Definitions for On the distributed compression of quantum information

QAlgorithms.Papers.Independent.Intakethird100.P11.DistributedAchievable

For finite types $A,B$ with decidable equality, an arbitrary real-valued function $F$ assigning a value to every distributed block code at every choice of natural parameters $n,d_A,d_B$, and real numbers $r_A,r_B$, distributed achievability means that for every $\delta>0$ and every $\varepsilon>0$, there exists a natural number $N$ such that for every natural number $n>N$, there exist positive natural numbers $d_A,d_B$ and a code $c$ of positive length $n$ satisfying

$$1-\varepsilon\le F(c), \qquad \frac{\log_2d_A}{n}\le r_A+\delta, \qquad \frac{\log_2d_B}{n}\le r_B+\delta.$$

Such a code consists of encoders from matrices on $A^n$ and $B^n$ to matrices of dimensions $d_A,d_B$, and a decoder from their joint compressed space back to matrices on $A^n\times B^n$; every map must have a finite Kraus representation $\Phi(X)=\sum_jK_jXK_j^\dagger$ with $\sum_jK_j^\dagger K_j=I$. The function $F$ has no assumed bounds or other properties, and the rates have no separately imposed nonnegativity condition. The threshold uses the strict inequality $N<n$, so length zero is never required.

QAlgorithms.Papers.Independent.Intakethird100.P11.DistributedBlockCode

For finite types $A,B$ with decidable equality and natural numbers $n,d_A,d_B$, a distributed block code consists of proofs that $n,d_A,d_B>0$, two maps $\mathcal E_A$ and $\mathcal E_B$ from complex matrices indexed respectively by $A^n$ and $B^n$ to matrices indexed by $\{0,\ldots,d_A-1\}$ and $\{0,\ldots,d_B-1\}$, and a map $\mathcal D$ from matrices indexed by $\{0,\ldots,d_A-1\}\times\{0,\ldots,d_B-1\}$ to matrices indexed by $A^n\times B^n$. Each of these three maps must admit a finite Kraus representation: for the corresponding input and output index sets, there are a natural number $m$ and rectangular complex matrices $K_0,\ldots,K_{m-1}$ such that

$$\sum_{j=0}^{m-1}K_j^\dagger K_j=I, \qquad \Phi(X)=\sum_{j=0}^{m-1}K_jXK_j^\dagger \quad\text{for every input matrix }X.$$

Here $A^n$ means functions from $\{0,\ldots,n-1\}$ to $A$, and similarly for $B^n$. The parameters $n,d_A,d_B$ may initially be any natural numbers, but no code exists when any of them is zero. No nonemptiness assumptions are imposed on $A,B$; the channel requirements also prevent such a code from existing if either type is empty.

QAlgorithms.Papers.Independent.Intakethird100.P11.DistributedBlockCode.output

For finite types $A,B$ with decidable equality, natural numbers $n,d_A,d_B$, and a distributed block code $c$, whose length and compressed dimensions are positive and whose encoders $\mathcal E_A,\mathcal E_B$ and decoder $\mathcal D$ each have a finite Kraus representation $\Phi(X)=\sum_jK_jXK_j^\dagger$ with $\sum_jK_j^\dagger K_j=I$, the output map sends every matrix $X$ indexed by $A^n\times B^n$ to $\mathcal D(Z)$. Explicitly, for compressed indices $u,u'<d_A$ and $v,v'<d_B$,

$$Z_{(u,v),(u',v')} = \left[ \mathcal E_B\!\left( \left[ \left[ \mathcal E_A\!\left( [X_{(a,b),(a',b')}]_{a,a'\in A^n} \right) \right]_{u,u'} \right]_{b,b'\in B^n} \right) \right]_{v,v'}.$$

Thus the definition first applies Alice’s encoder to each matrix block with Bob’s two indices fixed, then applies Bob’s encoder to each block with Alice’s compressed indices fixed, and finally applies the decoder. It is defined on every complex input matrix, without requiring that matrix to be a density matrix.

QAlgorithms.Papers.Independent.Intakethird100.P11.EnsembleIrreducible

For an arbitrary label type $I$, a finite coordinate type $X$, arbitrary real weights $p_i$, and arbitrary vectors $v_i\in\mathbb C^X$, ensemble irreducibility means that at least one label has strictly positive weight, and every subset $S\subseteq I$ containing a positive-weight label and excluding another positive-weight label has positive-weight labels $i\in S$ and $j\notin S$ such that

$$\langle v_i,v_j\rangle = \sum_{x\in X}\overline{v_i(x)}v_j(x)\ne0.$$

The label type need not be finite, and no normalization of the vectors, nonnegativity of all weights, or condition on their sum is required. With exactly one positive-weight label, the subset condition is vacuous; with no positive-weight labels, the predicate is false.

