Definitions for A triangle of dualities: reversibly decomposable quantum channels, source-channel duality, and time reversal

QAlgorithms.Papers.Independent.Intakethird100.P64.FeedbackSimulationCode.channel

For finite types $S,A,B$, a natural number $n$, and a feedback simulation code $P$, write $\mathcal B_j$ for the length-$j$ Boolean strings, $V=P.\mathrm{enc}$, $W=P.\mathrm{dec}$, $q=P.q$, $k=P.k$, $w_A=P.\mathrm{aliceWork}$, and $w_B=P.\mathrm{bobWork}$; these matrices satisfy $V^\dagger V=I$ and $W^\dagger W=I$. This defines a map $\mathcal N_P$ from matrices on $S^n$ to matrices on $A^n\times B^n$ whose entries, for every input matrix $\rho$, are

$$\begin{aligned} \mathcal N_P(\rho)((a,b),(a',b')) ={}&2^{-k} \sum_{s,s'\in S^n}\sum_{h,h'\in\mathcal B_k} \sum_{m,m'\in\mathcal B_q} \sum_{u=0}^{w_A}\sum_{v=0}^{w_B} \rho(s,s')\\ &\quad{}\cdot V(((a,m),u),(s,h)) \,\overline{V(((a',m'),u),(s',h'))}\\ &\quad{}\cdot W((b,v),(m,h)) \,\overline{W((b',v),(m',h'))}. \end{aligned}$$

Here $T^n$ denotes the functions from $\{0,\ldots,n-1\}$ to $T$. This formula expands the appended vector $2^{-k/2}\sum_h|h,h\rangle$, the two matrix actions, the summation over their work coordinates, and the intermediate message coordinates $m,m'$. It applies to arbitrary $\rho$, including matrices that are not density matrices, and includes zero values of $n,q,k,w_A,w_B$.

QAlgorithms.Papers.Independent.Intakethird100.P64.FeedbackSimulationCode

For finite types $S,A,B$ and any natural number $n$, this structure consists of natural numbers $q,k,w_A,w_B$, a complex matrix

$$V:\mathbb C^{\,S^n\times\mathcal B_k} \longrightarrow \mathbb C^{\,(A^n\times\mathcal B_q)\times\{0,\ldots,w_A\}},$$

and a complex matrix

$$W:\mathbb C^{\,\mathcal B_q\times\mathcal B_k} \longrightarrow \mathbb C^{\,B^n\times\{0,\ldots,w_B\}},$$

together with the identities $V^\dagger V=I$ and $W^\dagger W=I$ on their respective input spaces. Here $T^n$ means the functions from $\{0,\ldots,n-1\}$ to $T$, and $\mathcal B_j$ is the set of Boolean strings of length $j$. The fields $w_A,w_B$ are aliceWork and bobWork, so their associated work spaces have dimensions $w_A+1,w_B+1$. All four natural-number fields may be zero, $n=0$ is allowed, and no nonemptiness assumption is imposed on $S,A,B$.

QAlgorithms.Papers.Independent.Intakethird100.P64.RelativeIdentityTest

For a type $S$ equipped with a finite enumeration, $n\in\mathbb N$, an arbitrary complex matrix $\rho$ indexed by $S$, an arbitrary function $F$ from complex matrices indexed by $S^n$ to matrices with the same indices, and real numbers $\eta,\varepsilon$, this predicate requires the following: for every positive integer $d$ and every positive semidefinite matrix $\sigma$ indexed by $\{0,\ldots,d-1\}\times S^n$ with $\operatorname{tr}\sigma=1$,

$$\left\|\operatorname{tr}_{\{0,\ldots,d-1\}}\sigma-\rho^{\otimes n}\right\|_1\le\eta \quad\Longrightarrow\quad \|(\operatorname{id}_d\otimes F)(\sigma)-\sigma\|_1\le\varepsilon.$$

