Definitions for Fundamental limits to quantum channel discrimination

QAlgorithms.Papers.Independent.Intakethird100.P49.AdaptiveProtocol

For any natural numbers $d,e,o,n$, this structure consists of a natural number $r$ with $r>0$; an initial complex matrix $\rho_0$ on $\mathbb C^r\otimes\mathbb C^d$ that is positive semidefinite and has complex trace $1$; a function assigning to every $k\in\mathbb N$ a channel $\mathcal B_k$ from matrices on $\mathbb C^r\otimes\mathbb C^e$ to matrices on $\mathbb C^r\otimes\mathbb C^d$; a channel $\mathcal Z$ from matrices on $\mathbb C^r\otimes\mathbb C^d$ to complex $o\times o$ matrices; and a channel $\mathcal T$ from matrices on $\mathbb C^r\otimes\mathbb C^e$ to complex $o\times o$ matrices. Each channel requirement means that the corresponding function equals $X\mapsto\sum_{j=0}^{m-1}K_jXK_j^\dagger$ for some natural number $m$ and matrices $K_j$ satisfying $\sum_jK_j^\dagger K_j=I$. The parameter $n$ imposes no further condition on these fields, and the intermediate channels are specified for every natural index, including indices beyond $n$. The dimension parameters are not separately required to be positive, although the density and channel requirements can make some zero-dimensional choices impossible.

QAlgorithms.Papers.Independent.Intakethird100.P49.AdmissiblePrivateMeasure

Let $E$ assign a real number $E_{A:B}(\rho)$ to every complex matrix on $A\times B$, for all finite nonempty index types $A,B$ with decidable equality; let $g,h:\mathbb R\to\mathbb R$ and $\mathcal D\subseteq\mathbb R$. This predicate means that all the following hold jointly: $E_{A:B}(\rho)\ge0$ for every density matrix $\rho$, and $E_{C:D}(L(\rho))\le E_{A:B}(\rho)$ for every such $\rho$ and every finite local protocol $L$ from $A:B$ to $C:D$; these protocols, after identifying the bases with finite integer index sets, are generated by identities and finite local Kraus families satisfying $\sum_iK_i^\dagger K_i=1$, with an independently chosen subsequent protocol for each outcome and the resulting branches summed. Both $g(t)$ and $h(t)$ tend to zero as $t\to0$ through positive real values, and $\mathcal D\subseteq[0,\infty)$. For every four finite nonempty index types $A,B,C,D$ with decidable equality and density matrices $\rho$ on $A\times B$ and $\sigma$ on $C\times D$,

$$E_{(A\times C):(B\times D)}(\rho\boxtimes\sigma) \le E_{A:B}(\rho)+E_{C:D}(\sigma), \quad (\rho\boxtimes\sigma)_{(a,c,b,d),(a',c',b',d')} =\rho_{(a,b),(a',b')}\sigma_{(c,d),(c',d')}.$$

For every $k,s\in\mathbb N$, every density matrix $\sigma$ on $\{0,\ldots,s\}^2$, and every family of unitary matrices $U_a$ on that space indexed by $0\le a<2^k$, one has $k\le E(\gamma)$, with the bipartition $(a,x):(b,y)$, where

$$\gamma_{((a,x),(b,y)),((a',x'),(b',y'))} = \begin{cases} 2^{-k}(U_a\sigma U_{a'}^\dagger)_{(x,y),(x',y')}, &a=b,\ a'=b',\\ 0,&\text{otherwise}. \end{cases}$$

Finally, for every finite nonempty $A,B$ with decidable equality, density matrices $\rho,\sigma$ on $A\times B$, and $\gamma\in\mathcal D$ satisfying $\|\rho-\sigma\|_1\le\gamma$,

$$|E_{A:B}(\rho)-E_{A:B}(\sigma)| \le g(\gamma)\log_2(|A||B|)+h(\gamma).$$

Here a density matrix is positive semidefinite with trace $1$, and $\|X\|_1=\operatorname{Re}\operatorname{tr}\sqrt{X^\dagger X}$. The cases $k=0$ and $s=0$ are included; $\mathcal D$ may be empty, making the last condition vacuous, and no separate nonnegativity assumption on $g,h$ is imposed.

QAlgorithms.Papers.Independent.Intakethird100.P49.Ports

For natural numbers $d,M$, this denotes the set of functions $\{0,\ldots,M-1\}\to\{0,\ldots,d-1\}$, equivalently length-$M$ strings with symbols in a $d$-element set. When $M=0$, it contains exactly the empty string, including when $d=0$; when $d=0<M$, it is empty.

QAlgorithms.Papers.Independent.Intakethird100.P49.adaptiveProtocolOutput

For natural numbers $d,e,o,n$, a protocol $P$, and an arbitrary function $\mathcal E$ from complex $d\times d$ matrices to complex $e\times e$ matrices, this defines a complex $o\times o$ matrix by

$$\operatorname{adaptiveProtocolOutput}(P,\mathcal E)= \begin{cases} P.\mathrm{terminalZero}(P.\mathrm{initial}),&n=0,\\ P.\mathrm{terminal}\!\left(\widehat{\mathcal E}(\rho_{n-1})\right),&n>0, \end{cases} \qquad \rho_0=P.\mathrm{initial},\quad \rho_{k+1}=P.\mathrm{between}_k(\widehat{\mathcal E}(\rho_k)).$$

Here the memory dimension $r=P.\mathrm{memory}$ is positive, and $\widehat{\mathcal E}(X)_{(u,b),(v,c)}=\mathcal E((X_{(u,i),(v,j)})_{i,j})_{b,c}$. The initial matrix is positive semidefinite with trace $1$. Every intermediate map, the map $P.\mathrm{terminalZero}$ from the memory–input space, and the map $P.\mathrm{terminal}$ from the memory–output space has a finite Kraus representation $X\mapsto\sum_jK_jXK_j^\dagger$ with $\sum_jK_j^\dagger K_j=I$. No such assumption is made on $\mathcal E$. The zero-use branch does not apply $\mathcal E$; the positive-use branch applies it $n$ times, with $n-1$ intermediate maps.

