Definitions for Quantum rate distortion coding with auxiliary resources

QAlgorithms.Papers.Independent.Intakethird100.P27.DistortionSource

For finite types $A$ and $B$, a distortion source consists of a complex vector $\psi$ indexed by $A\times A$, satisfying $\|\psi\|=1$, and a positive semidefinite complex matrix $\Delta$ indexed by $A\times B$. There is no normalization or upper bound imposed on $\Delta$. The normalization of $\psi$ excludes empty $A$, but the definition does not require $B$ to be nonempty.

QAlgorithms.Papers.Independent.Intakethird100.P27.IidRegister

For any type $A$ and natural number $n$, the register $A^n$ is the type of functions from $\{0,\ldots,n-1\}$ to $A$. No finiteness assumption on $A$ is required. When $n=0$, this register has exactly one element, the empty function, including when $A$ is empty.

QAlgorithms.Papers.Independent.Intakethird100.P27.LossyAchievable

For finite types $A,B$, a source consisting of a unit vector $\psi\in\mathbb C^{A\times A}$ and a positive semidefinite matrix $\Delta$ on $A\times B$, an assistance label $a$ among $\mathrm{unassisted}$, $\mathrm{entanglement}$, and $\mathrm{classical}$, and real numbers $r,D$, lossy achievability means $r\ge0$, $D\ge0$, and the following: for every $\varepsilon>0$ and every $s>0$, there exists a natural number $n_0$ such that, for every natural number $n\ge n_0$ with $n\ge1$, there is a code with positive dimensions $q,c,h$, a positive semidefinite trace-one resource $\tau$ on $[h]\times[h]$, a finite Kraus trace-preserving encoder from matrices on $A^n\times[h]$ to matrices on $[c]\times[q]$, and a finite Kraus trace-preserving decoder from matrices on $([c]\times[q])\times[h]$ to matrices on $B^n$. The encoder output must have zero entries between distinct classical indices for every input matrix. The assistance restriction is $c=h=1$ for $\mathrm{unassisted}$, $c=1$ for $\mathrm{entanglement}$, and $h=1$ for $\mathrm{classical}$. Starting with the matrix whose entries are $\prod_i\psi(r_i,x_i)\overline{\psi(r'_i,x'_i)}\,\tau_{(u,v),(u',v')}$, apply the encoder to $x,u$, retain $r,v$, and then apply the decoder to the encoder output and $v$; call the resulting matrix on $A^n\times B^n$ $X$. The code must satisfy

$$\log_2 q\le n(r+s), \qquad \frac1n\sum_{i=0}^{n-1}\operatorname{Re}\operatorname{tr}(\Delta X_i)\le D+\varepsilon,$$

where $X_i$ is obtained by fixing the two retained indices at position $i$ and summing entries of $X$ over matching row and column indices at every other position. Only the quantum dimension appears in the rate bound; the classical and shared dimensions have no additional size bound.

QAlgorithms.Papers.Independent.Intakethird100.P27.LossyAssistance

The assistance type has exactly three distinct values, named $\mathrm{unassisted}$, $\mathrm{entanglement}$, and $\mathrm{classical}$, and equality between its values is decidable. This declaration supplies only these labels and does not itself attach resource restrictions to them.

QAlgorithms.Papers.Independent.Intakethird100.P27.LossyCode.Allowed

For finite types $A,B$, any natural number $n$, and a lossy code $P$ with positive quantum, classical, and shared dimensions $q,c,h$, a positive semidefinite trace-one shared resource on two $h$-dimensional registers, and finite Kraus trace-preserving encoder and decoder maps of the dimensions specified by the code, with the encoder output block diagonal in its classical index, the allowed-assistance predicate is defined as follows: $\mathrm{unassisted}$ requires $c=1$ and $h=1$; $\mathrm{entanglement}$ requires $c=1$; and $\mathrm{classical}$ requires $h=1$. Beyond the code’s existing requirements, these cases impose no further restriction on $q$, on the unrestricted dimension in the latter two cases, or on the shared resource.