Explicitly, $(\rho^{\otimes n})_{x,y}=\prod_{i=0}^{n-1}\rho_{x_i,y_i}$, the partial trace has entry $\sum_{a=0}^{d-1}\sigma_{(a,x),(a,y)}$, and $(\operatorname{id}_d\otimes F)(\sigma)_{(a,x),(b,y)}=F(\sigma_{(a,\cdot),(b,\cdot)})_{x,y}$; the trace norm is $\|M\|_1=\operatorname{Re}\operatorname{tr}\sqrt{M^\dagger M}$. There is no density-matrix assumption on $\rho$, channel assumption on $F$, or sign restriction on $\eta,\varepsilon$. A negative $\eta$ makes the implication vacuous. If $S$ is empty and $n>0$, no such trace-one $\sigma$ exists, so the predicate is vacuous; $n=0$ is also included, and its tensor-product matrix is $[1]$.

QAlgorithms.Papers.Independent.Intakethird100.P64.RelativeSimulationAchievable

For finite types $S,A,B$, an arbitrary complex matrix $U:(A\times B)\times S\to\mathbb C$, an arbitrary complex matrix $\rho$ on $S$, and real numbers $Q,E$, this proposition says that for every $\varepsilon>0$ and every $\delta>0$, there exist $\eta>0$ and a natural number $N$ such that, for every natural number $n\ge N$ with $n\ge1$, there is a code $P$ with natural numbers $q,k,w_A,w_B$ and matrices

$$V:\mathbb C^{S^n\times\mathcal B_k}\to \mathbb C^{(A^n\times\mathcal B_q)\times\{0,\ldots,w_A\}}, \qquad W:\mathbb C^{\mathcal B_q\times\mathcal B_k}\to \mathbb C^{B^n\times\{0,\ldots,w_B\}}, \quad V^\dagger V=W^\dagger W=I,$$

for which $q\le\lfloor n(Q+\delta)\rfloor_+$ and $k\le\lfloor n(E+\delta)\rfloor_+$, and the following test holds for every positive natural number $d$ and every positive semidefinite matrix $\sigma$ of trace $1$ on $\{0,\ldots,d-1\}\times S^n$:

$$\left\|\operatorname{Tr}_{\{0,\ldots,d-1\}}\sigma-\rho^{[n]}\right\|_1\le\eta \quad\Longrightarrow\quad \left\|(\mathrm{id}_d\otimes\mathcal N_P)(\sigma) -(\mathrm{id}_d\otimes\mathcal U_n)(\sigma)\right\|_1\le\varepsilon.$$

Here $T^n$ denotes length-$n$ strings over $T$, $\mathcal B_j=\{\mathrm{false},\mathrm{true}\}^j$, $\rho^{[n]}(s,s')=\prod_i\rho(s_i,s'_i)$, $U^{[n]}((a,b),s)=\prod_iU((a_i,b_i),s_i)$, and $\mathcal U_n(X)=U^{[n]}X(U^{[n]})^\dagger$. The code map $\mathcal N_P$ appends the outer product of $2^{-k/2}\sum_h|h,h\rangle$, applies $V$ to $S^n$ and the first appended register, applies $W$ to the resulting $\mathcal B_q$ register and the second appended register, and sums over equal work coordinates in the resulting matrix. Explicitly, its matrix entries are the sums of

$$2^{-k}X(s,s')\, V(((a,m),u),(s,h)) \overline{V(((a',m'),u),(s',h'))}\, W((b,v),(m,h)) \overline{W((b',v),(m',h'))}$$

over $s,s',h,h',m,m',u,v$ in their respective indexing sets. For any matrix map $\Phi$, its reference extension acts on each reference block by $(\mathrm{id}_d\otimes\Phi)(\sigma)((r,y),(r',y'))=\Phi(\sigma_{r,r'})(y,y')$, where $\sigma_{r,r'}(s,s')=\sigma((r,s),(r',s'))$; the partial trace sums the diagonal reference blocks, and $\|M\|_1=\operatorname{Re}\operatorname{tr}\sqrt{M^\dagger M}$, with the square root defined by continuous functional calculus. The natural-number floor $\lfloor x\rfloor_+$ is zero for negative $x$. Neither $U^\dagger U=I$, nor positivity or normalization of $\rho$, nor nonnegativity of $Q,E$ is assumed; if no density matrix satisfies the stated proximity condition, that conditional test is vacuous.