QAlgorithms.Papers.Independent.Intakethird100.P49.amortizedREE

For positive natural numbers $d,e$ and any function $\mathcal E$ from complex $d\times d$ matrices to complex $e\times e$ matrices, this defines the extended-real supremum

$$\sup_{\substack{a,b\in\mathbb N\\ \rho\ \mathrm{density\ on}\ ((\{0,\ldots,a\}\times\{0,\ldots,d-1\})\times\{0,\ldots,b\})}} \left[R_{\mathbb R}(\omega)-R_{\mathbb R}(\rho)\right],$$

where

$$\omega_{(x,(u,y)),(x',(u',y'))} =\mathcal E\!\left( \bigl[\rho_{((x,i),y),((x',j),y')}\bigr]_{i,j} \right)_{u,u'}.$$

The input bipartition is $(x,i):y$, and the output bipartition is $x:(u,y)$. For either bipartition, $R_{\mathbb R}(X)$ is the real conversion of the infimum, over finite convex combinations of product density matrices $\sigma$, of $D_{\mathrm{bits}}(X\|\sigma)$; $+\infty$ converts to $0$. Here $D_{\mathrm{bits}}=+\infty$ unless both arguments are positive semidefinite of trace $1$ and $\ker\sigma\subseteq\ker X$; otherwise it equals $\max\{0,\operatorname{Re}\operatorname{tr}[(t\mapsto t\ln t)(X)-X(\ln)(\sigma)]\}/\ln2$. All functions of matrices use continuous functional calculus. The ancillary parameters $a,b$ may be zero, giving one-dimensional ancillary registers. No linearity or channel assumption on $\mathcal E$ is imposed, so $\omega$ need not be a density matrix.

QAlgorithms.Papers.Independent.Intakethird100.P49.bellPorts

For natural numbers $d,M$, this defines a complex square matrix $B$ indexed by pairs of strings in $D^M$, where $D=\{0,\ldots,d-1\}$, by

$$B_{(x,y),(x',y')} = \prod_{i=0}^{M-1} \begin{cases} 1/d,&x_i=y_i\ \text{and }x'_i=y'_i,\\ 0,&\text{otherwise}. \end{cases}$$

For $d>0$, this entry is $d^{-M}$ when $x=y$ and $x'=y'$, and zero otherwise. When $M=0$, the index set is a singleton and its sole entry is the empty product $1$, including when $d=0$. When $d=0<M$, the matrix has no entries.

QAlgorithms.Papers.Independent.Intakethird100.P49.bellProbeBlockError

For natural numbers $d,e,n$ with $d\ne0$, and arbitrary functions $\mathcal E_0,\mathcal E_1$ from complex $d\times d$ matrices to complex $e\times e$ matrices, this defines

$$\frac{1-\tfrac12\|C_0^{[n]}-C_1^{[n]}\|_1}{2}, \qquad C_i((a,b),(a',b'))= \mathcal E_i(d^{-1}|a\rangle\langle a'|)_{b,b'}.$$

Here $a,a'<d$, $b,b'<e$, $|a\rangle\langle a'|$ is the matrix unit, $C^{[n]}(x,y)=\prod_{j=0}^{n-1}C(x_j,y_j)$, and $\|X\|_1=\operatorname{Re}\operatorname{Tr}\sqrt{X^\dagger X}$. Equivalently, $C_i$ is obtained by extending $\mathcal E_i$ blockwise to the second factor of the pure density matrix of $d^{-1/2}\sum_{a<d}|a,a\rangle$. The functions $\mathcal E_i$ are not assumed to be linear or channels, and $e=0$ is permitted. When $n=0$, each product is empty and equals $1$, so the two matrices coincide and the defined number is $1/2$.

QAlgorithms.Papers.Independent.Intakethird100.P49.capacityContinuity

For every real number $t$, this defines

$$c(t)=(1+t)\log_2(1+t)-t\log_2t.$$

There is no restriction $t\ge0$. The logarithms are the total real logarithms used by the code: $\log_2 0=0$, and for nonzero negative arguments $\log_2x=\ln|x|/\ln2$; in particular, $c(0)=0$.

QAlgorithms.Papers.Independent.Intakethird100.P49.choiPorts

For natural numbers $d,e$, with the explicit assumption $d\ne0$, any function $E$ from complex $d\times d$ matrices to complex $e\times e$ matrices, and any natural number $M$, this defines a complex matrix indexed by $D^M\times F^M$, where $D=\{0,\ldots,d-1\}$ and $F=\{0,\ldots,e-1\}$, as follows. Let $B^{r,s}$ be the $d\times d$ matrix whose sole nonzero entry is $1/d$ in position $(r,s)$. The entries are

$$C_{(x,y),(x',y')} = \prod_{i=0}^{M-1} E(B^{x_i,x'_i})_{y_i,y'_i}.$$

Each factor is the corresponding entry of the matrix obtained by applying the identity on one half of the normalized maximally entangled projector and $E$ on the other half. No linearity or channel assumption is made on $E$, so the factor $1/d$ remains inside its argument. When $M=0$, the matrix has one entry, equal to the empty product $1$; $e=0$ is allowed.

