Definitions for Channel simulation with quantum side information

QAlgorithms.Papers.Independent.Intakethird100.P42.CQFeedbackCode.IsClassical

Let $X$ be any type, let $B$ be a finite type, and let $y,n$ be arbitrary natural numbers, including zero. Let $P$ consist of natural numbers $c,m,h$, a function $e(x,l,a,s)\ge0$ for $x\in X^n$, $l\in\{0,1\}^c$, $a\in\{0,1\}^m$, and $s\in\{0,1\}^h$, satisfying $\sum_{a,s}e(x,l,a,s)=1$ for every $x,l$; matrices $M_{l,a,s}$ on $\mathbb C^{B^n}$, positive semidefinite and satisfying $\sum_sM_{l,a,s}=I$ for every $l,a$; and a decoding function $f(l,a,s)\in\{0,\ldots,y-1\}^n$. The predicate defined here is

$$\forall l,a,s,\ \forall u,v\in B^n,\qquad u\ne v\ \Longrightarrow\ M_{l,a,s}(u,v)=0.$$

Thus it requires every one of the specified measurement matrices to be diagonal. When $n=0$, $B^n$ has one element and this off-diagonal condition is vacuous. When $y=0<n$, no such $P$ exists because its decoding function would have a nonempty domain and an empty codomain.

QAlgorithms.Papers.Independent.Intakethird100.P42.CQFeedbackCode

For an arbitrary type $X$, a finite type $B$ with a specified finite enumeration, and natural numbers $y,n$, a code consists of natural numbers $c,h,p$, called its coin, message, and private bit counts; a real function $e(x,\ell,m,s)$, where $x\in X^{[n]}$, $\ell\in\{0,1\}^c$, $m\in\{0,1\}^h$, and $s\in\{0,1\}^p$, satisfying

$$e(x,\ell,m,s)\ge0,\qquad \sum_m\sum_s e(x,\ell,m,s)=1 \quad\text{for every }x,\ell;$$

matrices $F_{\ell,m,s}$ on $B^{[n]}$, such that for every $\ell,m$, every $F_{\ell,m,s}$ is positive semidefinite and $\sum_sF_{\ell,m,s}=I$; and a function $d(\ell,m,s)\in[y]^{[n]}$. Here $[k]=\{0,\ldots,k-1\}$, and words are functions on $[n]$. The type $X$ need not be finite, and $B$ may be empty. Zero bit counts are allowed and give singleton bitstring sets. Length $n=0$ is allowed and gives singleton word sets; if $y=0<n$, no such code exists because the required decoder would map a nonempty bitstring domain into an empty word type.

QAlgorithms.Papers.Independent.Intakethird100.P42.ClassicalSimulationAchievable

Let $X$ be a finite type, possibly empty, let $p:X\to\mathbb R$ be arbitrary, and let $Z,V$ be auxiliary data with positive finite output sizes $z,v$, respectively, and conditional probabilities $Z(b\mid x)$, $V(u\mid x)$, each nonnegative and summing to $1$ over its output for every $x$. For real numbers $R,C$, this predicate means $R\ge0$, $C\ge0$, and that for every $\varepsilon,\delta>0$ there is a natural $N$ such that every $n\ge N$ with $n\ge1$ admits the following data: natural numbers $c,m,h$; encoder weights $e(x,l,a,s)\ge0$, for $x\in X^n$, $l\in\{0,1\}^c$, $a\in\{0,1\}^m$, $s\in\{0,1\}^h$, satisfying $\sum_{a,s}e(x,l,a,s)=1$; diagonal positive semidefinite matrices $M_{l,a,t}$ on $\mathbb C^{\{0,\ldots,z-1\}^n}$, satisfying $\sum_tM_{l,a,t}=I$ for every $l,a$; and a decoding function $f(l,a,s)\in\{0,\ldots,v-1\}^n$. To specify the required error, put $p_n(x)=\prod_ip(x_i)$, $Z_n(b\mid x)=\prod_iZ(b_i\mid x_i)$, and $V_n(u\mid x)=\prod_iV(u_i\mid x_i)$. On the basis indexed by $(x,u,w,b)\in X^n\times\{0,\ldots,v-1\}^n\times\{0,\ldots,v-1\}^n\times\{0,\ldots,z-1\}^n$, let $T$ and $O$ be diagonal matrices with entries

$$T_{x,u,w,b}=\mathbf1_{\{u=w\}}\,p_n(x)V_n(u\mid x)Z_n(b\mid x),$$ $$O_{x,u,w,b} =p_n(x)2^{-c}Z_n(b\mid x) \sum_{l,a,s,t}e(x,l,a,s)\, \mathbf1_{\{f(l,a,s)=u\}}\mathbf1_{\{f(l,a,t)=w\}}\, M_{l,a,t}(b,b).$$

The required bounds are $\|T-O\|_1\le\varepsilon$, $m\le n(R+\delta)$, and $c\le n(C+\delta)$, where $\|M\|_1=\operatorname{Re}\operatorname{tr}\sqrt{M^\dagger M}$. No bound is imposed on $h$, and $c,m,h$ may be zero. No sign or normalization condition is imposed on $p$; the displayed matrices and error are used even when $p$ is not a probability law.

