Definitions for Quantum Markov chains, sufficiency of quantum channels, and Renyi information measures

QAlgorithms.Papers.Independent.Intakethird100.P52.IsStrictChannel

For finite, possibly empty types $H$ and $K$, with decidable equality on $H$, and an arbitrary function $N:\operatorname{Mat}_H(\mathbb C)\to\operatorname{Mat}_K(\mathbb C)$, this predicate means that $N$ has a finite Kraus representation and sends every positive definite matrix to a positive definite matrix. Explicitly, there exist $m\in\mathbb N$ and matrices $E_j\in\mathbb C^{K\times H}$, indexed by $0\le j<m$, such that $\sum_jE_j^\dagger E_j=I_H$ and $N(X)=\sum_jE_jXE_j^\dagger$ for every complex matrix $X$ on $H$; additionally, $X\succ0$ implies $N(X)\succ0$ for every such $X$. The quantified matrices need not have trace one, and the Kraus family is allowed to be empty.

QAlgorithms.Papers.Independent.Intakethird100.P52.IsSufficient

For finite, possibly empty types $H,K$ with decidable equality, arbitrary complex matrices $\rho,\sigma$ on $H$, and an arbitrary function $N:\operatorname{Mat}_H(\mathbb C)\to\operatorname{Mat}_K(\mathbb C)$, this predicate means that there exists a function $R:\operatorname{Mat}_K(\mathbb C)\to\operatorname{Mat}_H(\mathbb C)$ satisfying $R(N(\rho))=\rho$ and $R(N(\sigma))=\sigma$, where $R$ has a finite Kraus representation: for some $m\in\mathbb N$ and matrices $E_j\in\mathbb C^{H\times K}$, $\sum_jE_j^\dagger E_j=I_K$ and $R(Y)=\sum_jE_jYE_j^\dagger$ for every $Y$. No positivity or normalization is required of $\rho,\sigma$, and no channel or linearity assumption is imposed on $N$.

QAlgorithms.Papers.Independent.Intakethird100.P52.conditionalMaxInformation

For finite, possibly empty types $A,C,B$, each with decidable equality, and any complex matrix $\rho$ on $A\times(C\times B)$, set $\Sigma=\rho_{AC}\otimes I_B$, where $(\rho_{AC})_{(i,c),(j,d)}=\sum_b\rho_{(i,c,b),(j,d,b)}$, and put $D=\operatorname{Tr}_A\Sigma$, $Y=\operatorname{Tr}_A\rho$, with partial trace summing equal $A$-indices. Let $Q(X)$ and $J(X)$ be continuous functional calculus for $\sqrt{x}$ and $(\sqrt{x})^{-1}$, respectively, with inverse zero at zero, and define

$$R_\rho=Q(\Sigma)\left[I_A\otimes J(D)YJ(D)\right]Q(\Sigma).$$

The declaration defines

$$\inf\left\{\lambda\in\mathbb R: 2^\lambda R_\rho-\rho\ \text{is positive semidefinite}\right\}$$

as an extended-real infimum over the embedded real numbers. An empty feasible set gives $+\infty$, and a feasible set unbounded below gives $-\infty$. No positivity or normalization assumption on $\rho$ is required.

QAlgorithms.Papers.Independent.Intakethird100.P52.conditionalMinInformation

For finite, possibly empty types $A,C,B$, each with decidable equality, and any complex matrix $\rho$ on $A\times(C\times B)$, put $\Sigma=\rho_{AC}\otimes I_B$, where $(\rho_{AC})_{(i,c),(j,d)}=\sum_b\rho_{(i,c,b),(j,d,b)}$, and let $D=\operatorname{Tr}_A\Sigma$, $Y=\operatorname{Tr}_A\rho$, with partial trace summing equal $A$-indices. Write $Q(X)$ for continuous functional calculus for $x\mapsto\sqrt{x}$, and $J(X)$ for that for $x\mapsto(\sqrt{x})^{-1}$, whose value at zero is zero, and define