QAlgorithms.Papers.Independent.Intakethird100.P11.FiniteAchievable

For finite types $I,A,B$, with decidable equality on $A,B$, arbitrary real weights $p_i$, arbitrary vectors $v_i\in\mathbb C^{A\times B}$, and real rates $r_A,r_B$, finite achievability means that for every $\delta,\varepsilon>0$, there is a natural number $N$ such that for every natural number $n>N$, there exist positive dimensions $d_A,d_B$ and encoders $\mathcal E_A,\mathcal E_B$ and decoder $\mathcal D$ satisfying

$$1-\varepsilon\le \sum_{w\in I^n} \left(\prod_{t=0}^{n-1}p_{w(t)}\right) \operatorname{Re}\operatorname{tr}\!\left( \rho_w\,\mathcal D((\mathcal E_A\otimes\mathcal E_B)(\rho_w)) \right), \qquad \frac{\log_2d_A}{n}\le r_A+\delta, \qquad \frac{\log_2d_B}{n}\le r_B+\delta.$$

Here $V_w(a,b)=\prod_{t=0}^{n-1}v_{w(t)}(a(t),b(t))$, $(\rho_w)_{x,y}=V_w(x)\overline{V_w(y)}$, the encoders map matrices on $A^n,B^n$ to dimensions $d_A,d_B$, the tensor action applies them separately to those two sets of indices, and the decoder returns a matrix on $A^n\times B^n$. Each map must admit a finite Kraus representation $\Phi(X)=\sum_jK_jXK_j^\dagger$ with $\sum_jK_j^\dagger K_j=I$. The definition imposes no probability or normalization conditions on $p$ or $v$, and no separate sign restrictions on the rates.

QAlgorithms.Papers.Independent.Intakethird100.P11.bellSourceVectors

The Bell source vectors are the following four vectors in $\mathbb C^{\{\mathrm{false},\mathrm{true}\}\times\{\mathrm{false},\mathrm{true}\}}$, indexed in the displayed order by $0,1,2,3$:

$$\begin{aligned} v_0&=\frac{e_{(\mathrm{false},\mathrm{false})}+e_{(\mathrm{true},\mathrm{true})}}{\sqrt2}, & v_1&=\frac{e_{(\mathrm{false},\mathrm{false})}-e_{(\mathrm{true},\mathrm{true})}}{\sqrt2},\\ v_2&=\frac{e_{(\mathrm{false},\mathrm{true})}+e_{(\mathrm{true},\mathrm{false})}}{\sqrt2}, & v_3&=\frac{e_{(\mathrm{false},\mathrm{true})}-e_{(\mathrm{true},\mathrm{false})}}{\sqrt2}. \end{aligned}$$

Here $e_{(a,b)}$ has coordinate $1$ at $(a,b)$ and $0$ at every other coordinate.

QAlgorithms.Papers.Independent.Intakethird100.P11.erasureBasis

This defines four vectors $b_0,b_1,b_2,b_3$ in $\mathbb C^{\{0,1,2,3\}\times\{0,1,2,3\}}$ by

$$b_0=\frac{e_{00}+e_{33}}{\sqrt2},\qquad b_1=\frac{e_{03}+e_{30}}{\sqrt2},\qquad b_2=\frac{e_{11}+e_{22}}{\sqrt2},\qquad b_3=\frac{e_{21}+e_{12}}{\sqrt2},$$

where $e_{ab}$ is the vector whose coordinate at $(a,b)$ is $1$ and whose other coordinates are $0$.

QAlgorithms.Papers.Independent.Intakethird100.P11.erasureFidelity

Let $n,d_A,d_B$ be natural numbers and let $c$ be a distributed block code for two four-dimensional letter registers. Such a code includes $n,d_A,d_B>0$, separate encoding channels $E_A,E_B$ to dimensions $d_A,d_B$, and a decoding channel $D$ back to the two length-$n$ registers; each channel has a finite Kraus representation $X\mapsto\sum_jK_jXK_j^\dagger$ with $\sum_jK_j^\dagger K_j=1$. Put $\mathcal C=D\circ(E_A\otimes E_B)$, with the encoders acting separately. For a $4\times4$ unitary $U$, set

$$v(U)=\frac{ U_{00}(e_{00}+e_{33})+U_{10}(e_{03}+e_{30}) +U_{20}(e_{11}+e_{22})+U_{30}(e_{21}+e_{12}) }{\sqrt2},$$

where the $e_{ab}$ are coordinate unit vectors. For $w=(U_0,\ldots,U_{n-1})$, define $V_w(a,b)=\prod_{t=0}^{n-1}v(U_t)(a_t,b_t)$ and $P_w=V_wV_w^\dagger$. The defined real number is

$$\int_{U(4)^n} \operatorname{Re}\operatorname{tr}\!\left(P_w\,\mathcal C(P_w)\right) \,dH^{\otimes n}(w),$$

where $H$ is Haar measure on $U(4)$, normalized to have total mass $1$.