QAlgorithms.Papers.Independent.Intakethird100.P42.CoverEncoder

For a finite type $X$ and a natural number $M$, a cover encoder is a function $e:X\times\{0,\ldots,M-1\}\to\mathbb R$ satisfying, for every $x\in X$, $e(x,i)\ge0$ for every $i<M$ and $\sum_{i=0}^{M-1}e(x,i)=1$. No nonemptiness of $X$ or positivity of $M$ is required in the definition: if $X$ is empty these conditions are vacuous, whereas if $X$ is nonempty and $M=0$, no cover encoder exists.

QAlgorithms.Papers.Independent.Intakethird100.P42.ExactDistortionAchievable

For finite, finitely enumerated types $X,B$, arbitrary data $E=(p,\rho)$ with $p:X\to\mathbb R$ and complex matrices $\rho_x$ on $\mathbb C^B$, any real-valued function $d:X\times X\to\mathbb R$, and real numbers $D,R$, this property means that $R\ge0$ and that, for every real $\delta>0$, there exists a natural number $N$ such that every natural number $n\ge N$ with $n\ge1$ admits a natural number $b$, an encoder $e:X^n\to\{0,1\}^b$, and matrices $M_{m,z}$ on $\mathbb C^{B^n}$, positive semidefinite and summing to the identity over $z\in X^n$ for each message $m$, satisfying

$$\sum_{x,z\in X^n} \left(\prod_{i=0}^{n-1}p(x_i)\right) \operatorname{Re}\operatorname{tr}\!\left(M_{e(x),z}\rho_x^{(n)}\right) \frac{\sum_{i=0}^{n-1}d(x_i,z_i)}{n}\le D, \qquad b\le n(R+\delta),$$

where $(\rho_x^{(n)})_{u,v}=\prod_i(\rho_{x_i})_{u_i,v_i}$. The code may depend on $n$ and $\delta$; the distortion bound is exactly $D$. There are no validity assumptions on $E$, no sign assumptions on $d,D$, and no nonemptiness assumptions on $X,B$. The block length $n=0$ is excluded from the code requirement, while zero message bits are allowed.

QAlgorithms.Papers.Independent.Intakethird100.P42.ExactDistortionCode

For finite types $X$ and $B$, each equipped with a finite enumeration, and any natural number $n$, an exact distortion code consists of a natural number $b$ of message bits, a function $e:X^n\to\{0,1\}^b$, and matrices $M_{m,z}$ on $\mathbb C^{B^n}$ for every message $m\in\{0,1\}^b$ and reconstruction $z\in X^n$, where $X^n$ and $B^n$ mean functions from $\{0,\ldots,n-1\}$ into $X$ and $B$, respectively. For every message $m$, the matrices satisfy $M_{m,z}\succeq0$ for all $z$ and $\sum_{z\in X^n}M_{m,z}=I$; this requirement includes messages outside the image of $e$. Neither $n$ nor $b$ must be positive, and neither type must be nonempty. At $n=0$, each word type has one element, including when its alphabet is empty.

QAlgorithms.Papers.Independent.Intakethird100.P42.QuantumRDChoice

For finite types $X,B$, each equipped with a finite enumeration, a quantum rate-distortion choice consists of a positive natural number $a$, matrices $\tau_x$ on $\mathbb C^a$ for every $x\in X$, and matrices $M_z$ on $\mathbb C^{\{0,\ldots,a-1\}\times B}$ for every $z\in X$, satisfying

$$\tau_x\succeq0,\qquad \operatorname{tr}\tau_x=1 \quad(x\in X),\qquad M_z\succeq0\quad(z\in X),\qquad \sum_{z\in X}M_z=I.$$

Neither $X$ nor $B$ is required to be nonempty. If $X$ is empty, the output-state conditions are vacuous, but the decoder condition still requires its empty sum to equal the identity; consequently, when $B$ is nonempty, no such choice exists for empty $X$.

QAlgorithms.Papers.Independent.Intakethird100.P42.RDDatum

For finite types $X,B$ and a natural number $n$, a rate-distortion datum consists of a positive natural number $L$, a real function $V:X^n\times\{0,\ldots,L-1\}\to\mathbb R$ satisfying $V(y\mid x)\ge0$ and $\sum_{y<L}V(y\mid x)=1$ for every word $x\in X^n$, and matrices $D_{y,z}$ on $\mathbb C^{B^n}$, indexed by $y<L$ and $z\in X^n$, such that for each $y$, every $D_{y,z}$ is positive semidefinite and $\sum_{z\in X^n}D_{y,z}=I$. Here $X^n$ and $B^n$ are the functions from $\{0,\ldots,n-1\}$ into $X$ and $B$, respectively. The definition allows $n=0$ and empty types $X,B$, subject to these normalization and matrix conditions; for example, when $X^n$ is empty and $B^n$ is nonempty, the required empty sum cannot equal the identity, so no such datum exists.