QAlgorithms.Papers.Independent.Intakethird100.P49.dampingFidelity

For every pair of real numbers $p,q$, this defines

$$\operatorname{dampingFidelity}(p,q) =\frac{1+\sqrt{(1-p)(1-q)}+\sqrt{pq}}2.$$

There are no assumptions that $p$ or $q$ lie in $[0,1]$. Each real square root is the total real square-root function, which returns $0$ for a negative argument.

QAlgorithms.Papers.Independent.Intakethird100.P49.dampingSimulationError

For every natural number $M$ and real number $p$, this defines

$$\xi_M\left(\frac{1-p}{2}+\sqrt{1-p}\right).$$

The scalar $\xi_M$ is specified as follows. On indices $(a,u)$, where $a\in\{0,1\}$ and $u\in\{0,1\}^M$, define $S_i((a,u),(b,v))=1/2$ when $a=u_i$, $b=v_i$, and $u_j=v_j$ for every $j\ne i$, and define it to be $0$ otherwise. Set $S=\sum_{i<M}S_i$, $R=f(S)$ by continuous functional calculus for $f(x)=(\sqrt x)^{-1}$, and $Q_i=RS_iR+M^{-1}(I-RSR)$. Then

$$\xi_M= 1-\operatorname{Re}\!\left( 2^{-M}\sum_{i<M} \sum_{\substack{u,v\in\{0,1\}^M\\u_i=0,\ v_i=1\\u_j=v_j\ (j\ne i)}} Q_i((1,v),(0,u)) \right).$$

Thus $\xi_M$ is one minus the real $(0,1)$ entry obtained by applying the specified selection map to the matrix unit $|0\rangle\langle1|$. All inverses at zero are $0$; $M=0$ is included and gives $\xi_0=1$ through the empty sum. There is no restriction on $p$, and $\sqrt{1-p}=0$ when $p>1$. This declaration defines a real number without asserting that it is an error probability.

QAlgorithms.Papers.Independent.Intakethird100.P49.illuminationBlock

For every $d\in\mathbb N$ and arbitrary $b,\eta\in\mathbb R$, this defines a complex $(d+1)\times(d+1)$ matrix with real entries

$$B_{ij}=\frac1{(d+1)^2} \begin{cases} (1-\eta)(1-db)^2+\eta(1-db),&i=j=0,\\ \eta\sqrt{(1-db)b},&\text{exactly one of }i,j\text{ is }0,\\ (1-\eta)b^2+\eta b,&i=j\ne0,\\ \eta b,&i\ne j,\ i,j\ne0. \end{cases}$$

No positivity or range conditions are imposed. A negative square-root argument gives zero, while the denominator is always positive. For $d=0$, the matrix is $(1)$.

QAlgorithms.Papers.Independent.Intakethird100.P49.illuminationChannel

For every natural number $d$ and arbitrary real numbers $b,\eta$, this defines a function on complex $(d+1)\times(d+1)$ matrices by

$$\Lambda_{d,b,\eta}(X) =(1-\eta)\operatorname{tr}(X)\, \operatorname{diag}(1-db,b,\ldots,b)+\eta X.$$

The input $X$ is arbitrary, and no assumptions that the displayed function is a channel or that the diagonal matrix is positive are part of this definition. The case $d=0$ is included.

QAlgorithms.Papers.Independent.Intakethird100.P49.illuminationNoise

For every natural number $d$, including $d=0$, and every $b\in\mathbb R$, this defines the complex diagonal matrix

$$\tau_{d,b}=\operatorname{diag}(1-db,\underbrace{b,\ldots,b}_{d\text{ entries}})$$

on the indices $0,\ldots,d$. No positivity assumption on these entries is imposed. When $d=0$, the matrix is the $1\times1$ matrix $(1)$.

QAlgorithms.Papers.Independent.Intakethird100.P49.illuminationOmegaSquared

For every $d\in\mathbb N$ and $b,\eta\in\mathbb R$, put $D=d+1$, $\tau=\operatorname{diag}(1-db,b,\ldots,b)$, and define matrices on $\{0,\ldots,d\}^2$ by

$$(C_t)_{(i,a),(j,b')} =\frac{1-t}{D}\,\mathbf1_{i=j}\tau_{a b'} +\frac{t}{D}\,\mathbf1_{a=i}\mathbf1_{b'=j}.$$

These are obtained by applying $X\mapsto(1-t)\operatorname{tr}(X)\tau+tX$ to the second register of the projector onto $D^{-1/2}\sum_i|i,i\rangle$. The declaration defines

$$\Omega^2_{d,b,\eta} =\bigl(\sqrt{\phantom{x}}\,\bigr)(C_0)\,C_\eta\, \bigl(\sqrt{\phantom{x}}\,\bigr)(C_0),$$

where the square-root function is applied through the real continuous functional calculus. No positivity or parameter restrictions are assumed; negative eigenvalues of the Hermitian matrix $C_0$ contribute zero under the real square root. The case $d=0$ is included.