$$R_\rho=Q(\Sigma)\left[I_A\otimes J(D)YJ(D)\right]Q(\Sigma).$$

The declaration defines the extended-real quantity

$$-\ell_2\!\left( \left[\operatorname{Re}\operatorname{Tr} Q\!\left(Q(\rho)R_\rho Q(\rho)\right)\right]^2 \right),$$

where $\ell_2(x)=\log_2x$ for $x>0$ and $-\infty$ for $x\le0$. In particular, a zero squared expression gives $+\infty$. There is no positivity or normalization assumption on $\rho$.

QAlgorithms.Papers.Independent.Intakethird100.P52.conditionalPetzReference

For finite, possibly empty types $A,C,B$, each with decidable equality, and any complex matrix $\rho$ on $A\times(C\times B)$, define $\Sigma=\rho_{AC}\otimes I_B$, where $(\rho_{AC})_{(i,c),(j,d)}=\sum_b\rho_{(i,c,b),(j,d,b)}$, and let $D=\operatorname{Tr}_A\Sigma$, $Y=\operatorname{Tr}_A\rho$, where $(\operatorname{Tr}_A X)_{(c,b),(d,e)}=\sum_iX_{(i,c,b),(i,d,e)}$. If $Q(X)$ denotes continuous functional calculus for $x\mapsto\sqrt{x}$, and $J(X)$ denotes continuous functional calculus for $x\mapsto(\sqrt{x})^{-1}$, the declaration defines the matrix

$$R_\rho=Q(\Sigma)\left[I_A\otimes\bigl(J(D)YJ(D)\bigr)\right]Q(\Sigma).$$

No density-matrix or positivity hypothesis is imposed. The scalar square root and inverse are their total real operations, so the inverse-square-root function takes value zero at zero.

QAlgorithms.Papers.Independent.Intakethird100.P52.conditionalRenyiInformation

For finite types $A,C,B$, each with decidable equality and possibly empty, any real $a$, and any complex matrix $\rho$ indexed by $A\times(C\times B)$, define $\Sigma=\rho_{AC}\otimes I_B$, where

$$(\rho_{AC})_{(i,c),(j,d)}=\sum_b\rho_{(i,c,b),(j,d,b)}.$$

Write $\rho_{CB}=\operatorname{Tr}_A\rho$ and $\Sigma_{CB}=\operatorname{Tr}_A\Sigma$, with $(\operatorname{Tr}_A X)_{(c,b),(d,e)}=\sum_i X_{(i,c,b),(i,d,e)}$. For any square matrix $X$, let $S_t(X)$ denote continuous functional calculus applied to the real function that is $0$ at $x=0$ and $x^t$ otherwise, and set

$$K_a=\! S_{(1-a)/2}(\Sigma) \left[I_A\otimes \left( S_{(a-1)/2}(\Sigma_{CB})S_{1-a}(\rho_{CB})S_{(a-1)/2}(\Sigma_{CB}) \right)\right] S_{(1-a)/2}(\Sigma).$$

The declaration defines the extended-real value

$$\frac{1}{a-1}\, \ell_2\!\left(\operatorname{Re}\operatorname{Tr}(S_a(\rho)K_a)\right), \qquad \ell_2(x)= \begin{cases}\log_2x,&x>0,\\-\infty,&x\le0.\end{cases}$$

There are no positivity, trace, or parameter restrictions: $a=0,1$ are included, real division by zero is defined as zero, extended-real arithmetic is used, and $S_t$ assigns zero to a zero spectral value even when $t=0$.