QAlgorithms.Papers.Independent.Intakethird100.P42.SimulationAchievable

For finite types $X,B$ with specified finite enumerations, arbitrary data $p_x\in\mathbb R,\rho_x\in\mathbb C^{B\times B}$, an auxiliary system of positive size $y$ with $V(u\mid x)\ge0$ and $\sum_uV(u\mid x)=1$, and real numbers $R,C$, this property means that $R,C\ge0$ and, for every $\varepsilon,\delta>0$, there is $N\in\mathbb N$ such that every $n\ge N$ with $n\ge1$ admits a code with coin, message, and private bit counts $c,h,p$ satisfying

$$\|T-O\|_1\le\varepsilon,\qquad h\le n(R+\delta),\qquad c\le n(C+\delta).$$

The code has nonnegative weights $e(x,\ell,m,s)$ summing to $1$ over $m,s$ for every $x,\ell$, decoded words $d(\ell,m,s)\in[y]^{[n]}$, and matrices $F_{\ell,m,t}\succeq0$ on $B^{[n]}$ summing to $I$ over $t$ for every $\ell,m$; $\ell,m,s,t$ are bitstrings of lengths $c,h,p,p$. With $p_n(x)=\prod_i p_{x_i}$ and $(\rho_x^{(n)})_{b,b'}=\prod_i(\rho_{x_i})_{b_i,b'_i}$, the matrices on tuples $(x,u,v,b)$ have entries

$$\begin{aligned} T_{(x,u,v,b),(x',u',v',b')} &=\mathbf1_{\{x=x',\,u=u',\,v=v',\,u=v\}}\, p_n(x)\prod_iV(u_i\mid x_i)(\rho_x^{(n)})_{b,b'},\\ O_{(x,u,v,b),(x',u',v',b')} &=\mathbf1_{\{x=x',\,u=u',\,v=v'\}} \sum_{\ell,m,s,t}\mathbf1_{\{d(\ell,m,s)=u,\,d(\ell,m,t)=v\}}\, p_n(x)2^{-c}e(x,\ell,m,s) [\mathcal L_{F_{\ell,m,t}}(\rho_x^{(n)})]_{b,b'}. \end{aligned}$$

Here $\mathcal L_F(\sigma)$ relabels onto a finite ordinal, forms $\operatorname{cfc}(\sqrt{\cdot},F)\sigma\operatorname{cfc}(\sqrt{\cdot},F)$, and relabels back; $\|Z\|_1=\operatorname{Re}\operatorname{tr}\operatorname{cfc}(\sqrt{\cdot},Z^\dagger Z)$. The private bit count is unrestricted. No validity assumption on the ensemble data or nonemptiness assumption on either finite type is part of this definition; only positive block lengths are tested.

QAlgorithms.Papers.Independent.Intakethird100.P42.SimulationBasis

For arbitrary types $X,B$ and natural numbers $y,n$, this is the type

$$X^{[n]}\times\bigl([y]^{[n]}\times([y]^{[n]}\times B^{[n]})\bigr),$$

where $[k]=\{0,\ldots,k-1\}$ and $Z^{[n]}$ denotes functions from $[n]$ to $Z$. Thus an element consists of an $X$-word, two separate $[y]$-words, and a $B$-word, all of length $n$. No finiteness or nonemptiness assumptions on $X$ or $B$ are imposed. When $n=0$, every word type here is a singleton, even if its alphabet is empty; when $n>0$, an empty alphabet makes the corresponding word type empty.

QAlgorithms.Papers.Independent.Intakethird100.P42.SourceSimulation

For finite types $X,B$ with specified finite enumerations, arbitrary real weights $p_x$, complex matrices $\rho_x$ on $B$, an auxiliary system of positive size $y$ with probabilities $V(u\mid x)$, and real numbers $R,C$, this is the following property: $R,C\ge0$, and for every $\varepsilon,\delta>0$, all sufficiently large positive integers $n$ admit natural bit counts $c,h,p$, encoding weights $e(x,\ell,m,s)\ge0$ summing to $1$ over $m,s$, a decoder $d(\ell,m,s)\in[y]^{[n]}$, and matrices $F_{\ell,m,t}\succeq0$ on $B^{[n]}$ summing to $I$ over $t$, such that $h\le n(R+\delta)$, $c\le n(C+\delta)$, and $\|T-O\|_1\le\varepsilon$. Here $x\in X^{[n]}$, the bitstrings $\ell,m,s,t$ have lengths $c,h,p,p$, $p_n(x)=\prod_i p_{x_i}$, $(\rho_x^{(n)})_{b,b'}=\prod_i(\rho_{x_i})_{b_i,b'_i}$, and

$$\begin{aligned} T_{(x,u,v,b),(x',u',v',b')} &=\mathbf1_{\{x=x',\,u=u',\,v=v',\,u=v\}}\, p_n(x)\prod_iV(u_i\mid x_i)(\rho_x^{(n)})_{b,b'},\\ O_{(x,u,v,b),(x',u',v',b')} &=\mathbf1_{\{x=x',\,u=u',\,v=v'\}} \sum_{\ell,m,s,t}\mathbf1_{\{d(\ell,m,s)=u,\,d(\ell,m,t)=v\}}\, p_n(x)2^{-c}e(x,\ell,m,s) [\mathcal L_{F_{\ell,m,t}}(\rho_x^{(n)})]_{b,b'}. \end{aligned}$$

The matrix indices range over $X^{[n]}\times[y]^{[n]}\times[y]^{[n]}\times B^{[n]}$; $\mathcal L_F(\sigma)$ relabels onto a finite ordinal, forms $\operatorname{cfc}(\sqrt{\cdot},F)\sigma\operatorname{cfc}(\sqrt{\cdot},F)$, and relabels back, while $\|Z\|_1=\operatorname{Re}\operatorname{tr}\operatorname{cfc}(\sqrt{\cdot},Z^\dagger Z)$. There is no bound on $p$, no ensemble-validity hypothesis, and no nonemptiness hypothesis on $X$ or $B$.