QAlgorithms.Papers.Independent.Intakethird100.P49.meanSimulationError

For natural numbers $d,e,M$ and arbitrary functions $\mathcal E_0,\mathcal E_1$ from complex $d\times d$ matrices to complex $e\times e$ matrices, this defines

$$\frac{\|\mathcal E_0-\mathcal E_0\circ\mathcal P_{d,M}\|_\diamond+ \|\mathcal E_1-\mathcal E_1\circ\mathcal P_{d,M}\|_\diamond}{2}.$$

For a function $F$ of these matrix dimensions, the notation used here means $\|F\|_\diamond=\sup_{\|X\|_1\le1}\|\widehat F(X)\|_1$, where $X$ is a matrix on $\mathbb C^d\otimes\mathbb C^d$, $\widehat F(X)_{(a,b),(a',b')}=F((X_{(a,i),(a',j)})_{i,j})_{b,b'}$, and $\|X\|_1=\operatorname{Re}\operatorname{Tr}\sqrt{X^\dagger X}$. To define $\mathcal P_{d,M}$, for $i<M$ and $u,v\in[d]^M$, set $S_i((a,u),(b,v))=d^{-1}$ if $a=u_i$, $b=v_i$, and $u_j=v_j$ for all $j\ne i$, and set it to $0$ otherwise. Let $S=\sum_iS_i$, let $R=f(S)$ by continuous functional calculus for $f(x)=(\sqrt x)^{-1}$, and let $Q_i=RS_iR+M^{-1}(I-RSR)$. The simulation function is

$$\mathcal P_{d,M}(X)_{b,c} =d^{-M}\sum_{i<M}\sum_{a,a'<d} \sum_{\substack{u,v\in[d]^M\\u_i=b,\ v_i=c\\u_j=v_j\ (j\ne i)}} Q_i((a',v),(a,u))X_{a,a'}.$$

Zero dimensions and $M=0$ are included, empty products are $1$, and all inverses at zero are $0$; in particular $\mathcal P_{d,0}=0$. The functions $\mathcal E_i$ need not be linear or channels. The suprema are total real suprema, taking value $0$ when their defining sets are unbounded above.

QAlgorithms.Papers.Independent.Intakethird100.P49.opticalAntiBell

This definition specifies a vector in the complex Euclidean space indexed by $\{0,1\}\times\{0,1\}$: its coordinate at $(i,j)$ is $1/\sqrt2$ when $i\ne j$, and zero when $i=j$. Equivalently, the vector is

$$\frac{|0,1\rangle+|1,0\rangle}{\sqrt2},$$

where each displayed basis vector has coordinate one at its named index and zero elsewhere.

QAlgorithms.Papers.Independent.Intakethird100.P49.opticalChannel

For arbitrary real numbers $\eta,\delta$ and a Boolean value $s$, this definition gives a map from complex $2\times2$ matrices to complex $3\times3$ matrices. Put

$$a=\sqrt{\frac{(1+\delta)\eta}{2}}, \qquad b=\begin{cases} -\sqrt{\frac{(1-\delta)\eta}{2}},&s=\mathrm{true},\\ \sqrt{\frac{(1-\delta)\eta}{2}},&s=\mathrm{false}, \end{cases}$$

and define

$$\mathcal O_{\eta,\delta,s}(X) =K_0XK_0^\dagger+K_1XK_1^\dagger, \qquad K_0=\begin{pmatrix}1&0\\0&a\\0&b\end{pmatrix}, \quad K_1=\begin{pmatrix}0&\sqrt{1-\eta}\\0&0\\0&0\end{pmatrix}.$$

There are no restrictions on $\eta$ or $\delta$, and the definition does not assert Kraus completeness or trace preservation; each real square root is defined to be zero when its argument is negative.

QAlgorithms.Papers.Independent.Intakethird100.P49.opticalChoi

For arbitrary real numbers $\eta,\delta$ and a Boolean $s$, this defines a complex matrix on $\{0,1\}\times\{0,1,2\}$ by

$$C_{\eta,\delta,s}=(\operatorname{id}\otimes\Phi_{\eta,\delta,s})(|\chi\rangle\langle\chi|), \qquad |\chi\rangle=\frac{|0,1\rangle+|1,0\rangle}{\sqrt2},$$

where $\Phi(X)=K_0XK_0^\dagger+K_1XK_1^\dagger$, with

$$K_0=\begin{pmatrix} 1&0\\ 0&\sqrt{(1+\delta)\eta/2}\\ 0&\varepsilon_s\sqrt{(1-\delta)\eta/2} \end{pmatrix}, \qquad K_1=\begin{pmatrix}0&\sqrt{1-\eta}\\0&0\\0&0\end{pmatrix}, \quad \varepsilon_s=\begin{cases}-1&s=\mathrm{true},\\1&s=\mathrm{false}.\end{cases}$$

Here $|v\rangle\langle v|$ has entries $v_i\overline{v_j}$, and the identity extension applies $\Phi$ separately to each block indexed by the first register. No range assumptions on $\eta,\delta$ are imposed; real square roots of negative numbers are zero.

QAlgorithms.Papers.Independent.Intakethird100.P49.opticalPsi

For arbitrary $\eta,\delta\in\mathbb R$ and Boolean $s$, this defines the vector in $\mathbb C^{\{0,1\}\times\{0,1,2\}}$

$$\psi_{\eta,\delta,s} =\frac{|1,0\rangle+\sqrt{(1+\delta)\eta/2}\,|0,1\rangle +\varepsilon_s\sqrt{(1-\delta)\eta/2}\,|0,2\rangle} {\sqrt{1+\eta}}, \qquad \varepsilon_s=\begin{cases}-1&s=\mathrm{true},\\1&s=\mathrm{false}.\end{cases}$$

All other coordinates are zero. There are no parameter restrictions or normalization hypotheses. Real square roots of negative numbers and division by zero are interpreted as zero; in particular, the vector is zero when $\eta\le-1$.