QAlgorithms.Papers.Independent.Intakethird100.P52.conditionalRenyiTilde

For finite, possibly empty types $A,C,B$, each with decidable equality, any real $a$, and any complex matrix $\rho$ on $A\times(C\times B)$, set $\Sigma=\rho_{AC}\otimes I_B$, where $(\rho_{AC})_{(i,c),(j,d)}=\sum_b\rho_{(i,c,b),(j,d,b)}$, and set $\rho_{CB}=\operatorname{Tr}_A\rho$, $\Sigma_{CB}=\operatorname{Tr}_A\Sigma$, where partial trace sums over equal $A$-indices. Let $S_t(X)$ be continuous functional calculus for the function equal to $0$ at zero and $x^t$ otherwise. Define

$$\widetilde K_a= S_{(1-a)/(2a)}(\Sigma) \left[I_A\otimes \left( S_{(a-1)/(2a)}(\Sigma_{CB}) S_{(1-a)/a}(\rho_{CB}) S_{(a-1)/(2a)}(\Sigma_{CB}) \right)\right] S_{(1-a)/(2a)}(\Sigma).$$

The declaration defines

$$\frac{1}{a-1}\, \ell_2\!\left( \operatorname{Re}\operatorname{Tr} S_a\!\left(S_{1/2}(\rho)\widetilde K_aS_{1/2}(\rho)\right) \right)$$

as an extended real, where $\ell_2(x)=\log_2x$ for $x>0$ and $\ell_2(x)=-\infty$ otherwise. No positivity or normalization of $\rho$, or exclusion of $a=0,1$, is assumed; real division by zero is zero, and the functional calculus function is zero at a zero spectral value for every exponent.

QAlgorithms.Papers.Independent.Intakethird100.P52.conditionalSigma

For arbitrary types $A,C$, a finite, possibly empty type $B$ with decidable equality, and any complex matrix $\rho$ indexed by $A\times(C\times B)$, this declaration defines a matrix on the same index set by

$$\Sigma_{(a,c,b),(a',c',b')}= \begin{cases} \displaystyle\sum_{u\in B} \rho_{(a,c,u),(a',c',u)},&b=b',\\ 0,&b\ne b'. \end{cases}$$

Thus each diagonal block in the $B$ register is the matrix obtained by summing over equal $B$ indices of $\rho$. Neither $A$ nor $C$ is required to be finite, and no positivity, Hermiticity, or normalization is assumed. If $B$ is empty, the matrix has an empty index set.

QAlgorithms.Papers.Independent.Intakethird100.P52.deltaMax

For finite, possibly empty types $H,K$ with decidable equality, arbitrary complex matrices $\rho,\sigma$ on $H$, and any function $N:\operatorname{Mat}_H(\mathbb C)\to\operatorname{Mat}_K(\mathbb C)$, write $F_f(X)$ for continuous functional calculus and define $(D_N(Y))_{ij}=\sum_{k,l\in K}\overline{N(E_{ij})_{kl}}Y_{kl}$ using matrix units. Set $S=F_{\sqrt{\phantom{x}}}(\sigma)$, $T=F_{x\mapsto(\sqrt{x})^{-1}}(N(\sigma))$, and $\tau=S D_N(TN(\rho)T)S$. This declaration defines

$$\inf\left\{\ell\in\mathbb R:\;2^\ell\tau-\rho\succeq0\right\},$$

where the real numbers in this set are embedded into the extended reals before taking the infimum. No positivity, normalization, channel, or linearity assumptions are imposed; the inverse square-root function takes value zero at zero. An empty feasible set gives $+\infty$, whereas a feasible set unbounded below gives $-\infty$; in particular, an empty $H$ gives $-\infty$.