QAlgorithms.Papers.Independent.Intakethird100.P42.blockRD

For finite types $X,B$, arbitrary data $E=(p,\rho)$ with $p:X\to\mathbb R$ and complex matrices $\rho_a$ on $B$, an arbitrary real function $d:X\times X\to\mathbb R$, a real threshold $D$, and a natural number $n$, this definition takes an infimum in $[0,\infty]$ over the following quantities. Choose any $L>0$, nonnegative conditional weights $V(u\mid x)$ with $\sum_{u<L}V(u\mid x)=1$ for every $x\in X^n$, and positive semidefinite matrices $D_{u,z}$ on $B^n$ with $\sum_{z\in X^n}D_{u,z}=I$, subject to

$$\sum_{x\in X^n}\sum_{u<L}\sum_{z\in X^n} p_n(x)V(u\mid x)\operatorname{Re}\operatorname{tr}(D_{u,z}\rho_x^{(n)}) \frac{\sum_{i<n}d(x_i,z_i)}{n}\le D,$$

where $p_n(x)=\prod_{i<n}p(x_i)$ and $(\rho_x^{(n)})_{a,b}=\prod_{i<n}(\rho_{x_i})_{a_i,b_i}$. For each such choice set $J(u,x)=p_n(x)V(u\mid x)$, $r(u)=\sum_xJ(u,x)$, $\bar\rho=\sum_xp_n(x)\rho_x^{(n)}$, and $\sigma_u=r(u)^{-1}\sum_xJ(u,x)\rho_x^{(n)}$. The quantity entered into the infimum is

$$\max\!\left\{0,\, \frac{H(p_n)+H(r)-H(J)-S(\bar\rho)+\sum_{u<L}r(u)S(\sigma_u)}{n}\right\},$$

embedded as a finite extended nonnegative real, where $H(a)=-\sum_za(z)\log_2a(z)$ and $S(A)=-\operatorname{Re}\operatorname{tr}(f(A))$ for $f(t)=t\log_2t$, evaluated by continuous functional calculus. No probability or density assumptions on $E$, or sign assumptions on $d,D$, are imposed. All divisions are total, so $\sigma_u=0$ if $r(u)=0$, and the displayed quotient is $0$ when $n=0$. The infimum of an empty feasible set is $+\infty$; at $n=0$, every feasible datum contributes $0$.

QAlgorithms.Papers.Independent.Intakethird100.P42.bookProbability

For a finite type $Y$ with a specified finite enumeration, an arbitrary real function $q:Y\to\mathbb R$, natural numbers $n,M$, and any predicate $\mathcal E$ on functions $b:[M]\to Y^{[n]}$, this defines the real number

$$\sum_{b:[M]\to Y^{[n]}} \begin{cases} \displaystyle\prod_{i\in[M]}\prod_{j\in[n]}q\bigl(b(i)(j)\bigr),&\mathcal E(b),\\ 0,&\neg\mathcal E(b). \end{cases}$$

There is no assumption that $q$ is nonnegative or sums to $1$. Both $n=0$ and $M=0$ are included: in either case there is exactly one book and its product weight is $1$, so the result is $1$ or $0$ according to the predicate on that book. If $Y$ is empty and both $n,M$ are positive, there are no books and the sum is $0$.

QAlgorithms.Papers.Independent.Intakethird100.P42.coveringError

For finite types $X,Y$, arbitrary real functions $q:Y\to\mathbb R$ and $P:Y\times X\to\mathbb R$, a natural number $M$, an indexed book $b:\{0,\ldots,M-1\}\to Y$, and a function $e(x,i)\ge0$ with $\sum_{i<M}e(x,i)=1$ for every $x\in X$, the covering error is

$$\sum_{i=0}^{M-1}\sum_{x\in X} \left|\frac{P(b_i,x)}{M} -\left(\sum_{y\in Y}q(y)P(y,x)\right)e(x,i)\right|.$$

No probability assumptions on $q$ or $P$ are imposed. Division by $0$ has value $0$, and the outer sum is empty when $M=0$, giving error $0$ whenever the supplied encoder exists; such an encoder cannot exist for $M=0$ and nonempty $X$.

QAlgorithms.Papers.Independent.Intakethird100.P42.coveringMarginal

For arbitrary types $X,Y$ with $Y$ finite, functions $q:Y\to\mathbb R$ and $P:Y\times X\to\mathbb R$, and $x\in X$, the covering marginal at $x$ is $\sum_{y\in Y}q(y)P(y,x)$. Neither function is required to be nonnegative or normalized, and $X$ need not be finite. If $Y$ is empty, the value is $0$.

QAlgorithms.Papers.Independent.Intakethird100.P42.diagonalCQSource

For any type $X$, any function $p:X\to\mathbb R$, and auxiliary data $Z$ consisting of a positive natural number $z$ and numbers $Z(u\mid x)\ge0$, indexed by $x\in X$ and $0\le u<z$, with $\sum_{u<z}Z(u\mid x)=1$ for every $x$, this definition constructs data with label type $X$, matrix index type $\{0,\ldots,z-1\}$, weight function $p$, and matrices

$$\rho_x(u,v)= \begin{cases} Z(u\mid x),&u=v,\\ 0,&u\ne v, \end{cases}$$

viewing the real entries as complex numbers. There is no finiteness or nonemptiness assumption on $X$, and no nonnegativity or normalization assumption on $p$.