QAlgorithms.Papers.Independent.Intakethird100.P27.LossyCode.initial

For finite types $A,B$, any natural number $n$, a lossy code $P$ with positive dimensions $q,c,h$, shared density matrix $\tau$ on two $h$-dimensional registers, finite Kraus trace-preserving encoder and decoder, and encoder output block diagonal in its classical index, and a source consisting of a unit vector $\psi\in\mathbb C^{A\times A}$ and a positive semidefinite matrix $\Delta$ on $A\times B$, the initial matrix is indexed by $A^n\times((A^n\times\{0,\ldots,h-1\})\times\{0,\ldots,h-1\})$ and has entries

$$\Omega_{(r,((x,u),v)),(r',((x',u'),v'))} = \left(\prod_{i=0}^{n-1}\psi(r_i,x_i)\overline{\psi(r'_i,x'_i)}\right) \tau_{(u,v),(u',v')}.$$

Thus it is the product of $n$ copies of the matrix $[\psi(a,x)\overline{\psi(a',x')}]$ and the shared resource, with the displayed register ordering. When $n=0$, the product is one and the entries reduce to those of $\tau$.

QAlgorithms.Papers.Independent.Intakethird100.P27.LossyCode

For finite types $A,B$ and any natural number $n$, a lossy code consists of positive natural numbers $q,c,h$, representing its quantum, classical, and shared dimensions; a positive semidefinite matrix $\tau$ of trace one on $\{0,\ldots,h-1\}^2$; an encoder from matrices on $A^n\times\{0,\ldots,h-1\}$ to matrices on $\{0,\ldots,c-1\}\times\{0,\ldots,q-1\}$; and a decoder from matrices on $(\{0,\ldots,c-1\}\times\{0,\ldots,q-1\})\times\{0,\ldots,h-1\}$ to matrices on $B^n$. Each map must admit a finite Kraus representation: it sends every matrix $X$ to $\sum_j K_jXK_j^\dagger$, with $\sum_jK_j^\dagger K_j=I$ on its input space. Moreover, for every encoder input matrix $X$, every distinct classical indices $m,m'$, and all quantum indices $w,w'$, the encoder output satisfies $\operatorname{Enc}(X)_{(m,w),(m',w')}=0$. Here $A^n$ and $B^n$ denote length-$n$ tuples; $n=0$ is allowed, with each tuple register then a singleton.

QAlgorithms.Papers.Independent.Intakethird100.P27.LossyCode.output

For finite types $A,B$, any natural number $n$, a lossy code $P$ with positive quantum, classical, and shared dimensions $q,c,h$, shared density matrix $\tau$, finite Kraus trace-preserving encoder $\operatorname{Enc}:A^n\times[h]\to[c]\times[q]$ and decoder $\operatorname{Dec}:([c]\times[q])\times[h]\to B^n$, where these arrows denote maps between matrix spaces and the encoder is block diagonal in its classical output index, and a source $(\psi,\Delta)$ with $\|\psi\|=1$ and $\Delta\ge0$, the output is a matrix on $A^n\times B^n$. To define its entries, for each $r,r'\in A^n$ form the matrix $Y_{r,r'}$ on $A^n\times[h]\times[h]$ with entries

$$Y_{r,r'}{}_{((x,u),v),((x',u'),v')} = \prod_{i=0}^{n-1}\psi(r_i,x_i)\overline{\psi(r'_i,x'_i)} \,\tau_{(u,v),(u',v')}.$$

Apply the encoder to its $A^n\times[h]$ matrix blocks while retaining the second shared register, producing $Z_{r,r'}$ with entries $Z_{r,r'}{}_{((m,w),v),((m',w'),v')}=\operatorname{Enc}([Y_{r,r'}{}_{((x,u),v),((x',u'),v')}]_{(x,u),(x',u')})_{(m,w),(m',w')}$; the output entry at $(r,b),(r',b')$ is then $\operatorname{Dec}(Z_{r,r'})_{b,b'}$. Here $[d]=\{0,\ldots,d-1\}$; $n=0$ is allowed and all products over its positions equal one.