QAlgorithms.Papers.Independent.Intakethird100.P52.deltaMin

For finite, possibly empty types $H,K$ with decidable equality, arbitrary complex matrices $\rho,\sigma$ on $H$, and any function $N:\operatorname{Mat}_H(\mathbb C)\to\operatorname{Mat}_K(\mathbb C)$, write $F_f(X)$ for continuous functional calculus and define $(D_N(Y))_{ij}=\sum_{k,l\in K}\overline{N(E_{ij})_{kl}}Y_{kl}$ using matrix units. Set $S=F_{\sqrt{\phantom{x}}}(\sigma)$, $T=F_{x\mapsto(\sqrt{x})^{-1}}(N(\sigma))$, and $\tau=S D_N(TN(\rho)T)S$. The declaration defines $-L(r^2)$, where

$$r=\operatorname{Re}\operatorname{Tr} F_{\sqrt{\phantom{x}}}\!\left( F_{\sqrt{\phantom{x}}}(\rho)\,\tau\,F_{\sqrt{\phantom{x}}}(\rho) \right), \qquad L(x)= \begin{cases} \log_2x,&x>0,\\ -\infty,&x\le0. \end{cases}$$

No positivity, normalization, channel, or linearity assumptions are imposed. The inverse square-root function takes value zero at zero under real inversion, and the defined extended-real value is $+\infty$ whenever $r=0$, including the empty-$H$ case.

QAlgorithms.Papers.Independent.Intakethird100.P52.deltaTilde

For finite, possibly empty types $H,K$ with decidable equality, any $a\in\mathbb R$, arbitrary complex matrices $\rho,\sigma$ on $H$, and any function $N:\operatorname{Mat}_H(\mathbb C)\to\operatorname{Mat}_K(\mathbb C)$, let $S_t(X)$ be continuous functional calculus for the function equal to $0$ at zero and $\operatorname{rpow}(x,t)$ otherwise, and define $(D_N(Y))_{ij}=\sum_{k,l\in K}\overline{N(E_{ij})_{kl}}Y_{kl}$ using matrix units. Set

$$B= S_{(1-a)/(2a)}(\sigma)\, D_N\!\left( S_{(a-1)/(2a)}(N(\sigma))\,S_{(1-a)/a}(N(\rho))\, S_{(a-1)/(2a)}(N(\sigma)) \right)\, S_{(1-a)/(2a)}(\sigma).$$

The declaration defines $\frac1{a-1}L\!\left(\operatorname{Re}\operatorname{Tr}S_a(S_{1/2}(\rho)BS_{1/2}(\rho))\right)$ in the extended reals, where $L(x)=\log_2x$ for $x>0$ and $L(x)=-\infty$ otherwise, and the real prefactor is embedded before multiplication. No positivity, normalization, channel, linearity, or parameter restrictions are imposed. Division by zero is zero, so $a=0$ is included, and at $a=1$ the resulting extended-real value is zero.

QAlgorithms.Papers.Independent.Intakethird100.P52.jointTrace

For any finite, possibly empty type $H$ with decidable equality, any real $a$, and any complex square matrices $P,Q$ indexed by $H$, let $S_t(X)$ denote continuous functional calculus for the real function equal to zero at zero and $x^t$ otherwise. The declaration defines the real number

$$\operatorname{Re}\operatorname{Tr} S_{1/a}\!\left(S_{(a-1)/2}(Q)\,P\,S_{(a-1)/2}(Q)\right).$$

No positivity or normalization hypotheses are imposed. The parameter $a=0$ is included through real division by zero being zero, and the function defining $S_0$ has value zero at zero.

QAlgorithms.Papers.Independent.Intakethird100.P52.minRelative

For a finite, possibly empty type $H$ with decidable equality and arbitrary complex matrices $\rho,\tau$ on $H$, let $F_f(X)$ denote continuous functional calculus with the real function $f$, and put

$$r=\operatorname{Re}\operatorname{Tr} F_{\sqrt{\phantom{x}}}\!\left( F_{\sqrt{\phantom{x}}}(\rho)\,\tau\,F_{\sqrt{\phantom{x}}}(\rho) \right).$$

This declaration defines $-L(r^2)$ in the extended reals, where $L(x)=\log_2x$ when $x>0$ and $L(x)=-\infty$ when $x\le0$. Thus the value is $-\log_2(r^2)$ if $r\ne0$ and $+\infty$ if $r=0$. No positivity or trace normalization is assumed; in particular, the empty-index case gives $+\infty$.