QAlgorithms.Papers.Independent.Intakethird100.P49.pbtChannel

For natural numbers $d,M$, let $D=\{0,\ldots,d-1\}$. On $D\times D^M$, define $A_i[(r,x),(s,y)]=1/d$ if $r=x_i$, $s=y_i$, and $x_j=y_j$ for all $j\ne i$, and zero otherwise; put $S=\sum_iA_i$, $R=f(S)$ using continuous functional calculus with $f(t)=(\sqrt t)^{-1}$, and $H_i=RA_iR+M^{-1}(I-RSR)$. This declaration defines a function $\mathcal T_{d,M}$ from complex $D\times D$ matrices to such matrices, whose entries can be written

$$\mathcal T_{d,M}(X)_{a,b} = d^{-M} \sum_{\substack{i<M,\ r,s\in D,\ x,y\in D^M\\ x_i=a,\ y_i=b,\ x_j=y_j\ \forall j\ne i}} (H_i)_{(s,y),(r,x)}X_{r,s}.$$

Here $d^{-M}$ means $(d^{-1})^M$, and all reciprocals are total, taking value zero at zero. There is no restriction on $X$, and the definition includes $d=0$ and $M=0$. For $M=0$, this function is the zero map; for $d=0$, its input and output matrices have empty index sets.

QAlgorithms.Papers.Independent.Intakethird100.P49.pbtEntanglementFidelity

For natural numbers $d,M$, with the explicit assumption $d\ne0$, put $D=\{0,\ldots,d-1\}$. On $D\times D^M$, let $A_i[(r,x),(s,y)]=1/d$ when $r=x_i$, $s=y_i$, and $x_j=y_j$ for all $j\ne i$, and zero otherwise; set $S=\sum_iA_i$, $R=f(S)$ by continuous functional calculus for $f(t)=(\sqrt t)^{-1}$, and $H_i=RA_iR+M^{-1}(I-RSR)$. The declared real number, obtained by evaluating the normalized Choi matrix against the vector whose $(r,s)$-coordinate is $d^{-1/2}$ if $r=s$ and zero otherwise, is explicitly

$$\frac{1}{d^{M+2}} \operatorname{Re} \sum_{\substack{r,s\in D,\ i<M,\ x,y\in D^M\\ x_i=r,\ y_i=s,\ x_j=y_j\ \forall j\ne i}} (H_i)_{(s,y),(r,x)}.$$

The index $M$ may be zero, in which case the sum and the declared number are zero. In the functional calculus, the reciprocal of $\sqrt t$ is zero for $t\le0$.

QAlgorithms.Papers.Independent.Intakethird100.P49.pbtError

For natural numbers $d,M$, put $D=\{0,\ldots,d-1\}$. Define $A_i[(r,x),(s,y)]$, for $r,s\in D$ and $x,y\in D^M$, to be $1/d$ when $r=x_i$, $s=y_i$, and $x_j=y_j$ for every $j\ne i$, and zero otherwise. Set $S=\sum_iA_i$, $R=f(S)$ by continuous functional calculus with $f(t)=(\sqrt t)^{-1}$, $H_i=RA_iR+M^{-1}(I-RSR)$, and

$$\mathcal T(X)_{a,b} = d^{-M}\! \sum_{\substack{i<M,\ r,s\in D,\ x,y\in D^M\\ x_i=a,\ y_i=b,\ x_j=y_j\ \forall j\ne i}} (H_i)_{(s,y),(r,x)}X_{r,s}.$$

For a complex matrix $Z$ indexed by $D\times D$, let $Z^{r,s}$ be the matrix with entries $Z^{r,s}_{u,v}=Z_{(r,u),(s,v)}$, and define $\widehat\Delta(Z)_{(r,a),(s,b)}=Z_{(r,a),(s,b)}-\mathcal T(Z^{r,s})_{a,b}$. The declared real number is

$$\sup\left\{\|\widehat\Delta(Z)\|_1:\|Z\|_1\le1\right\}, \qquad \|Y\|_1=\operatorname{Re}\operatorname{Tr}\sqrt{Y^\dagger Y},$$

with the square root interpreted by continuous functional calculus. The supremum includes arbitrary complex matrices $Z$, without positivity or Hermiticity requirements, and has no factor $1/2$. Zero sizes are included; at $M=0$, $\mathcal T$ is zero. All reciprocals use the convention $0^{-1}=0$.

QAlgorithms.Papers.Independent.Intakethird100.P49.pbtNumber

For natural numbers $d,M$, this defines the number to be $0$ if $d<2$. If $d\ge2$, put $D=\{0,\ldots,d-1\}$, and on $D\times D^M$ define $A_i[(r,x),(s,y)]=1/d$ when $r=x_i$, $s=y_i$, and $x_j=y_j$ for all $j\ne i$, and zero otherwise; set $S=\sum_iA_i$, $R=f(S)$ using continuous functional calculus with $f(t)=(\sqrt t)^{-1}$, and $H_i=RA_iR+M^{-1}(I-RSR)$. In this case the number is

$$1-\operatorname{Re}\left[ d^{-M} \sum_{\substack{i<M,\ x,y\in D^M\\ x_i=0,\ y_i=1,\ x_j=y_j\ \forall j\ne i}} (H_i)_{(1,y),(0,x)} \right].$$

Equivalently, the subtracted quantity is the real part of the $(0,1)$ entry obtained by applying the port map to the matrix having a single $1$ in entry $(0,1)$. Thus $M=0$ gives $1$ when $d\ge2$. Reciprocals are total, with $0^{-1}=0$.