QAlgorithms.Papers.Independent.Intakethird100.P42.dilutionError

For any finite type $Y$, any function $q:Y\to\mathbb R$, natural numbers $n,M$, and list of indexed words $b:\{0,\ldots,M-1\}\to Y^n$, where $Y^n$ denotes the functions $\{0,\ldots,n-1\}\to Y$, the dilution error is

$$\sum_{y\in Y^n}\left|\prod_{j=0}^{n-1}q(y_j)-\frac{\#\{i<M:b_i=y\}}{M}\right|.$$

No nonnegativity or normalization of $q$ is assumed. Empty products are $1$, empty sums are $0$, and division by $0$ is defined as $0$; thus the empirical term is $0$ when $M=0$. When $n=0$, there is exactly one empty word, and the error is $0$ for $M>0$ and $1$ for $M=0$.

QAlgorithms.Papers.Independent.Intakethird100.P42.empiricalLaw

For a finite type $Y$ with a specified finite enumeration, any natural number $M$, any function $b:[M]\to Y$, and any $y\in Y$, this defines

$$\frac{|\{i\in[M]:b(i)=y\}|}{M},$$

with numerator and denominator regarded as real numbers. The definition includes $M=0$: the numerator is then zero, and real division by zero is defined to be zero, so the value is $0$ for every $y$. No positive-size assumption is imposed.

QAlgorithms.Papers.Independent.Intakethird100.P42.exactDistortionExpected

For finite, finitely enumerated types $X,B$, arbitrary data $E=(p,\rho)$ consisting of a real function $p:X\to\mathbb R$ and complex matrices $\rho_x$ on $\mathbb C^B$, an arbitrary function $d:X\times X\to\mathbb R$, a natural number $n$, and an exact distortion code $P$, the defined real number is

$$\sum_{x,z\in X^n} \left(\prod_{i=0}^{n-1}p(x_i)\right) \operatorname{Re}\operatorname{tr}\!\left(M_{e(x),z}\rho_x^{(n)}\right) \frac{\sum_{i=0}^{n-1}d(x_i,z_i)}{n}, \qquad (\rho_x^{(n)})_{u,v} =\prod_{i=0}^{n-1}(\rho_{x_i})_{u_i,v_i}.$$

Here $X^n,B^n$ are length-$n$ word types, $P$ has some number $b\ge0$ of message bits, $e:X^n\to\{0,1\}^b$, and, for every message $m$, $M_{m,z}\succeq0$ and $\sum_zM_{m,z}=I$ on $\mathbb C^{B^n}$. No normalization or positivity of $p$ or $\rho_x$, and no sign condition on $d$, is assumed. Empty alphabets and $n=0$ are allowed; at $n=0$, empty products equal $1$, and the distortion factor is $0/0=0$, so the defined value is $0$.

QAlgorithms.Papers.Independent.Intakethird100.P42.exactDistortionRate

For finite, finitely enumerated types $X,B$, arbitrary data $E=(p,\rho)$ with real weights $p(x)$ and complex matrices $\rho_x$ on $\mathbb C^B$, any $d:X\times X\to\mathbb R$, and any real $D$, the defined value is the infimum in $[0,\infty]$ of all nonnegative real rates $R$ having the following property: for every $\delta>0$, all sufficiently large positive integers $n$ admit an encoder $e:X^n\to\{0,1\}^b$, for some integer $b\ge0$, and matrices $M_{m,z}$ on $\mathbb C^{B^n}$ with $M_{m,z}\succeq0$ and $\sum_{z\in X^n}M_{m,z}=I$ for every message $m$, such that

$$b\le n(R+\delta),\qquad \sum_{x,z\in X^n} \left(\prod_i p(x_i)\right) \operatorname{Re}\operatorname{tr}\!\left(M_{e(x),z}\rho_x^{(n)}\right) \frac{\sum_i d(x_i,z_i)}{n}\le D, \quad (\rho_x^{(n)})_{u,v}=\prod_i(\rho_{x_i})_{u_i,v_i}.$$

More precisely, each such $R$ enters the infimum through its nonnegative extended-real image, which equals $R$ because $R\ge0$. If no such rate exists, the value is $+\infty$; attainment is not part of this definition. Empty alphabets are permitted, and no distribution, density-matrix, or nonnegativity assumptions on $E,d,D$ are imposed.

QAlgorithms.Papers.Independent.Intakethird100.P42.feedbackMeasurementBranch

For a finite type $B$ with a specified finite enumeration and arbitrary complex matrices $F,\rho$ indexed by $B$, choose the enumeration equivalence $e:B\to[|B|]$, and write $\widetilde F_{ij}=F_{e^{-1}(i),e^{-1}(j)}$ and $\widetilde\rho_{ij}=\rho_{e^{-1}(i),e^{-1}(j)}$. The resulting matrix has entries

$$\bigl(\operatorname{cfc}(\sqrt{\cdot},\widetilde F)\, \widetilde\rho\, \operatorname{cfc}(\sqrt{\cdot},\widetilde F)\bigr)_{e(a),e(b)} \qquad(a,b\in B),$$

where $\operatorname{cfc}$ is the continuous functional calculus and the square root is the real square-root function, which is zero on negative inputs. No positivity, Hermiticity, trace, or normalization hypothesis is imposed on either matrix; the functional-calculus expression uses its total definition even outside its usual hypotheses. The empty finite type $B$ is also allowed.