QAlgorithms.Papers.Independent.Intakethird100.P52.outputReference

For finite, possibly empty types $H,K$, each with decidable equality, any real $a$, any complex square matrices $\rho,\sigma$ indexed by $H$, and any function $N$ from such matrices to complex square matrices indexed by $K$, let $S_t(X)$ denote continuous functional calculus for the function equal to $0$ at zero and $x^t$ otherwise. The declaration defines the matrix on $K$

$$S_{1/a}\!\left( S_{(a-1)/2}(N(\sigma))\, N\!\left(S_{(1-a)/2}(\sigma)S_a(\rho)S_{(1-a)/2}(\sigma)\right)\, S_{(a-1)/2}(N(\sigma)) \right).$$

There are no assumptions that $N$ is linear or a channel, or that either matrix is positive or normalized. The definition includes $a=0$, using $1/0=0$; $S_0$ still assigns zero at a zero spectral value.

QAlgorithms.Papers.Independent.Intakethird100.P52.renyiDifference

For finite, possibly empty types $H,K$ with decidable equality, any $a\in\mathbb R$, arbitrary complex matrices $\rho,\sigma$ on $H$, and any function $N:\operatorname{Mat}_H(\mathbb C)\to\operatorname{Mat}_K(\mathbb C)$, let $S_t(X)$ be continuous functional calculus for the function equal to $0$ at zero and $\operatorname{rpow}(x,t)$ otherwise, and define $(D_N(Y))_{ij}=\sum_{k,l\in K}\overline{N(E_{ij})_{kl}}Y_{kl}$ using matrix units. With $L(x)=\log_2x$ for $x>0$ and $L(x)=-\infty$ for $x\le0$, this declaration defines the extended-real value

$$\frac1{a-1}\, L\!\left( \operatorname{Re}\operatorname{Tr}\!\left[ S_a(\rho)\,S_{(1-a)/2}(\sigma)\, D_N\!\left( S_{(a-1)/2}(N(\sigma))\,S_{1-a}(N(\rho))\,S_{(a-1)/2}(N(\sigma)) \right)\, S_{(1-a)/2}(\sigma) \right]\right).$$

The real prefactor is embedded into the extended reals before multiplication. There are no positivity, normalization, channel, linearity, or parameter restrictions; at $a=1$ the prefactor is zero, so the defined value is zero, including when the logarithm is $-\infty$.

QAlgorithms.Papers.Independent.Intakethird100.P52.renyiKernel

For finite, possibly empty types $H,K$ with decidable equality, any $a\in\mathbb R$, arbitrary complex matrices $\rho,\sigma$ on $H$, and any function $N:\operatorname{Mat}_H(\mathbb C)\to\operatorname{Mat}_K(\mathbb C)$, define $S_t(X)$ by continuous functional calculus with the real function that is $0$ at $x=0$ and $\operatorname{rpow}(x,t)$ otherwise. Define $D_N(Y)$ entrywise by $(D_N(Y))_{ij}=\sum_{k,l\in K}\overline{N(E_{ij})_{kl}}Y_{kl}$, where $E_{ij}$ has entry $1$ at $(i,j)$ and zero elsewhere. The declaration defines the matrix

$$S_{(1-a)/2}(\sigma)\, D_N\!\left( S_{(a-1)/2}(N(\sigma))\,S_{1-a}(N(\rho))\,S_{(a-1)/2}(N(\sigma)) \right)\, S_{(1-a)/2}(\sigma).$$

There are no positivity, normalization, channel, or linearity assumptions. Every real $a$, including $a=1$, is included; $S_0$ uses the function $0$ at zero and $1$ elsewhere, rather than the constant function $1$.