QAlgorithms.Papers.Independent.Intakethird100.P27.averageDistortion

For finite types $A,B$, a source consisting of a unit vector $\psi\in\mathbb C^{A\times A}$ and a positive semidefinite matrix $\Delta$ on $A\times B$, any natural number $n$, and any complex matrix $X$ on $A^n\times B^n$, the average distortion is

$$\frac1n\sum_{i=0}^{n-1}\operatorname{Re}\operatorname{tr}(\Delta X_i), \qquad (X_i)_{(a,b),(a',b')}= \sum_{\substack{u_i=a,\ v_i=b,\ u'_i=a',\ v'_i=b'\\ \forall j\ne i,\ (u_j,v_j)=(u'_j,v'_j)}} X_{(u,v),(u',v')},$$

where the inner sum ranges over $u,u'\in A^n$ and $v,v'\in B^n$. No density-matrix assumption is imposed on $X$. For $n=0$, the numerator is an empty sum and real division by zero is defined to be zero, so the average distortion is zero.

QAlgorithms.Papers.Independent.Intakethird100.P27.conditionalOutputInformation

For finite types $R,A,B,C$, any function $N$ from complex matrices on $A$ to complex matrices on $C$, and any vector $\phi\in\mathbb C^{R\times(A\times B)}$, define the matrix $X$ on $R\times(C\times B)$ by

$$X_{(r,(c,b)),(r',(c',b'))} = N\!\left( \left[\phi_{(r,(a,b))}\, \overline{\phi_{(r',(a',b'))}}\right]_{a,a'\in A} \right)_{c,c'}.$$

The declaration defines the real number

$$H(X_{RB})+H(X_{CB})-H(X_B)-H(X),$$

where

$$(X_{RB})_{(r,b),(r',b')}=\sum_cX_{(r,(c,b)),(r',(c,b'))}, \quad (X_{CB})_{(c,b),(c',b')}=\sum_rX_{(r,(c,b)),(r,(c',b'))}, \quad (X_B)_{b,b'}=\sum_c\sum_rX_{(r,(c,b)),(r,(c,b'))},$$

and $H(Y)=-\operatorname{Re}\operatorname{tr}(f(Y))$, with $f(x)=x\log_2x$ applied by continuous functional calculus. No normalization of $\phi$, channel property of $N$, or nonemptiness of the finite types is assumed; the formula also covers empty index types.

QAlgorithms.Papers.Independent.Intakethird100.P27.formationBlockValue

For finite types $A,B$, a source $S$ with unit vector $\psi\in\mathbb C^{A\times A}$ and positive semidefinite distortion matrix $\Delta$ on $A\times B$, any real $D$, and any $k\in\mathbb N$, define

$$\max(k^{-1},0)\, \inf_{\substack{N\text{ a channel}\\ \bar d_S(X_N)\le D}} \max(E_F(X_N),0) \quad\text{in }[0,\infty].$$

Here a channel from matrices on $A^k$ to matrices on $B^k$ means a map $N(Z)=\sum_jK_jZK_j^\dagger$ for finitely many matrices $K_j$, with $\sum_jK_j^\dagger K_j=I$; its output $X_N$ is specified by

$$X_N((a,b),(a',b')) = N\!\left( \left(\prod_{i<k}\psi(a_i,x_i)\overline{\psi(a'_i,y_i)}\right)_{x,y} \right)(b,b').$$