QAlgorithms.Papers.Independent.Intakethird100.P64.RoutingCode

For a type $S$ equipped with a finite enumeration and $n\in\mathbb N$, this structure consists of two functions $\mathcal E,\mathcal D$ from complex matrices indexed by $S^n$ to matrices with the same indices, together with a finite Kraus representation for each: there exist natural numbers $a,b$, matrices $E_0,\ldots,E_{a-1}$ and $D_0,\ldots,D_{b-1}$, such that

$$\sum_{j=0}^{a-1}E_j^\dagger E_j=I,\qquad \mathcal E(X)=\sum_{j=0}^{a-1}E_jXE_j^\dagger\quad\text{for every }X,$$

and

$$\sum_{j=0}^{b-1}D_j^\dagger D_j=I,\qquad \mathcal D(X)=\sum_{j=0}^{b-1}D_jXD_j^\dagger\quad\text{for every }X.$$

No further relationship between the two functions is required. The type $S$ may be empty; $n=0$ is allowed and gives a singleton tuple index set. The Kraus-family sizes may be zero, in which case their sums are zero and the displayed completeness equality still must hold.

QAlgorithms.Papers.Independent.Intakethird100.P64.SlepianWolfCode.error

Let $R,S,A,B$ be types equipped with finite enumerations, possibly empty, and let $n\in\mathbb N$. A code $P$ supplies natural numbers $q,l,w_A,w_B$ and complex matrices

$$V:\mathbb C^{A^n}\longrightarrow \mathbb C^{(\mathcal B_l\times\mathcal B_q)\times\{0,\ldots,w_A\}}, \qquad W:\mathbb C^{\mathcal B_q\times B^n}\longrightarrow \mathbb C^{(\mathcal B_l\times S^n)\times\{0,\ldots,w_B\}},$$

satisfying $V^\dagger V=I$ and $W^\dagger W=I$, where $\mathcal B_j=\{0,1\}^j$. For any matrix $U:\mathbb C^S\to\mathbb C^{A\times B}$ and vector $\phi\in\mathbb C^{R\times S}$, define

$$v_n(r,a,b)=\prod_{i=0}^{n-1}\sum_{s\in S}U_{(a_i,b_i),s}\phi(r_i,s), \qquad t_n(r,u,v,s)= \begin{cases} \displaystyle 2^{-l/2}\prod_{i=0}^{n-1}\phi(r_i,s_i),&u=v,\\ 0,&u\ne v. \end{cases}$$

For each work-index pair $\alpha\in\{0,\ldots,w_A\}$, $\beta\in\{0,\ldots,w_B\}$, define a matrix from $A^n\times B^n$ to $\mathcal B_l\times(\mathcal B_l\times S^n)$ by

$$K_{\alpha\beta}[(u,(v,s)),(a,b)] = \sum_{m\in\mathcal B_q}V_{((u,m),\alpha),a}\, W_{((v,s),\beta),(m,b)}.$$

The declared error is

$$\left\| \sum_{\alpha,\beta}(I_{R^n}\otimes K_{\alpha\beta})v_nv_n^\dagger (I_{R^n}\otimes K_{\alpha\beta})^\dagger -t_nt_n^\dagger \right\|_1, \qquad \|M\|_1=\operatorname{Re}\operatorname{tr}\sqrt{M^\dagger M},$$

with the square root defined by continuous functional calculus. No isometry assumption on $U$ or normalization assumption on $\phi$ is made. All four code parameters may be zero, and $n=0$ is included, with empty products equal to $1$.