QAlgorithms.Papers.Independent.Intakethird100.P52.renyiReference

For finite, possibly empty types $H,K$ with decidable equality, any $a\in\mathbb R$, arbitrary complex matrices $\rho,\sigma$ on $H$, and any function $N:\operatorname{Mat}_H(\mathbb C)\to\operatorname{Mat}_K(\mathbb C)$, let $S_t(X)$ denote continuous functional calculus for the function equal to $0$ at zero and $\operatorname{rpow}(x,t)$ otherwise, and define $(D_N(Y))_{ij}=\sum_{k,l\in K}\overline{N(E_{ij})_{kl}}Y_{kl}$ using matrix units $E_{ij}$. This declaration defines

$$S_{1/(1-a)}\!\left( S_{(1-a)/2}(\sigma)\, D_N\!\left( S_{(a-1)/2}(N(\sigma))\,S_{1-a}(N(\rho))\,S_{(a-1)/2}(N(\sigma)) \right)\, S_{(1-a)/2}(\sigma) \right).$$

There are no positivity, normalization, channel, or linearity assumptions. In particular $a=1$ is allowed: the outer exponent is then zero under the convention $1/0=0$, and $S_0$ uses the function $0$ at zero and $1$ elsewhere.

QAlgorithms.Papers.Independent.Intakethird100.P52.sandwichedKernel

For finite, possibly empty types $H,K$ with decidable equality, any $a\in\mathbb R$, arbitrary complex matrices $\rho,\sigma$ on $H$, and any function $N:\operatorname{Mat}_H(\mathbb C)\to\operatorname{Mat}_K(\mathbb C)$, let $S_t(X)$ be continuous functional calculus for the real function equal to $0$ at zero and $\operatorname{rpow}(x,t)$ elsewhere, and put $(D_N(Y))_{ij}=\sum_{k,l\in K}\overline{N(E_{ij})_{kl}}Y_{kl}$, with $E_{ij}$ the matrix unit. The declaration defines

$$S_{(1-a)/(2a)}(\sigma)\, D_N\!\left( S_{(a-1)/(2a)}(N(\sigma))\, S_{(1-a)/a}(N(\rho))\, S_{(a-1)/(2a)}(N(\sigma)) \right)\, S_{(1-a)/(2a)}(\sigma).$$

No positivity, normalization, channel, or linearity assumptions are imposed. The definition includes $a=0$, where each displayed exponent is zero because real division by zero is defined as zero; $S_0$ uses the function $0$ at zero and $1$ elsewhere. Negative exponents likewise retain the explicitly assigned value zero at a zero spectral argument.

QAlgorithms.Papers.Independent.Intakethird100.P52.sandwichedReference

For finite, possibly empty types $H,K$ with decidable equality, any $a\in\mathbb R$, arbitrary complex matrices $\rho,\sigma$ on $H$, and any function $N:\operatorname{Mat}_H(\mathbb C)\to\operatorname{Mat}_K(\mathbb C)$, let $S_t(X)$ denote continuous functional calculus for the function equal to $0$ at zero and $\operatorname{rpow}(x,t)$ otherwise, and set $(D_N(Y))_{ij}=\sum_{k,l\in K}\overline{N(E_{ij})_{kl}}Y_{kl}$ using matrix units $E_{ij}$. The defined matrix is

$$S_{a/(1-a)}\!\left( S_{(1-a)/(2a)}(\sigma)\, D_N\!\left( S_{(a-1)/(2a)}(N(\sigma))\,S_{(1-a)/a}(N(\rho))\, S_{(a-1)/(2a)}(N(\sigma)) \right)\, S_{(1-a)/(2a)}(\sigma) \right).$$

No positivity, normalization, channel, or linearity assumptions are imposed. Both $a=0$ and $a=1$ are included, with every division by zero evaluated as zero; an exponent zero invokes the function $0$ at zero and $1$ elsewhere.