QAlgorithms.Papers.Independent.Intakethird100.P42.feedbackTraceError

For finite types $X,B$ with specified finite enumerations, arbitrary ensemble data $p_x\in\mathbb R,\rho_x\in\mathbb C^{B\times B}$, an auxiliary system with positive size $y$, probabilities $V(u\mid x)\ge0$ summing to $1$ over $u\in[y]$, a natural number $n$, and a code, this is

$$\operatorname{Re}\operatorname{tr} \operatorname{cfc}\!\left(\sqrt{\cdot},(T-O)^\dagger(T-O)\right).$$

Here both matrices are indexed by $(x,u,v,b)\in X^{[n]}\times[y]^{[n]}\times[y]^{[n]}\times B^{[n]}$. Write $p_n(x)=\prod_i p_{x_i}$ and $(\rho_x^{(n)})_{b,b'}=\prod_i(\rho_{x_i})_{b_i,b'_i}$. The target entry is zero unless $x=x',u=u',v=v',u=v$, and otherwise is $p_n(x)\prod_iV(u_i\mid x_i)(\rho_x^{(n)})_{b,b'}$. To define $O$, the code supplies bit counts $c,h,p$, nonnegative encoding weights $e(x,\ell,m,s)$ with $\sum_{m,s}e(x,\ell,m,s)=1$, decoded words $d(\ell,m,s)$, and positive semidefinite matrices $F_{\ell,m,t}$ on $B^{[n]}$ with $\sum_tF_{\ell,m,t}=I$. Its entry is zero unless $x=x',u=u',v=v'$, and otherwise equals

$$\sum_{\ell,m,s,t} \mathbf1_{\{d(\ell,m,s)=u,\ d(\ell,m,t)=v\}}\, p_n(x)2^{-c}e(x,\ell,m,s) \bigl[\mathcal L_{F_{\ell,m,t}}(\rho_x^{(n)})\bigr]_{b,b'},$$

where $\ell,m,s,t$ are bitstrings of lengths $c,h,p,p$, and $\mathcal L_F(\sigma)$ is obtained by relabeling onto a finite ordinal, applying the functional-calculus expression $\operatorname{cfc}(\sqrt{\cdot},F)\sigma\operatorname{cfc}(\sqrt{\cdot},F)$, and relabeling back. No probability or density-matrix assumptions on $p_x,\rho_x$, and no nonemptiness assumptions on $X,B$, are imposed. The definition includes $n=0$, when products are empty products and word sets are singleton.

QAlgorithms.Papers.Independent.Intakethird100.P42.fqrsQuantum

For arbitrary finite types $A,C,B,R$, possibly empty, and any vector $\psi\in\mathbb C^{R\times(A\times(C\times B))}$, form $\rho_{uv}=\psi_u\overline{\psi_v}$. For a register collection $U$, obtain $\rho_U$ by summing over equal complementary labels, so that $(\rho_U)_{u,u'}=\sum_w\psi_{(u,w)}\overline{\psi_{(u',w)}}$, with coordinates in their original order. Define $S(U)=-\operatorname{Re}\operatorname{tr}(f(\rho_U))$, where $f(t)=t\log_2t$ is applied by continuous functional calculus and $0\log_20=0$. The assigned real number is exactly

$$\frac{S(C)+S(RB)-S(RCB)}{2}.$$

No normalization or nonzero condition is imposed on $\psi$, and the definition also applies when a register type is empty.

QAlgorithms.Papers.Independent.Intakethird100.P42.printedQuantumCost

For arbitrary finite types $A,C,B,R$, possibly empty, and any vector $\psi\in\mathbb C^{R\times(A\times(C\times B))}$, let $\rho_{uv}=\psi_u\overline{\psi_v}$. For any collection $U$ of registers, let $(\rho_U)_{u,u'}=\sum_w\psi_{(u,w)}\overline{\psi_{(u',w)}}$, summing over the complementary labels with coordinates restored to their original order, and put $S(U)=-\operatorname{Re}\operatorname{tr}(f(\rho_U))$, where $f(t)=t\log_2t$ is applied by continuous functional calculus, with $0\log_20=0$. The definition assigns

$$\frac{S(AC)+S(RA)-S(A)-S(RAC)}{2}.$$

This formula requires neither $\|\psi\|=1$ nor $\psi\ne0$, and includes empty register types.

QAlgorithms.Papers.Independent.Intakethird100.P42.productConditional

For arbitrary types $X,Y$, any real function $P:Y\times X\to\mathbb R$, any natural number $n$, and words $y:\{0,\ldots,n-1\}\to Y$ and $x:\{0,\ldots,n-1\}\to X$, the product conditional is $\prod_{i=0}^{n-1}P(y_i,x_i)$. No finiteness, nonemptiness, nonnegativity, or normalization assumptions are imposed on $X,Y,P$. For $n=0$, the product is $1$.

QAlgorithms.Papers.Independent.Intakethird100.P42.quantumRDCost

For finite, finitely enumerated types $X,B$, arbitrary data $E=(p,\rho)$ with $p:X\to\mathbb R$ and complex matrices $\rho_x$ on $\mathbb C^B$, and a quantum rate-distortion choice $P$, let $a>0$ be its output dimension and let $\tau_x$ be its positive semidefinite, trace-one output matrices on $\mathbb C^a$. Write

$$\bar\tau=\sum_xp(x)\tau_x,\qquad \bar\rho=\sum_xp(x)\rho_x,\qquad \Omega=\sum_xp(x)(\tau_x\otimes\rho_x), \qquad S(A)=-\operatorname{Re}\operatorname{tr}\!\bigl(f(A)\bigr), \quad f(t)=t\log_2t,$$

where $f(A)$ is evaluated by continuous functional calculus. The defined cost is

$$\left[S(\bar\tau)-\sum_xp(x)S(\tau_x)\right] -\left[S(\bar\tau)+S(\bar\rho)-S(\Omega)\right].$$

The choice $P$ also includes matrices $M_z\succeq0$ on $\mathbb C^{a\times B}$ with $\sum_zM_z=I$, although these matrices do not occur in the cost. No validity assumptions on $E$ or nonemptiness assumptions on $X,B$ are made, and no sign condition on the resulting cost is included.