QAlgorithms.Papers.Independent.Intakethird100.P49.pbtPOVM

For natural numbers $d,M$ and $i\in\{0,\ldots,M-1\}$, let $\mathcal H=\{0,\ldots,d-1\}\times\{0,\ldots,d-1\}^{M}$, and define matrices on $\mathcal H$ by $(A_j)_{(a,x),(b,y)}=1/d$ when $a=x_j$, $b=y_j$, and $x_\ell=y_\ell$ for every $\ell\ne j$, and zero otherwise. Set

$$S=\sum_{j=0}^{M-1}A_j,\qquad R=f(S),\quad f(t)=(\sqrt t)^{-1},$$

where $f(S)$ uses continuous functional calculus. This declaration defines the matrix

$$H_i=RA_iR+\frac1M(I-RSR).$$

The scalar function uses the real square root and the total reciprocal, so $f(t)=0$ for $t\le0$. The index $i$ requires $M>0$; $d=0$ is allowed and gives an empty matrix. This definition itself imposes no additional condition asserting that the resulting family is a measurement.

QAlgorithms.Papers.Independent.Intakethird100.P49.pbtSelection

For natural numbers $d,e,M$, put $D=\{0,\ldots,d-1\}$, $F=\{0,\ldots,e-1\}$, and $\mathcal H=D\times D^M$. On $\mathcal H$, define $A_i[(a,x),(b,y)]$ to be $1/d$ if $a=x_i$, $b=y_i$, and $x_j=y_j$ for all $j\ne i$, and zero otherwise; set $S=\sum_iA_i$, $R=f(S)$ by continuous functional calculus for $f(t)=(\sqrt t)^{-1}$, and $H_i=RA_iR+M^{-1}(I-RSR)$. This declaration defines a function from complex matrices indexed by $\mathcal H\times F^M$ to complex matrices indexed by $\{\ast\}\times F$: for any input matrix $X$,

$$\mathcal S(X)_{(\ast,a),(\ast,a')} = \sum_{\substack{i<M,\ p,q\in\mathcal H,\ b,c\in F^M\\ b_i=a,\ c_i=a',\ b_j=c_j\ \forall j\ne i}} (H_i)_{q,p}\,X_{(p,b),(q,c)}.$$

In particular, the matrix entry of $H_i$ occurs in the order $q,p$. No density, positivity, or normalization condition is required of $X$. All sizes may be zero; when $M=0$, the sum is empty and the function returns the zero matrix. Reciprocals, including those in $f$, are total and take value zero at zero.

QAlgorithms.Papers.Independent.Intakethird100.P49.pbtSignal

For natural numbers $d,M$ and an index $i\in\{0,\ldots,M-1\}$, this defines a complex square matrix $A_i$ indexed by pairs $(a,x)$, where $a\in\{0,\ldots,d-1\}$ and $x\in\{0,\ldots,d-1\}^{M}$, by

$$(A_i)_{(a,x),(b,y)} = \begin{cases} 1/d,&a=x_i,\ b=y_i,\ \text{and }x_j=y_j\text{ for every }j\ne i,\\ 0,&\text{otherwise}. \end{cases}$$

There is no positivity assumption on $d$ or $M$. Supplying $i$ requires $M>0$; if $d=0$, the matrix has an empty index set. The division is complex division, with inverse zero defined to be zero.

QAlgorithms.Papers.Independent.Intakethird100.P49.pbtSimulationInput

For natural numbers $d,e$, with the explicit assumption $d\ne0$, any function $E$ from complex $d\times d$ matrices to complex $e\times e$ matrices, any natural number $M$, and any complex $d\times d$ matrix $X$, this defines a complex matrix indexed by $(D\times D^M)\times F^M$, where $D=\{0,\ldots,d-1\}$ and $F=\{0,\ldots,e-1\}$, by

$$Y_{((a,x),y),((a',x'),y')} = X_{a,a'}\prod_{i=0}^{M-1}E(B^{x_i,x'_i})_{y_i,y'_i},$$

where $B^{r,s}$ has entry $1/d$ in position $(r,s)$ and zero elsewhere. Thus the product consists of entries of the normalized Choi matrix defined by applying $E$ to blocks of the normalized maximally entangled projector. Neither $E$ nor $X$ is assumed to satisfy linearity, positivity, or normalization conditions. When $M=0$, both string registers are singletons and this matrix is $X$ under the resulting reindexing; $e=0$ is allowed.

QAlgorithms.Papers.Independent.Intakethird100.P49.privateCapacityBound

For every positive natural number $d$, every natural number $M$, and any function $\mathcal E$ on complex $d\times d$ matrices, this defines

$$M R_{\mathbb R}(C_{\mathcal E}) +\frac{2d(d-1)}M\log_2d +(1+t)\log_2(1+t)-t\log_2t, \qquad t=\frac{d(d-1)}M.$$

Here $d-1$ is real subtraction, and the matrix $C_{\mathcal E}$ on $\{0,\ldots,d-1\}^2$ is given by

$$(C_{\mathcal E})_{(i,a),(j,b)} =\mathcal E(d^{-1}|i\rangle\langle j|)_{a,b}.$$

The value $R_{\mathbb R}(X)$ is the real conversion of the infimum over separable density matrices $\sigma$ of $D_{\mathrm{bits}}(X\|\sigma)$, with $+\infty$ converted to zero; separable density matrices are finite convex combinations of product density matrices. The divergence is infinite unless both arguments are density matrices and $\ker\sigma\subseteq\ker X$, and otherwise equals $\max\{0,\operatorname{Re}\operatorname{tr}[(x\mapsto x\ln x)(X)-X(\ln)(\sigma)]\}/\ln2$, with continuous functional calculus. No channel assumption on $\mathcal E$ is imposed. The case $M=0$ is allowed: total division by zero gives zero, so the entire expression is zero.