For a matrix $X$ on $A^k\times B^k$, $\bar d_S(X)=k^{-1}\sum_{i<k}\operatorname{Re}\operatorname{Tr}(\Delta X_i)$, where $X_i$ sums over equal row and column indices outside position $i$. The real quantity $E_F(X)$ is the infimum over all finite lists of nonnegative weights $p_j$ summing to $1$ and unit vectors $u_j\in\mathbb C^{A^k\times B^k}$ satisfying $X=\sum_jp_ju_ju_j^\dagger$, of

$$\sum_jp_jS_2\!\left( \left(\sum_{b\in B^k}u_j(a,b)\overline{u_j(a',b)}\right)_{a,a'} \right), \qquad S_2(H)=-\operatorname{Re}\operatorname{Tr}\bigl((x\mapsto x\log_2x)(H)\bigr).$$

The function of a matrix here is given by continuous functional calculus, with $0\log_2 0=0$; the real infimum defining $E_F$ is $0$ if its defining set is empty. The outer constrained infimum is instead taken in $[0,\infty]$ and is $+\infty$ if there is no feasible channel. Inverses at zero are zero, and multiplication in $[0,\infty]$ has $0\cdot\infty=0$; consequently the defined block value is $0$ when $k=0$, including when the constraint is infeasible.

QAlgorithms.Papers.Independent.Intakethird100.P27.iidChannelOutput

For finite types $A,B$, let $S$ contain a unit vector $\psi\in\mathbb C^{A\times A}$ and a positive semidefinite matrix $\Delta$ on $A\times B$. For any natural number $k$, including zero, and any matrix function $N$ from matrices on $A^k$ to matrices on $B^k$, this defines the matrix $X$ on $A^k\times B^k$ with entries

$$X((a,b),(a',b')) = N\!\left( \left( \prod_{i=0}^{k-1} \psi(a_i,x_i)\overline{\psi(a'_i,y_i)} \right)_{x,y\in A^k} \right)(b,b').$$

Here $A^k$ and $B^k$ mean functions from $\operatorname{Fin}(k)$ to the respective types. There is no channel assumption on $N$, and $\Delta$ does not enter this definition. For $k=0$, both function spaces have one element and the empty product is $1$, so the output is the one-by-one matrix obtained by applying $N$ to the one-by-one matrix $[1]$.

QAlgorithms.Papers.Independent.Intakethird100.P27.isotropicConvexUpper

For every real $D$, this defines

$$U(D)= \inf_{\substack{m\in\mathbb N,\ m>0,\ p_i\ge0,\ \sum_{i=1}^m p_i=1\\ 0\le d_i\le1,\ \sum_{i=1}^m p_id_i\le D}} \ \sum_{i=1}^m p_i C(d_i) \quad\text{in }[0,\infty],$$

where $C(d)=0$ for $d>3/4$, and $C(d)=E_{\mathrm p}((1-d)P+\frac d3(I-P))$ for $d\le3/4$. The matrix $P$ on two two-dimensional registers has entries $P_{(i,j),(k,l)}=1/2$ when $i=j$ and $k=l$, and zero otherwise. For a matrix $\rho$ on these two registers, $E_{\mathrm p}(\rho)$ is the infimum over positive finite ancillary dimensions $u,v$ and unit vectors $\xi$ on $(\{0,1\}\times\{1,\ldots,u\})\times(\{0,1\}\times\{1,\ldots,v\})$ whose marginal on the original registers is $\rho$, of the nonnegative part of $H(\operatorname{Tr}_{\{0,1\}\times\{1,\ldots,v\}}|\xi\rangle\langle\xi|)$. Here $(|\xi\rangle\langle\xi|)_{xy}=\xi_x\overline{\xi_y}$, and $H(X)=-\operatorname{Re}\operatorname{tr}(f(X))$, with $f(x)=x\log_2x$ applied by continuous functional calculus. Zero weights and repeated $d_i$ are allowed, and the weighted mean distortion need only be at most $D$. Products and infima use extended nonnegative arithmetic, including $0\cdot(+\infty)=0$ and $\inf\varnothing=+\infty$; in particular, when $D<0$, the defining feasible set is empty.

