsto\sum_jK_jXK_j^\dagger$ with finitely many Kraus matrices satisfying $\sum_jK_j^\dagger K_j=I$. The tensor power is determined by $\Phi_V^{\otimes n}(|x\rangle\langle x'|)=\bigotimes_i\Phi_V(|x_i\rangle\langle x'_i|)$, and $\mathrm{id}\otimes\Phi$ applies $\Phi$ to each matrix block indexed by the reference register. The logarithm is $\log_2x=\ln x/\ln2$. The sequence includes $n=0$, with the empty tensor product and total division $x/0=0$, although the conditions concern $n\to\infty$. No isometry condition on $V$ is assumed; the supremum is taken in $\mathbb R$, with its total supremum convention, including value $0$ for the empty set.

QAlgorithms.Papers.Independent.Intakethird100.P50.linearCoefficient

For any three real polynomials $P_0,P_1,P_2$ and any index $i\in\{0,1,2\}$, this defines the coefficient of the degree-one monomial in $P_i$. It is $0$ when that monomial is absent; no noise or probability condition is assumed.

QAlgorithms.Papers.Independent.Intakethird100.P50.maxEntry

For types $A,B$ and any complex matrix $X$ indexed by $A\times B$, this defines

$$\sup\{\,|X_{i,j}|:i\in A,\ j\in B\,\}$$

as a real supremum, where $|\cdot|$ is the complex modulus. For finite nonempty $A,B$, this is the largest entry modulus; if either indexing type is empty, the defining set is empty and the value is $0$. The definition itself imposes no finiteness or boundedness hypothesis and uses the total real supremum operation.

QAlgorithms.Papers.Independent.Intakethird100.P50.modifiedWeights

For any three real polynomials $P_0,P_1,P_2$, any three real numbers $a_0,a_1,a_2$, and any real $p$, set $q_i=P_i(p+a_ip^2)$. This defines the four real weights

$$(1-q_0-q_1-q_2,\ q_0,\ q_1,\ q_2),$$

indexed in that order by $0,1,2,3$. No sign, normalization, or noise hypotheses are imposed on the inputs; individual weights may be negative.

QAlgorithms.Papers.Independent.Intakethird100.P50.pairCoefficient

For any three real polynomials $P_0,P_1,P_2$, let $c_i$ be the coefficient of the degree-one monomial in $P_i$, with absent coefficients equal to $0$. This defines

$$\left|c_0c_1+c_0c_2+c_1c_2\right|.$$

The absolute value applies to the entire sum, and no assumptions are imposed on the polynomials.

QAlgorithms.Papers.Independent.Intakethird100.P50.pauliDilation

For any four real numbers $w_0,w_1,w_2,w_3$, this defines a complex matrix $V$ whose rows are indexed by pairs $(b,k)\in\{0,1\}\times\{0,1,2,3\}$ and whose columns are indexed by $a\in\{0,1\}$, by

$$V_{(b,k),a}=\sqrt{w_k}\,(P_k)_{b,a}, \qquad (P_0,P_1,P_2,P_3)= \left( \begin{pmatrix}1&0\\0&1\end{pmatrix}, \begin{pmatrix}0&1\\1&0\end{pmatrix}, \begin{pmatrix}0&-\mathrm i\\\mathrm i&0\end{pmatrix}, \begin{pmatrix}1&0\\0&-1\end{pmatrix} \right).$$

The real square root is regarded as a complex number. No nonnegativity or normalization assumption is imposed on $w$; the real square root used here is $0$ for negative inputs.

QAlgorithms.Papers.Independent.Intakethird100.P50.pauliKrausOperator

This defines four complex $2\times2$ matrices, indexed by $0,1,2,3$, with rows and columns indexed by $0,1$:

$$P_0=\begin{pmatrix}1&0\\0&1\end{pmatrix},\qquad P_1=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad P_2=\begin{pmatrix}0&-\mathrm i\\ \mathrm i&0\end{pmatrix},\qquad P_3=\begin{pmatrix}1&0\\0&-1\end{pmatrix},$$

where $\mathrm i^2=-1$.