QAlgorithms.Papers.Independent.Intakethird100.P49.protocolInput

For natural numbers $d,e,o,n$, a protocol $P$, and an arbitrary function $\mathcal E$ from complex $d\times d$ matrices to complex $e\times e$ matrices, this defines a sequence of matrices on $\mathbb C^r\otimes\mathbb C^d$, with $r=P.\mathrm{memory}>0$, by

$$\rho_0=P.\mathrm{initial},\qquad \rho_{k+1}=P.\mathrm{between}_k\!\left(\widehat{\mathcal E}(\rho_k)\right) \quad(k\in\mathbb N),$$

where $\widehat{\mathcal E}(X)_{(u,b),(v,c)}=\mathcal E((X_{(u,i),(v,j)})_{i,j})_{b,c}$. The protocol supplies a positive semidefinite initial matrix of trace $1$, all the intermediate channels, and terminal channels from the memory–input and memory–output spaces to an $o$-dimensional output space; the terminal channels are unused here. Each channel is a function with a finite representation $\sum_jK_jXK_j^\dagger$ satisfying $\sum_jK_j^\dagger K_j=I$. The function $\mathcal E$ itself need not be a channel or linear. The sequence is defined for every natural $k$, without restricting $k$ to the protocol parameter $n$.

QAlgorithms.Papers.Independent.Intakethird100.P49.qubitNumberFormula

For every natural number $M$, this defines the real number

$$\frac{M+2}{3\,2^{M-1}} +\frac13\sum_{k=0}^{\lfloor(M\mathbin{\dotminus}1)/2\rfloor} \frac{s_k(s_k+1)}{2^{M-4}}\binom Mk \frac{M+2-\sqrt{(M+2)^2-(2s_k+1)^2}} {(M+2)^2-(2s_k+1)^2}, \qquad s_k=\frac{M-1}{2}-k.$$

In the sum’s upper limit, $M\mathbin{\dotminus}1=\max(M-1,0)$ is natural-number subtraction and the division by $2$ is natural-number division. In $s_k$, in the exponents $M-1$ and $M-4$, and elsewhere in the summand, $M$ and $k$ have first been converted to real numbers, so those subtractions are ordinary real subtraction and the powers of $2$ are real powers. The definition includes $M=0$. Division by zero has value $0$, and the real square root of a negative argument has value $0$.

QAlgorithms.Papers.Independent.Intakethird100.P49.realREE

For any finite nonempty index types $A,B$ with decidable equality and any complex matrix $\rho$ on $A\times B$, this defines a real number by first taking

$$R(\rho)=\inf_{\sigma\ \mathrm{separable\ density}}D_{\mathrm{bits}}(\rho\|\sigma) \quad\text{in }[0,\infty],$$

then converting $R(\rho)$ to a real number, with $+\infty$ converted to $0$. A separable density matrix here is a finite sum $\sigma=\sum_jw_j(A_j\otimes B_j)$, where $w_j\ge0$, $\sum_jw_j=1$, and every $A_j,B_j$ is positive semidefinite with trace $1$. The quantity $D_{\mathrm{bits}}$ is $+\infty$ unless both arguments are density matrices and $\ker\sigma\subseteq\ker\rho$; otherwise it is

$$\frac{\max\!\left\{0,\operatorname{Re}\operatorname{tr} \left[(x\mapsto x\ln x)(\rho)-\rho\,(\ln)(\sigma)\right]\right\}}{\ln2},$$

using continuous functional calculus and the total real logarithm, for which $\ln0=0$. There is no density hypothesis on the input $\rho$; in particular, a non-density input gives the real value $0$.

QAlgorithms.Papers.Independent.Intakethird100.P49.simulated

For natural numbers $d,e$, any function $E$ from complex $d\times d$ matrices to complex $e\times e$ matrices, and any natural number $M$, this defines the function $X\mapsto E(\mathcal T(X))$. Here, writing $D=\{0,\ldots,d-1\}$, define $A_i[(r,x),(s,y)]=1/d$ on $D\times D^M$ when $r=x_i$, $s=y_i$, and $x_j=y_j$ for all $j\ne i$, and zero otherwise; put $S=\sum_iA_i$, $R=f(S)$ by continuous functional calculus for $f(t)=(\sqrt t)^{-1}$, and $H_i=RA_iR+M^{-1}(I-RSR)$. Then

$$\mathcal T(X)_{a,b} = d^{-M} \sum_{\substack{i<M,\ r,s\in D,\ x,y\in D^M\\ x_i=a,\ y_i=b,\ x_j=y_j\ \forall j\ne i}} (H_i)_{(s,y),(r,x)}X_{r,s}.$$

No linearity or channel assumption is imposed on $E$. In particular, when $M=0$, the resulting function is the constant function $X\mapsto E(0)$, which is not required to be zero. The sizes $d,e$ may also be zero, and all reciprocals use $0^{-1}=0$.