QAlgorithms.Papers.Independent.Intakethird100.P27.isotropicEntropy

For every real $D$, with no restriction to a probability range, define $p_0=1-D$ and $p_1=p_2=p_3=D/3$. The defined real number is

$$-\sum_{i=0}^{3}p_i\log_2p_i = -(1-D)\log_2(1-D)-3\frac D3\log_2\frac D3.$$

The logarithm is the total real function $\log_2x=\log x/\log2$, including its value $0$ at $x=0$; thus a zero coefficient contributes zero. For $D\notin[0,1]$, the same formula is used without requiring the four coefficients to be nonnegative.

QAlgorithms.Papers.Independent.Intakethird100.P27.isotropicStrategyCost

For every real $D$, this defines an extended nonnegative real number

$$C(D)= \begin{cases} E_{\mathrm p}\!\left((1-D)P+\dfrac D3(I-P)\right),&D\le \dfrac34,\\[2mm] 0,&D>\dfrac34, \end{cases}$$

where $P$ is the matrix on $\{0,1\}\times\{0,1\}$ whose entry $P_{(i,j),(k,l)}$ is $1/2$ when $i=j$ and $k=l$, and is zero otherwise. Here $E_{\mathrm p}(\rho)$ is the infimum in $[0,\infty]$ over positive natural dimensions $u,v$ and unit vectors $\xi$ on $(\{0,1\}\times\{1,\ldots,u\})\times(\{0,1\}\times\{1,\ldots,v\})$ whose marginal on the two $\{0,1\}$ registers equals $\rho$, of the nonnegative part of $H(\operatorname{Tr}_{\{0,1\}\times\{1,\ldots,v\}}|\xi\rangle\langle\xi|)$. The matrix $|\xi\rangle\langle\xi|$ has entries $\xi_x\overline{\xi_y}$, and $H(X)=-\operatorname{Re}\operatorname{tr}(f(X))$, with $f(x)=x\log_2x$ applied by continuous functional calculus. An empty infimum is $+\infty$. No restriction on $D$ is imposed: the first branch includes $D=3/4$ and every negative $D$, and the second includes every $D>3/4$.

QAlgorithms.Papers.Independent.Intakethird100.P27.letterMarginal

For finite types $A,B$, any natural number $n$, any complex matrix $X$ indexed by $A^n\times B^n$, and any $i\in\{0,\ldots,n-1\}$, the letter marginal at $i$ is the matrix on $A\times B$ given by

$$(X_i)_{(a,b),(a',b')}= \sum_{\substack{u,u'\in A^n,\ v,v'\in B^n\\ u_i=a,\ v_i=b,\ u'_i=a',\ v'_i=b'\\ \forall j\ne i,\ (u_j,v_j)=(u'_j,v'_j)}} X_{(u,v),(u',v')}.$$

Here $A^n$ and $B^n$ are the sets of length-$n$ tuples. No positivity or trace assumption is imposed on $X$. When $n=0$, there is no index $i$, so there is no instance of this definition with such an argument.

QAlgorithms.Papers.Independent.Intakethird100.P27.observableDistortion

For finite types $A,B$, a source consisting of a unit vector $\psi\in\mathbb C^{A\times A}$ and a positive semidefinite matrix $\Delta$ on $A\times B$, and any function $N$ from complex matrices on $A$ to complex matrices on $B$, the observable distortion is $\operatorname{Re}\operatorname{tr}(\Delta T)$, where

$$T_{(a,b),(a',b')}= N\!\left(\left[\psi(a,x)\overline{\psi(a',x')}\right]_{x,x'\in A}\right)_{b,b'}.$$

The definition imposes no channel hypothesis on $N$, and $\Delta$ need not be bounded by the identity.