QAlgorithms.Papers.Independent.Intakethird100.P50.polynomialWeights

For any three real polynomials $P_0,P_1,P_2$ and any real $p$, this defines the four real weights

$$\left(1-P_0(p)-P_1(p)-P_2(p),\ P_0(p),\ P_1(p),\ P_2(p)\right),$$

indexed in that order by $0,1,2,3$. No assumptions on the polynomials or on $p$ are imposed, and the weights need not be nonnegative.

QAlgorithms.Papers.Independent.Intakethird100.P50.positiveCyclicShift

For every natural number $d$, this defines a complex $d\times d$ matrix $S$, with row and column indices $i,j\in\{0,\ldots,d-1\}$, by

$$S_{i,j}= \begin{cases} 1,&i=(j+1)\bmod d,\\ 0,&\text{otherwise}. \end{cases}$$

For $d=0$, the matrix has no rows or columns, so no remainder expression is evaluated at an entry; for $d=1$, it is the $1\times1$ matrix with entry $1$.

QAlgorithms.Papers.Independent.Intakethird100.P50.privateCorrection

For every natural number $e$ and every real number $\eta$, this defines

$$\eta\log_2(e-1)+4\eta\log_2e +2H_2\!\left(\frac{\eta}{2}\right) +4\left(1+\frac{\eta}{2}\right) H_2\!\left(\frac{\eta}{2+\eta}\right),$$

where $\log_2x=\ln x/\ln2$ and $H_2(x)=[-x\ln x-(1-x)\ln(1-x)]/\ln2$, using the total real logarithm. There are no restrictions on $\eta$ and no lower bound on $e$. In particular, $e=0,1$ and $\eta=-2$ are included: $e-1$ is real subtraction, $\ln0=0$, negative logarithm arguments are interpreted through absolute value, and division by zero is $0$.

QAlgorithms.Papers.Independent.Intakethird100.P50.privateLowNoiseLeading

For every natural number $e$ and every real number $\varepsilon$, this defines

$$-9\varepsilon^{3/2}\log_2\varepsilon +2\left( \frac3{\ln2}+4\log_2e+\log_2(e-1) \right)\varepsilon^{3/2}.$$

The exponent $3/2$ uses the total real-power operation, and $\log_2x=\ln x/\ln2$ uses the total real logarithm, including $\ln0=0$ and $\ln x=\ln|x|$ for negative $x$. All real $\varepsilon$ and all natural $e$, including $e=0,1$, are allowed; $e-1$ denotes real subtraction.

QAlgorithms.Papers.Independent.Intakethird100.P50.privatePolynomialLeading

For every natural number $e$ and all real numbers $c,r,p$, this defines

$$3crp^{\,r-1}(-p\log_2p) +cp^r\left( -3\log_2c+3+\frac3{\ln2} +\log_2(e-1)+4\log_2e \right).$$

The logarithm means $\log_2x=\ln x/\ln2$, with $\ln0=0$ and $\ln x=\ln|x|$ for negative $x$; the powers with exponents $r-1$ and $r$ are total real powers. The quantities $r-1$ and $e-1$ are real differences. The definition includes arbitrary real $c,r,p$, including nonpositive values, and every natural $e$, including $0$ and $1$.

QAlgorithms.Papers.Independent.Intakethird100.P50.quantumCorrection

For every natural number $e$ and every real number $\eta$, this defines

$$\frac{\eta}{2}\log_2(e-1)+\eta\log_2e +H_2\!\left(\frac{\eta}{2}\right) +\left(1+\frac{\eta}{2}\right) H_2\!\left(\frac{\eta}{2+\eta}\right),$$

where $\log_2x=\ln x/\ln2$ and $H_2(x)=[-x\ln x-(1-x)\ln(1-x)]/\ln2$, using the total real logarithm. No conditions such as $e\ge2$ or $0\le\eta\le2$ are imposed. The subtraction $e-1$ occurs in $\mathbb R$; logarithms at zero have value $0$, logarithms at negative arguments use $\ln x=\ln|x|$, and division by zero is $0$, including when $\eta=-2$.

QAlgorithms.Papers.Independent.Intakethird100.P50.quantumLowNoiseLeading

For every natural number $e$ and every real number $\varepsilon$, this defines

$$-3\varepsilon^{3/2}\log_2\varepsilon +2\left( \frac1{\ln2}+\log_2e+\frac12\log_2(e-1) \right)\varepsilon^{3/2}.$$

Here $3/2$ is a real exponent and the power is the total real-power operation; $\log_2x=\ln x/\ln2$ uses the total real logarithm, with $\ln0=0$ and $\ln x=\ln|x|$ for negative $x$. There is no nonnegativity or smallness condition on $\varepsilon$, and no restriction excluding $e=0,1$; $e-1$ is computed in $\mathbb R$.