QAlgorithms.Papers.Independent.Intakethird100.P42.quantumRDExpected

For finite, finitely enumerated types $X,B$, arbitrary data $E=(p,\rho)$ consisting of $p:X\to\mathbb R$ and complex matrices $\rho_x$ on $\mathbb C^B$, any function $d:X\times X\to\mathbb R$, and a quantum rate-distortion choice $P$, the defined real number is

$$\sum_{x,z\in X}p(x)\, \operatorname{Re}\operatorname{tr}\!\left(M_z(\tau_x\otimes\rho_x)\right)d(x,z).$$

Here $P$ supplies an integer $a>0$, positive semidefinite matrices $\tau_x$ on $\mathbb C^a$ with trace $1$, and positive semidefinite matrices $M_z$ on $\mathbb C^{a\times B}$ satisfying $\sum_zM_z=I$. The tensor product has entries $(\tau_x\otimes\rho_x)_{(i,u),(j,v)}=(\tau_x)_{i,j}(\rho_x)_{u,v}$. No probability or density-matrix conditions are imposed on $E$, and $d$ may take negative values. Empty $X$ or $B$ is permitted whenever such a choice $P$ exists; if $X$ is empty, the displayed sum is $0$.

QAlgorithms.Papers.Independent.Intakethird100.P42.quantumRDValue

For finite, finitely enumerated types $X,B$, arbitrary data $E=(p,\rho)$ with real weights $p(x)$ and complex matrices $\rho_x$ on $\mathbb C^B$, any $d:X\times X\to\mathbb R$, and any real $D$, consider all choices of an integer $a>0$, matrices $\tau_x\succeq0$ on $\mathbb C^a$ with $\operatorname{tr}\tau_x=1$, and matrices $M_z\succeq0$ on $\mathbb C^{a\times B}$ with $\sum_zM_z=I$, for which

$$\sum_{x,z}p(x)\operatorname{Re}\operatorname{tr}\!\left(M_z(\tau_x\otimes\rho_x)\right)d(x,z)\le D.$$

For each choice, set $\bar\tau=\sum_xp(x)\tau_x$, $\bar\rho=\sum_xp(x)\rho_x$, $\Omega=\sum_xp(x)(\tau_x\otimes\rho_x)$, and

$$C=\left[S(\bar\tau)-\sum_xp(x)S(\tau_x)\right] -\left[S(\bar\tau)+S(\bar\rho)-S(\Omega)\right], \qquad S(A)=-\operatorname{Re}\operatorname{tr}\!\bigl((t\mapsto t\log_2t)(A)\bigr),$$

using continuous functional calculus. The defined value is the infimum in $[0,\infty]$ of the numbers $\max(C,0)$ from these choices. Thus negative costs are mapped to $0$, and an empty set of admissible choices gives $+\infty$. Neither attainment, validity of $E$, nonnegativity of $d,D$, nor nonemptiness of $X,B$ is assumed.

QAlgorithms.Papers.Independent.Intakethird100.P42.rdExpected

For finite types $X,B$, data $E$ consisting of an arbitrary real function $p:X\to\mathbb R$ and arbitrary complex matrices $\rho_a$ on $B$, an arbitrary function $d:X\times X\to\mathbb R$, a natural number $n$, and a rate-distortion datum consisting of $L>0$, nonnegative normalized conditional weights $V(y\mid x)$ for $y<L$, and positive semidefinite matrices $D_{y,z}$ on $B^n$ satisfying $\sum_{z\in X^n}D_{y,z}=I$, the expected-distortion expression is

$$\sum_{x\in X^n}\sum_{y<L}\sum_{z\in X^n} \left(\prod_{i<n}p(x_i)\right)V(y\mid x)\, \operatorname{Re}\operatorname{tr}(D_{y,z}\rho_x^{(n)})\, \frac{\sum_{i<n}d(x_i,z_i)}{n}, \qquad (\rho_x^{(n)})_{a,b}=\prod_{i<n}(\rho_{x_i})_{a_i,b_i}.$$

No validity assumption on $E$ is imposed: $p$ need not be a probability function and the matrices $\rho_a$ need not be density matrices. The function $d$ may take arbitrary real values. Empty products are $1$, empty sums are $0$, and division by $0$ is $0$, so this expression is $0$ at $n=0$.

QAlgorithms.Papers.Independent.Intakethird100.P42.redistributionReceiverInformation

For arbitrary finite types $A,C,B,R$, possibly empty, and any vector $\psi\in\mathbb C^{R\times(A\times(C\times B))}$, form the matrix $\rho_{uv}=\psi_u\overline{\psi_v}$. For a collection $U$ of these registers, let $\rho_U$ be the matrix obtained by summing $\rho$'s entries over equal labels in all complementary registers; equivalently, $(\rho_U)_{u,u'}=\sum_w\psi_{(u,w)}\overline{\psi_{(u',w)}}$, with the coordinates placed in their original order. Write $S(U)=-\operatorname{Re}\operatorname{tr}(f(\rho_U))$, where $f(t)=t\log_2t$ is applied by continuous functional calculus, with $0\log_20=0$. The definition assigns the real number

$$S(C)+S(B)-S(CB).$$

There is no assumption that $\|\psi\|=1$; the same formula applies to the zero vector and to empty register types.