QAlgorithms.Papers.Independent.Intakethird100.P27.outputWithSide

For arbitrary types $R,A,B,C$, any function $N$ from complex matrices on $A$ to complex matrices on $C$, and any vector $\phi\in\mathbb C^{R\times(A\times B)}$ equipped with the stated Euclidean-space structure, this defines a matrix $X$ on $R\times(C\times B)$ by

$$X_{(r,(c,b)),(r',(c',b'))} = N\!\left( \left[\phi_{(r,(a,b))}\, \overline{\phi_{(r',(a',b'))}}\right]_{a,a'\in A} \right)_{c,c'}.$$

Thus $N$ is applied separately to each matrix block obtained by fixing the two reference indices and the two $B$-indices. There are no finiteness, nonemptiness, normalization, linearity, positivity, or channel hypotheses.

QAlgorithms.Papers.Independent.Intakethird100.P27.purificationBlockValue

For finite types $A,B$, a source $S$ containing a unit vector $\psi\in\mathbb C^{A\times A}$ and a positive semidefinite distortion matrix $\Delta$ on $A\times B$, any real $D$, and any $k\in\mathbb N$, define

$$\max(k^{-1},0)\, \inf_{\substack{N\text{ a channel}\\\bar d_S(X_N)\le D}}\mathcal P(X_N) \quad\text{in }[0,\infty].$$

Here $N$ ranges over maps from matrices on $A^k$ to matrices on $B^k$ admitting finite Kraus representations $N(Z)=\sum_jK_jZK_j^\dagger$ with $\sum_jK_j^\dagger K_j=I$, and

$$X_N((a,b),(a',b')) = N\!\left( \left(\prod_{i<k}\psi(a_i,x_i)\overline{\psi(a'_i,y_i)}\right)_{x,y} \right)(b,b').$$

The distortion is $\bar d_S(X)=k^{-1}\sum_{i<k}\operatorname{Re}\operatorname{Tr}(\Delta X_i)$, where $X_i$ sums over equal row and column indices outside position $i$. For a matrix $X$ on $A^k\times B^k$, $\mathcal P(X)$ is the infimum in $[0,\infty]$ of $\max(S_2(H_\phi),0)$ over positive integers $u,v$ and unit vectors $\phi$ on $(A^k\times\operatorname{Fin}(u))\times(B^k\times\operatorname{Fin}(v))$ whose matrix $\phi\phi^\dagger$, after summing over equal auxiliary indices, equals $X$. The matrix $H_\phi$ is obtained by summing $\phi\phi^\dagger$ over equal $B^k\times\operatorname{Fin}(v)$ indices, and $S_2(H)=-\operatorname{Re}\operatorname{Tr}((x\mapsto x\log_2x)(H))$, with $0\log_2 0=0$. Either infimum is $+\infty$ if its indexing set is empty. Zero inverses and $0\cdot\infty$ are zero, so the defined block value is $0$ for $k=0$, irrespective of feasibility.

QAlgorithms.Papers.Independent.Intakethird100.P27.purificationCost

For finite types $A,B$ and any complex matrix $\rho$ on $A\times B$, this defines the infimum in $[0,\infty]$ of $\max(S_2(H_\phi),0)$ over all positive integers $u,v$ and unit vectors $\phi\in\mathbb C^{(A\times\operatorname{Fin}(u))\times(B\times\operatorname{Fin}(v))}$ such that

$$\rho((a,b),(a',b')) = \sum_{\alpha<u,\ \beta<v} \phi((a,\alpha),(b,\beta)) \overline{\phi((a',\alpha),(b',\beta))}.$$

Here $H_\phi$ is the matrix on $A\times\operatorname{Fin}(u)$ obtained by summing $\phi\phi^\dagger$ over equal $B\times\operatorname{Fin}(v)$ indices, and

$$S_2(H_\phi) =-\operatorname{Re}\operatorname{Tr}\bigl((x\mapsto x\log_2x)(H_\phi)\bigr),$$

with continuous functional calculus and $0\log_2 0=0$. The auxiliary dimensions have no upper bound. The definition does not assume that $\rho$ is a density matrix and does not assert attainment. If no such purification exists, including when either underlying type is empty, the value is $+\infty$.