QAlgorithms.Papers.Independent.Intakethird100.P50.quantumPolynomialLeading

For every natural number $e$ and all real numbers $c,r,p$, this defines

$$crp^{\,r-1}(-p\log_2p) +cp^r\left( -\log_2c+1+\frac1{\ln2} +\log_2e+\frac12\log_2(e-1) \right).$$

Here $\log_2x=\ln x/\ln2$, with the total real logarithm, so $\ln0=0$ and $\ln x=\ln|x|$ for negative $x$. The exponents $r-1$ and $r$ use the total real-power operation, and both subtractions $r-1$ and $e-1$ occur in $\mathbb R$. No positivity, integrality, or smallness assumptions are imposed on $c,r,p$, and $e=0,1$ are included.

QAlgorithms.Papers.Independent.Intakethird100.P50.quditPauliDilation

For every natural number $d$ and every real function $w$ on $\{0,\ldots,d-1\}^2$, this defines a complex matrix $V_w$ with columns indexed by $a\in\{0,\ldots,d-1\}$ and rows indexed by $(b,(j,k))\in\{0,\ldots,d-1\}^3$, by

$$(V_w)_{(b,(j,k)),a}=\sqrt{w(j,k)}\,(S_d^jC_d^k)_{b,a}, \qquad (S_d)_{u,v}= \begin{cases}1&u=(v+1)\bmod d,\\0&\text{otherwise},\end{cases} \qquad (C_d)_{u,v}= \begin{cases}e^{2\pi i u/d}&u=v,\\0&u\ne v.\end{cases}$$

The square root is the real square root, subsequently regarded as complex, and the exponents $j,k$ are the natural representatives of the indices. There is no assumption that the weights are nonnegative or sum to one, and no isometry assertion; negative weights contribute zero because the real square root of a negative number is zero. The definition includes $d=0$, when both matrix index sets are empty, and $d=1$.

QAlgorithms.Papers.Independent.Intakethird100.P50.recoveryDegrader

For finite types $A,B,E,F$ and arbitrary complex matrices $V:B\times E\leftarrow A$ and $W:A\times F\leftarrow B$, this defines a map from complex matrices on $B$ to complex matrices on $E$. Set

$$Z_{(a,(e,f)),a'}=\sum_{b\in B}W_{(a,f),b}V_{(b,e),a'}, \qquad \sigma_{a,a'}=\sum_{f\in F}(W\rho W^\dagger)_{(a,f),(a',f)}.$$

Its value at any complex matrix $\rho$ on $B$ is the matrix

$$\mathcal D_{V,W}(\rho)_{e,e'} =\sum_{f\in F}\sum_{a\in A} (Z\sigma Z^\dagger)_{(a,(e,f)),(a,(e',f))}.$$

Thus the definition first traces out $F$ from $W\rho W^\dagger$, applies conjugation by the displayed $Z$, and traces out its $A$ and $F$ coordinates. No isometry, positivity, normalization, or channel hypothesis is imposed on $V,W,\rho$, and the definition itself makes no channel assertion. Empty finite types are included, with empty sums equal to zero.

QAlgorithms.Papers.Independent.Intakethird100.P50.xzDilation

For every real $p$, this defines a complex matrix with entries

$$V_{(b,k),a}=\sqrt{w_k}\,(P_k)_{b,a}, \qquad (w_0,w_1,w_2,w_3)=\bigl((1-p)^2,\ p-p^2,\ p^2,\ p-p^2\bigr),$$

where $a,b\in\{0,1\}$, $k\in\{0,1,2,3\}$, and

$$P_0=\begin{pmatrix}1&0\\0&1\end{pmatrix},\quad P_1=\begin{pmatrix}0&1\\1&0\end{pmatrix},\quad P_2=\begin{pmatrix}0&-\mathrm i\\\mathrm i&0\end{pmatrix},\quad P_3=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$

No restriction on $p$ is imposed; the real square root of a negative weight is taken to be $0$.