QAlgorithms.Papers.Independent.Intakethird100.P42.senderMutual

For arbitrary finite types $A,C,B,R$, possibly empty, and any vector $\psi\in\mathbb C^{R\times(A\times(C\times B))}$, set $\rho_{uv}=\psi_u\overline{\psi_v}$. For each collection $U$ of registers, define its reduced matrix by $(\rho_U)_{u,u'}=\sum_w\psi_{(u,w)}\overline{\psi_{(u',w)}}$, summing over all complementary register labels and restoring the original coordinate order. Let $S(U)=-\operatorname{Re}\operatorname{tr}(f(\rho_U))$, with $f(t)=t\log_2t$ applied by continuous functional calculus and $0\log_20=0$. The definition assigns

$$S(C)+S(A)-S(AC).$$

The vector need not be normalized, and zero vectors and empty register types are included.

QAlgorithms.Papers.Independent.Intakethird100.P42.simulationOutput

For an arbitrary type $X$, a finite type $B$ with a specified finite enumeration, arbitrary real weights $p_x$ and complex matrices $\rho_x$ on $B$, natural numbers $y,n$, and a code, let its bit counts be $c,h,p$, its encoding numbers be $e(x,\ell,m,s)\ge0$ with $\sum_{m,s}e(x,\ell,m,s)=1$, its decoder be $d(\ell,m,s)\in[y]^{[n]}$, and its matrices $F_{\ell,m,t}$ on $B^{[n]}$ satisfy $F_{\ell,m,t}\succeq0$ and $\sum_tF_{\ell,m,t}=I$ for every $\ell,m$; the indices $\ell,m,s,t$ range respectively over bitstrings of lengths $c,h,p,p$. Put $p_n(x)=\prod_i p_{x_i}$ and $(\rho_x^{(n)})_{b,b'}=\prod_i(\rho_{x_i})_{b_i,b'_i}$. For matrices $F,\sigma$ on $B^{[n]}$, let $\mathcal L_F(\sigma)$ mean: relabel both matrices along the specified equivalence $B^{[n]}\simeq[|B^{[n]}|]$, form $\operatorname{cfc}(\sqrt{\cdot},F)\sigma\operatorname{cfc}(\sqrt{\cdot},F)$ in those coordinates, and relabel back. The defined output matrix is

$$O_{(x,u,v,b),(x',u',v',b')}= \begin{cases} \displaystyle \sum_{\ell,m,s,t} \mathbf1_{\{d(\ell,m,s)=u,\ d(\ell,m,t)=v\}}\, p_n(x)\,2^{-c}\,e(x,\ell,m,s)\, \bigl[\mathcal L_{F_{\ell,m,t}}(\rho_x^{(n)})\bigr]_{b,b'}, &x=x',\ u=u',\ v=v',\\ 0,&\text{otherwise}. \end{cases}$$

Thus $s$ supplies the encoding weight and the first decoded word, while the independently summed $t$ supplies the measurement matrix and the second decoded word; equality of those two words is not required. No validity hypothesis is imposed on the ensemble data. Zero lengths and zero bit counts are permitted, with empty products equal to $1$ and zero-length bitstring sets singleton.

QAlgorithms.Papers.Independent.Intakethird100.P42.simulationTarget

For arbitrary types $X,B$, data consisting of a real function $p:X\to\mathbb R$ and complex matrices $\rho_x$ on $B$, an auxiliary system consisting of a positive natural number $y$ and numbers $V(u\mid x)\ge0$ with $\sum_{u\in[y]}V(u\mid x)=1$ for every $x$, and any natural number $n$, this defines a matrix indexed by tuples $(x,u,v,b)\in X^{[n]}\times([y]^{[n]}\times([y]^{[n]}\times B^{[n]}))$. Its entries are

$$T_{(x,u,v,b),(x',u',v',b')}= \begin{cases} \displaystyle \left(\prod_{i\in[n]}p(x_i)V(u_i\mid x_i)\right) \left(\prod_{i\in[n]}(\rho_{x_i})_{b_i,b'_i}\right), &x=x',\ u=u',\ v=v',\ u=v,\\ 0,&\text{otherwise}. \end{cases}$$

Real coefficients in this expression are regarded as complex numbers. The functions $p,\rho_x$ are not assumed to form valid ensemble data: neither probability normalization nor density-matrix conditions are required. Neither $X$ nor $B$ is assumed finite or nonempty. Empty products equal $1$, so at $n=0$ this is the one-by-one matrix with entry $1$.

QAlgorithms.Papers.Independent.Intakethird100.P42.wordDistortion

For any type $X$, any function $d:X\times X\to\mathbb R$, any natural number $n$, and words $x,z:\{0,\ldots,n-1\}\to X$, the word distortion is $\frac1n\sum_{i=0}^{n-1}d(x_i,z_i)$. No sign, symmetry, diagonal, or metric conditions are imposed on $d$, and $X$ need not be finite. When $n=0$, both the empty sum and division by $0$ yield $0$.