QAlgorithms.Papers.Independent.Intakethird100.P27.purificationValues

For finite types $A,B$ and an arbitrary complex matrix $\rho$ on $A\times B$, this defines the set of all $r\in[0,\infty]$ for which there exist positive integers $u,v$ and a unit vector $\phi\in\mathbb C^{(A\times\operatorname{Fin}(u))\times(B\times\operatorname{Fin}(v))}$ satisfying

$$\rho((a,b),(a',b')) = \sum_{\alpha<u}\sum_{\beta<v} \phi((a,\alpha),(b,\beta)) \overline{\phi((a',\alpha),(b',\beta))}, \qquad r=\max(S_2(H_\phi),0),$$

where

$$H_\phi((a,\alpha),(a',\alpha')) = \sum_{b\in B}\sum_{\beta<v} \phi((a,\alpha),(b,\beta)) \overline{\phi((a',\alpha'),(b,\beta))},$$

and $S_2(H)=-\operatorname{Re}\operatorname{Tr}((x\mapsto x\log_2x)(H))$, using continuous functional calculus and $0\log_2 0=0$. No density assumption on $\rho$ is a binder; membership requires the displayed normalized purification. There is no upper bound on $u,v$. If either $A$ or $B$ is empty, no such unit vector exists and the set is empty.

QAlgorithms.Papers.Independent.Intakethird100.P27.rateDistortion

For finite types $A,B$, a source $S$ consists of a vector $\psi\in\mathbb C^{A\times A}$ with $\|\psi\|=1$ and a positive semidefinite matrix $\Delta$ on $A\times B$. For an assistance choice $a\in\{\text{unassisted},\text{entanglement},\text{classical}\}$ and any real $D$, this defines a value in $[0,\infty]$ by

$$\inf_{\substack{r\in\mathbb R\\ r\text{ satisfies the conditions below}}}\max(r,0).$$

The conditions on $r$ are $r\ge0$, $D\ge0$, and the following: for every $\varepsilon>0$ and $s>0$, there is $n_0\in\mathbb N$ such that for every $n\ge n_0$ with $n\ge1$, there is a code with positive integer quantum, classical, and shared dimensions $q,c,h$; a positive semidefinite trace-one shared matrix $\tau$ on $\operatorname{Fin}(h)\times\operatorname{Fin}(h)$; an encoder from $A^n\times\operatorname{Fin}(h)$ to $\operatorname{Fin}(c)\times\operatorname{Fin}(q)$; and a decoder from $(\operatorname{Fin}(c)\times\operatorname{Fin}(q))\times\operatorname{Fin}(h)$ to $B^n$. Both maps must admit finite Kraus representations $\Phi(X)=\sum_j K_jXK_j^\dagger$ with $\sum_jK_j^\dagger K_j=I$. Every encoder output must have zero entries between distinct classical labels. The assistance restrictions are $c=h=1$ for unassisted codes, $c=1$ for entanglement assistance, and $h=1$ for classical assistance. Apply the encoder and decoder to the second register of $n$ copies of $\psi$, together with $\tau$, leaving the first $A^n$ register untouched, and call the resulting matrix $X$ on $A^n\times B^n$. The required bounds are

$$\log_2q\le n(r+s), \qquad \frac1n\sum_{i=0}^{n-1}\operatorname{Re}\operatorname{Tr}(\Delta X_i)\le D+\varepsilon,$$

where $X_i$ is obtained by summing over equal row and column indices at all positions other than $i$. The infimum of an empty family is $+\infty$; in particular the value is $+\infty$ when $D<0$. No positive lower bound on the block length beyond $n\ge1$, and no bound on the classical or shared dimensions beyond the stated assistance restrictions, is imposed.