QAlgorithms.Papers.Independent.Intakethird100.P64.SlepianWolfCode

For finite types $S,A,B$ and any natural number $n$, this structure consists of natural numbers $q,l,w_A,w_B$, a complex matrix

$$V:\mathbb C^{A^n}\longrightarrow \mathbb C^{(\mathcal B_l\times\mathcal B_q)\times\{0,\ldots,w_A\}},$$

and a complex matrix

$$W:\mathbb C^{\mathcal B_q\times B^n}\longrightarrow \mathbb C^{(\mathcal B_l\times S^n)\times\{0,\ldots,w_B\}},$$

together with the identities $V^\dagger V=I$ and $W^\dagger W=I$ on their respective input spaces. Here $T^n$ is the set of functions from $\{0,\ldots,n-1\}$ to $T$, $\mathcal B_j$ is the set of length-$j$ Boolean strings, and $w_A,w_B$ are the fields aliceWork and bobWork, whose work spaces have dimensions $w_A+1,w_B+1$. Zero values of all natural-number fields and $n=0$ are included, and the finite indexing types are not assumed nonempty.

QAlgorithms.Papers.Independent.Intakethird100.P64.SourceRoutingAchievable

For types $R,S$ equipped with finite enumerations, possibly empty, and an arbitrary vector $\phi\in\mathbb C^{R\times S}$, put

$$\rho_{s,t}=\sum_{r\in R}\phi(r,s)\overline{\phi(r,t)},\qquad D_n[(r,s),(r',t)] =\prod_{i=0}^{n-1}\phi(r_i,s_i)\, \overline{\prod_{i=0}^{n-1}\phi(r'_i,t_i)},\qquad (\rho^{\otimes n})_{s,t}=\prod_{i=0}^{n-1}\rho_{s_i,t_i}.$$

This predicate means that for every $\varepsilon>0,\delta>0$, there is $N\in\mathbb N$ such that for every $n\ge N$ with $n\ge1$, there are channels $\mathcal E,\mathcal D$ on matrices indexed by $S^n$ satisfying $n\le n(1+\delta)$ and, for every $\eta>0$,

$$\left\|\operatorname{tr}_{R^n}(\operatorname{id}_{R^n}\otimes\mathcal E)(D_n) -\rho^{\otimes n}\right\|_1\le\eta,$$

and the following implication for every channel $F$ on that same matrix space: if for every positive integer $d$ and every positive semidefinite trace-one matrix $\sigma$ on $\{0,\ldots,d-1\}\times S^n$,

$$\left\|\operatorname{tr}_{\{0,\ldots,d-1\}}\sigma-\rho^{\otimes n}\right\|_1\le\eta \ \Longrightarrow\ \|(\operatorname{id}_d\otimes F)(\sigma)-\sigma\|_1\le\varepsilon,$$

then

$$\|(\operatorname{id}_{R^n}\otimes(\mathcal D\circ F\circ\mathcal E))(D_n)-D_n\|_1 \le\varepsilon.$$

Here a channel means a function $T(X)=\sum_{j=0}^{a-1}K_jXK_j^\dagger$ for some natural number $a$ and matrices satisfying $\sum_jK_j^\dagger K_j=I$; partial traces sum diagonal blocks in the traced index, and $(\operatorname{id}_Z\otimes T)(X)_{(z,s),(z',t)}=T(X_{(z,\cdot),(z',\cdot)})_{s,t}$. The norm is $\|M\|_1=\operatorname{Re}\operatorname{tr}\sqrt{M^\dagger M}$. No normalization of $\phi$ is assumed. The chosen channels precede the quantifier over $\eta$, so the first norm bound must hold for every positive $\eta$, forcing that difference to be zero; the numerical inequality involving $\delta$ is also explicitly required.