QAlgorithms.Papers.Independent.Intakethird100.P49.symmetricAdaptiveError

For natural numbers $d,e,n$ and arbitrary functions $\mathcal E_0,\mathcal E_1$ from complex $d\times d$ matrices to complex $e\times e$ matrices, this defines a real number equal to $1/2$ when $n=0$. For $n>0$, it is

$$\inf_P\frac{1-a_P(\mathcal E_0)+a_P(\mathcal E_1)}2.$$

Here $P$ ranges over the following data: a positive memory dimension $r$; a positive semidefinite matrix $\rho_0$ on $\mathbb C^r\otimes\mathbb C^d$ with complex trace $1$; a channel $\mathcal B_k$ from matrices on $\mathbb C^r\otimes\mathbb C^e$ to matrices on $\mathbb C^r\otimes\mathbb C^d$ for every natural number $k$; and an effect $T$ on $\mathbb C^r\otimes\mathbb C^e$, meaning $T\ge0$ and $I-T\ge0$. A channel here has a finite Kraus representation $X\mapsto\sum_jK_jXK_j^\dagger$ with $\sum_jK_j^\dagger K_j=I$. For any function $\mathcal E$, its extension to the memory register is defined blockwise by $\widehat{\mathcal E}(X)_{(u,b),(v,c)}=\mathcal E((X_{(u,i),(v,j)})_{i,j})_{b,c}$. Set $\rho_{k+1}^{\mathcal E}=\mathcal B_k(\widehat{\mathcal E}(\rho_k^{\mathcal E}))$, starting at $\rho_0$; then $a_P(\mathcal E)=\operatorname{Re}\operatorname{Tr}(T\widehat{\mathcal E}(\rho_{n-1}^{\mathcal E}))$. Neither $\mathcal E_0$ nor $\mathcal E_1$ is assumed to be a channel or linear, so these scalars are not required to be probabilities. Zero dimensions are allowed; for example, $d=0$ makes the required initial density matrix impossible. The infimum is the total real infimum, which is $0$ for an empty set or a set unbounded below.

QAlgorithms.Papers.Independent.Intakethird100.P49.testOutput

For natural numbers $d,e,n$, an adaptive test $P$, and an arbitrary function $\mathcal E$ from complex $d\times d$ matrices to complex $e\times e$ matrices, this defines a matrix on $\mathbb C^r\otimes\mathbb C^e$, where $r=P.\mathrm{memory}>0$, by

$$\operatorname{testOutput}(P,\mathcal E)=\widehat{\mathcal E}(\rho_{n-1}^{\mathcal E}),\qquad \rho_0^{\mathcal E}=P.\mathrm{initial},\qquad \rho_{k+1}^{\mathcal E}=P.\mathrm{between}_k\!\left(\widehat{\mathcal E}(\rho_k^{\mathcal E})\right).$$

The blockwise extension is $\widehat{\mathcal E}(X)_{(u,b),(v,c)}=\mathcal E((X_{(u,i),(v,j)})_{i,j})_{b,c}$. The test supplies an initial positive semidefinite matrix of trace $1$, channels $P.\mathrm{between}_k$ for every $k\in\mathbb N$, and an effect $T\ge0$ with $I-T\ge0$; the effect is not used in this definition. Each supplied channel has a finite Kraus representation $X\mapsto\sum_jK_jXK_j^\dagger$ with $\sum_jK_j^\dagger K_j=I$. No channel assumption is imposed on $\mathcal E$. The subtraction $n-1$ is natural-number subtraction, so when $n=0$ this definition still applies $\widehat{\mathcal E}$ once to the initial matrix.

QAlgorithms.Papers.Independent.Intakethird100.P49.universalBound

For natural numbers $d,e,n,M$ with $d\ne0$, and arbitrary functions $\mathcal E_0,\mathcal E_1$ from complex $d\times d$ matrices to complex $e\times e$ matrices, this defines

$$\frac{1-n\Delta_{d,M}-\tfrac12\left\|C_0^{[nM]}-C_1^{[nM]}\right\|_1}{2}.$$

Here $\|X\|_1=\operatorname{Re}\operatorname{Tr}\sqrt{X^\dagger X}$, $C_i$ is the matrix with entries $C_i((a,b),(a',b'))=\mathcal E_i(d^{-1}|a\rangle\langle a'|)_{b,b'}$, and $C^{[r]}(x,y)=\prod_{j=0}^{r-1}C(x_j,y_j)$. The scalar $\Delta_{d,M}$ is defined as $\|\mathrm{id}-\mathcal P_{d,M}\|_\diamond$, where for any function $F$ on $d$-dimensional matrices,

$$\|F\|_\diamond=\sup_{\|X\|_1\le1}\|\widehat F(X)\|_1,\qquad \widehat F(X)_{(a,b),(a',b')}= F((X_{(a,i),(a',j)})_{i,j})_{b,b'}.$$

The function $\mathcal P_{d,M}$ is specified by the following finite matrices and sums. Write $[d]=\{0,\ldots,d-1\}$, let $u,v\in[d]^M$, and, for $i<M$, set

$$S_i((a,u),(b,v))= \begin{cases} d^{-1},&a=u_i,\ b=v_i,\ \text{and }u_j=v_j\text{ for every }j\ne i,\\ 0,&\text{otherwise}. \end{cases}$$

Set $S=\sum_{i<M}S_i$, $R=f(S)$ by continuous functional calculus with $f(x)=(\sqrt x)^{-1}$, and $Q_i=RS_iR+M^{-1}(I-RSR)$. Then

$$\mathcal P_{d,M}(X)_{b,c} =d^{-M}\sum_{i<M}\sum_{a,a'<d} \sum_{\substack{u,v\in[d]^M\\u_i=b,\ v_i=c\\u_j=v_j\ (j\ne i)}} Q_i((a',v),(a,u))X_{a,a'}.$$

All inverses are total, with $0^{-1}=0$, including in the functional calculus. No channel assumptions are imposed on $\mathcal E_i$, no positivity assumption is imposed on $e$, and $n,M$ may be zero. Empty products equal $1$, the $M=0$ sum gives $\mathcal P_{d,0}=0$, and the defined bound at $n=0$ is $1/2$. This declaration defines a scalar; it asserts no inequality.