QAlgorithms.Papers.Independent.Intakethird100.P27.regularizedPurification

For finite types $A,B$, a source $S$ containing a unit vector $\psi\in\mathbb C^{A\times A}$ and a positive semidefinite matrix $\Delta$ on $A\times B$, and any real $D$, this defines the extended nonnegative number

$$\liminf_{k\to\infty}b_{k+1} = \sup_{K\in\mathbb N}\inf_{k\ge K}b_{k+1}.$$

For each positive integer $j$,

$$b_j=\frac1j \inf_{\substack{N\text{ a channel}\\ j^{-1}\sum_{i<j}\operatorname{Re}\operatorname{Tr}(\Delta(X_N)_i)\le D}} \mathcal P(X_N), \qquad X_N((a,b),(a',b')) = N\!\left( \left(\prod_{i<j}\psi(a_i,x_i)\overline{\psi(a'_i,y_i)}\right)_{x,y} \right)(b,b').$$

The channel $N$ maps matrices on $A^j$ to matrices on $B^j$ and must admit a finite representation $N(Z)=\sum_\ell K_\ell ZK_\ell^\dagger$ with $\sum_\ell K_\ell^\dagger K_\ell=I$. Each $(X_N)_i$ sums over equal row and column indices at every position except $i$. For any matrix $X$ on $A^j\times B^j$, $\mathcal P(X)$ is the infimum in $[0,\infty]$ of $\max(S_2(H_\phi),0)$ over positive integers $u,v$ and unit vectors $\phi$ on $(A^j\times\operatorname{Fin}(u))\times(B^j\times\operatorname{Fin}(v))$ whose matrix $\phi\phi^\dagger$, after summing over equal auxiliary indices, equals $X$; $H_\phi$ sums that matrix over equal $B^j\times\operatorname{Fin}(v)$ indices, and $S_2(H)=-\operatorname{Re}\operatorname{Tr}((x\mapsto x\log_2x)(H))$, with $0\log_2 0=0$. Empty infima are $+\infty$. Only positive block lengths enter this limit inferior, and the definition does not assert that the sequence converges or that any infimum is attained.

QAlgorithms.Papers.Independent.Intakethird100.P27.sourceFidelity

For arbitrary finite types $R,A$, any vector $v\in\mathbb C^{R\times A}$, and any function $N$ from complex matrices on $A$ to complex matrices on $A$, define $Q_{r,s}$ by

$$Q_{r,s}(x,y)=v(r,x)\overline{v(s,y)}.$$

The defined real number is

$$\operatorname{Re}\sum_{r,s\in R}\sum_{a,b\in A} \overline{v(r,a)}\,N(Q_{r,s})(a,b)\,v(s,b).$$

Thus $N$ is applied separately to each $A$-matrix block of the matrix $vv^\dagger$, and the resulting matrix is evaluated against $v$. Neither normalization of $v$ nor linearity, positivity, or trace preservation of $N$ is assumed. Empty finite types are included, in which case the displayed sum is zero.

QAlgorithms.Papers.Independent.Intakethird100.P27.testOutput

For finite types $A,B$, a source $S=(\psi,\Delta)$ with $\|\psi\|=1$ and $\Delta\ge0$, and any function $N$ from complex $A\times A$ matrices to complex $B\times B$ matrices, the test output is the matrix $T$ indexed by $A\times B$ whose entries are

$$T_{(a,b),(a',b')}= N\!\left(\left[\psi(a,x)\,\overline{\psi(a',x')}\right]_{x,x'\in A}\right)_{b,b'}.$$

Thus $N$ is applied separately to each block obtained by fixing the first $A$ index on each side of the matrix $[\psi(a,x)\overline{\psi(a',x')}]$. No linearity, positivity, or channel hypothesis is imposed on $N$.