QAlgorithms.Papers.Independent.Intakethird100.P64.SourceTransferAchievable

For types $R,S,A,B$ equipped with finite enumerations, possibly empty, any complex matrix $U:\mathbb C^S\to\mathbb C^{A\times B}$, vector $\phi\in\mathbb C^{R\times S}$, and real numbers $Q,E$, this predicate means that for every $\varepsilon>0$ and $\delta>0$, there is $N\in\mathbb N$ such that for every $n\ge N$ with $n\ge1$, there exist natural numbers $q,l,w_A,w_B$ and matrices

$$V:\mathbb C^{A^n}\to \mathbb C^{(\mathcal B_l\times\mathcal B_q)\times\{0,\ldots,w_A\}}, \qquad W:\mathbb C^{\mathcal B_q\times B^n}\to \mathbb C^{(\mathcal B_l\times S^n)\times\{0,\ldots,w_B\}},$$

where $\mathcal B_j=\{0,1\}^j$, such that $V^\dagger V=I$, $W^\dagger W=I$, and

$$q\le \lfloor n(Q+\delta)\rfloor_+,\qquad \lfloor n(E-\delta)\rfloor_+\le l,\qquad \left\|\sum_{\alpha,\beta}(I\otimes K_{\alpha\beta})v_nv_n^\dagger (I\otimes K_{\alpha\beta})^\dagger-t_nt_n^\dagger\right\|_1\le\varepsilon.$$

Here $\lfloor x\rfloor_+=\max(0,\lfloor x\rfloor)$, $\|M\|_1=\operatorname{Re}\operatorname{tr}\sqrt{M^\dagger M}$, the work indices run through $\alpha=0,\ldots,w_A$, $\beta=0,\ldots,w_B$, and

$$\begin{aligned} v_n(r,a,b)&=\prod_{i=0}^{n-1}\sum_{s\in S}U_{(a_i,b_i),s}\phi(r_i,s),\\ t_n(r,u,v,s)&=2^{-l/2}\mathbf1_{\{u=v\}}\prod_{i=0}^{n-1}\phi(r_i,s_i),\\ K_{\alpha\beta}[(u,(v,s)),(a,b)] &=\sum_{m\in\mathcal B_q}V_{((u,m),\alpha),a} W_{((v,s),\beta),(m,b)}. \end{aligned}$$

No normalization of $\phi$, isometry of $U$, or nonnegativity of $Q,E$ is required. In particular, a nonpositive argument of either natural-number floor produces zero; the entanglement condition is only the displayed lower bound on $l$. The code parameters may be zero, while the quantified block lengths are positive.

QAlgorithms.Papers.Independent.Intakethird100.P64.appendBell

For any type $X$, any natural number $k$, and any complex matrix $\rho$ on $X$, this defines a matrix on $(X\times\mathcal B_k)\times\mathcal B_k$, where $\mathcal B_k$ is the set of length-$k$ Boolean strings, by

$$\Omega_\rho((x,u,v),(x',u',v')) = \begin{cases} 2^{-k}\rho(x,x'),&u=v\ \text{and}\ u'=v',\\ 0,&\text{otherwise}. \end{cases}$$

Equivalently, it multiplies each entry of $\rho$ by the corresponding entry of $\beta_k\beta_k^\dagger$, where $\beta_k(u,v)=2^{-k/2}$ when $u=v$, and zero otherwise. No positivity or trace condition on $\rho$ is required. For $k=0$, each Boolean-string register has one value and the added factor is $1$.

QAlgorithms.Papers.Independent.Intakethird100.P64.distributedSource

For arbitrary types $R,S,A,B$, with $S$ equipped with a finite enumeration, any complex matrix $U$ with rows indexed by $A\times B$ and columns by $S$, any vector $\phi$ in the complex Euclidean space indexed by $R\times S$, and $n\in\mathbb N$, define

$$w(r,a,b)=\sum_{s\in S}U_{(a,b),s}\phi(r,s), \qquad v_n(r,a,b)=\prod_{i=0}^{n-1}w(r_i,a_i,b_i).$$

The declaration defines the matrix indexed by $R^n\times(A^n\times B^n)$ whose $(p,p')$-entry is $v_n(p)\overline{v_n(p')}$. Neither $U^\dagger U=I$ nor normalization of $\phi$ is assumed. Empty types are allowed; in particular, if $S$ is empty, each sum defining $w$ is zero. When $n=0$, the product is $1$ and the matrix is $[1]$.

QAlgorithms.Papers.Independent.Intakethird100.P64.feedbackEntropyA

For finite types $R,A,B$ and any complex vector $\psi$ indexed by $R\times(A\times B)$, this defines the real number $H(\rho_A)$, where

$$\rho_A(a,a')=\sum_{r\in R}\sum_{b\in B} \psi(r,(a,b))\,\overline{\psi(r,(a',b))}, \qquad H(M)=-\operatorname{Re}\operatorname{tr}\!\left[f(M)\right], \quad f(x)=x\log_2 x.$$

Here $f(M)$ is evaluated using continuous functional calculus. Thus the matrix whose entropy is taken is obtained from the outer product $\psi\psi^\dagger$ by summing over equal $R$- and $B$-coordinates. The vector need not have norm one, and the finite types may be empty; the definition also applies to the zero vector.

QAlgorithms.Papers.Independent.Intakethird100.P64.feedbackOutputVector

For arbitrary types $R,S,A,B$, with $S$ finite, a complex matrix $U$ with rows indexed by $A\times B$ and columns indexed by $S$, and a complex coordinate vector $\phi$ indexed by $R\times S$, this defines the vector $\psi$ indexed by $R\times(A\times B)$ through

$$\psi(r,(a,b))=\sum_{s\in S}U((a,b),s)\,\phi(r,s).$$

There is no normalization assumption on $\phi$ and no isometry assumption on $U$. If $S$ is empty, the sum is empty and the output is zero.

QAlgorithms.Papers.Independent.Intakethird100.P64.isometryPower

For arbitrary types $S,A,B$, any complex matrix $U$ with rows indexed by $A\times B$ and columns by $S$, and any natural number $n$, this defines the matrix $U^{[n]}$ with rows indexed by $A^n\times B^n$ and columns by $S^n$ through

$$U^{[n]}((a,b),s)=\prod_{i=0}^{n-1}U((a_i,b_i),s_i),$$

where $T^n$ means the functions from $\{0,\ldots,n-1\}$ to $T$. There is no finiteness or isometry hypothesis on $U$ or its indexing types. When $n=0$, the product is $1$, and each indexing function space has one element, so the result is the one-by-one matrix with entry $1$.

QAlgorithms.Papers.Independent.Intakethird100.P64.localOperation

For arbitrary types $X,Y,W$, with $X$ and $W$ finite, and any complex matrix $V$ whose rows are indexed by $Y\times W$ and columns by $X$, this defines a map $\mathcal L_V$ from complex matrices on $X$ to complex matrices on $Y$ by

$$\mathcal L_V(\rho)(y,y') =\sum_{w\in W}(V\rho V^\dagger)((y,w),(y',w)) =\sum_{w\in W}\sum_{x,x'\in X} V((y,w),x)\rho(x,x')\overline{V((y',w),x')}.$$

The input $\rho$ is arbitrary, and $V$ is not required to satisfy an isometry identity. If $X$ or $W$ is empty, the resulting map is zero.

QAlgorithms.Papers.Independent.Intakethird100.P64.mutualAB

For finite types $R,A,B$ and any complex vector $\psi$ indexed by $R\times(A\times B)$, this defines

$$H(\rho_A)+H(\rho_B)-H(\rho_{AB}),$$

where

$$\begin{aligned} \rho_A(a,a')&=\sum_{r,b}\psi(r,(a,b))\,\overline{\psi(r,(a',b))},\\ \rho_B(b,b')&=\sum_{r,a}\psi(r,(a,b))\,\overline{\psi(r,(a,b'))},\\ \rho_{AB}((a,b),(a',b'))&=\sum_r\psi(r,(a,b))\,\overline{\psi(r,(a',b'))}. \end{aligned}$$

The real-valued function used here is $H(M)=-\operatorname{Re}\operatorname{tr}[f(M)]$, where $f(x)=x\log_2x$ is applied by continuous functional calculus. The definition includes zero and unnormalized vectors and permits empty finite indexing types.

QAlgorithms.Papers.Independent.Intakethird100.P64.mutualRA

For finite types $R,A,B$ and any complex vector $\psi$ indexed by $R\times(A\times B)$, this defines

$$H(\rho_R)+H(\rho_A)-H(\rho_{RA}),$$

where

$$\begin{aligned} \rho_R(r,r')&=\sum_{a,b}\psi(r,(a,b))\,\overline{\psi(r',(a,b))},\\ \rho_A(a,a')&=\sum_{r,b}\psi(r,(a,b))\,\overline{\psi(r,(a',b))},\\ \rho_{RA}((r,a),(r',a'))&=\sum_b\psi(r,(a,b))\,\overline{\psi(r',(a',b))}. \end{aligned}$$

For each matrix $M$, $H(M)=-\operatorname{Re}\operatorname{tr}[f(M)]$, with $f(x)=x\log_2x$ evaluated by continuous functional calculus. No normalization of $\psi$ or nonemptiness of the finite types is assumed.

QAlgorithms.Papers.Independent.Intakethird100.P64.mutualRB

For finite types $R,A,B$ and any complex vector $\psi$ indexed by $R\times(A\times B)$, this defines

$$H(\rho_R)+H(\rho_B)-H(\rho_{RB}),$$

where

$$\begin{aligned} \rho_R(r,r')&=\sum_{a,b}\psi(r,(a,b))\,\overline{\psi(r',(a,b))},\\ \rho_B(b,b')&=\sum_{r,a}\psi(r,(a,b))\,\overline{\psi(r,(a,b'))},\\ \rho_{RB}((r,b),(r',b'))&=\sum_a\psi(r,(a,b))\,\overline{\psi(r',(a,b'))}. \end{aligned}$$

Here $H(M)=-\operatorname{Re}\operatorname{tr}[f(M)]$, with $f(x)=x\log_2x$ interpreted through continuous functional calculus. There is no unit-norm hypothesis on $\psi$, and no nonemptiness hypothesis on the finite types.

QAlgorithms.Papers.Independent.Intakethird100.P64.sourceDensity

For arbitrary types $R,X$, a vector $v\in\mathbb C^{R\times X}$ in the specified Euclidean space, and $n\in\mathbb N$, write $R^n$ and $X^n$ for tuples indexed by $\{0,\ldots,n-1\}$. This defines the matrix with entries

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

There is no normalization assumption on $v$ and no finiteness or nonemptiness assumption on $R,X$. For $n=0$, both tuple sets are singletons, the products are $1$, and the resulting matrix is the one-by-one matrix $[1]$.

QAlgorithms.Papers.Independent.Intakethird100.P64.sourceTarget

For arbitrary types $R,S$, any vector $\phi$ in the complex Euclidean space indexed by $R\times S$, and natural numbers $n,l$, let $\mathcal B_l=\{0,1\}^l$. This defines the outer-product matrix $tt^\dagger$, indexed by $R^n\times(\mathcal B_l\times(\mathcal B_l\times S^n))$, where

$$t(r,u,v,s)= \begin{cases} \displaystyle \frac{1}{\sqrt{2^l}}\prod_{i=0}^{n-1}\phi(r_i,s_i),&u=v,\\ 0,&u\ne v. \end{cases}$$

Thus its entry at $p,p'$ is $t(p)\overline{t(p')}$. No normalization, finiteness, or nonemptiness assumptions are imposed on $\phi,R,S$. For $n=0$, the product is $1$; for $l=0$, both bit-string registers are singletons and their equality condition always holds.

QAlgorithms.Papers.Independent.Intakethird100.P64.sourceToAlice

For every type $S$, including an empty or infinite type, this defines the complex matrix with rows indexed by $S\times\{\ast\}$ and columns indexed by $S$, whose entries are

$$J_{(a,\ast),s}= \begin{cases} 1,&a=s,\\ 0,&a\ne s. \end{cases}$$

No finiteness assumption on $S$ is imposed.