QAlgorithms.Papers.Independent.Intakethird100.P11.erasureVector

For every complex $4\times4$ unitary matrix $U$, this defines a vector in $\mathbb C^{\{0,1,2,3\}\times\{0,1,2,3\}}$ using its column indexed by $0$:

$$v(U)=\frac1{\sqrt2}\Bigl[ U_{00}(e_{00}+e_{33})+ U_{10}(e_{03}+e_{30})+ U_{20}(e_{11}+e_{22})+ U_{30}(e_{21}+e_{12}) \Bigr],$$

where $e_{ab}$ has coordinate $1$ at $(a,b)$ and $0$ elsewhere.

QAlgorithms.Papers.Independent.Intakethird100.P11.finiteFidelity

For finite types $I,A,B$, with decidable equality on $A,B$, arbitrary real weights $p_i$, arbitrary vectors $v_i\in\mathbb C^{A\times B}$, natural numbers $n,d_A,d_B$, and a distributed block code $c$, the finite fidelity is

$$\sum_{w\in I^n} \left(\prod_{t=0}^{n-1}p_{w(t)}\right) \operatorname{Re}\operatorname{tr}\!\left(\rho_w\,\mathcal C(\rho_w)\right), \quad V_w(a,b)=\prod_{t=0}^{n-1}v_{w(t)}(a(t),b(t)), \quad (\rho_w)_{x,y}=V_w(x)\overline{V_w(y)}.$$

Here $\mathcal C=\mathcal D\circ(\mathcal E_A\otimes\mathcal E_B)$ applies the two encoders separately to the Alice and Bob indices and then the decoder. The code requires $n,d_A,d_B>0$, and each of its three maps has a finite Kraus representation $\Phi(X)=\sum_jK_jXK_j^\dagger$ with $\sum_jK_j^\dagger K_j=I$. Neither nonnegative or normalized weights nor normalized vectors are required. In particular, if $I$ is empty, the sum is empty because $n>0$, and its value is zero.

QAlgorithms.Papers.Independent.Intakethird100.P11.hiddenAlice

For every real number $\alpha$, this defines three vectors in $\mathbb C^2$:

$$a_0=e_0,\qquad a_1=\sqrt\alpha\,e_0+\sqrt{1-\alpha}\,e_1,\qquad a_2=e_1,$$

where $e_0,e_1$ are coordinate unit vectors and the real square roots are regarded as complex scalars. There is no restriction $\alpha\in[0,1]$; the real square-root function used here is $0$ on negative inputs, and no normalization condition is part of this definition.

QAlgorithms.Papers.Independent.Intakethird100.P11.hiddenBob

For every real number $\beta$, this defines three vectors in $\mathbb C^3$:

$$b_0=\sqrt{1-\beta}\,e_0+\sqrt\beta\,e_1,\qquad b_1=e_1,\qquad b_2=\sqrt{1-\beta}\,e_2+\sqrt\beta\,e_0,$$

where $e_0,e_1,e_2$ are coordinate unit vectors and the real square roots are regarded as complex scalars. The parameter is not restricted to $[0,1]$; square roots of negative real inputs are $0$, and the definition imposes no normalization requirement.

QAlgorithms.Papers.Independent.Intakethird100.P11.hiddenBobAfter

For every real number $\beta$, this defines three vectors in $\mathbb C^3$:

$$\widetilde b_0=\sqrt{1-\beta}\,e_0+\sqrt\beta\,e_1,\qquad \widetilde b_1=e_1,\qquad \widetilde b_2=e_0,$$

where $e_0,e_1,e_2$ are coordinate unit vectors. The square roots are real and then embedded into $\mathbb C$, with value $0$ on negative inputs. No restriction on $\beta$ or normalization requirement is imposed.

QAlgorithms.Papers.Independent.Intakethird100.P11.hiddenComparison

For every pair of real numbers $\alpha,\beta$, define vectors

$$(a_0,a_1,a_2)= \left(e_0,\sqrt\alpha\,e_0+\sqrt{1-\alpha}\,e_1,e_1\right) \quad\text{in }\mathbb C^2,$$ $$(b_0,b_1,b_2)= \left(\sqrt{1-\beta}\,e_0+\sqrt\beta\,e_1,\ e_1,\ \sqrt{1-\beta}\,e_2+\sqrt\beta\,e_0\right) \quad\text{in }\mathbb C^3.$$

With $e_k$ denoting coordinate unit vectors, this defines the real number

$$\frac12\left[ H\!\left(\frac13\sum_{i=0}^2a_ia_i^\dagger\right) + H\!\left(\frac13\sum_{i=0}^2b_ib_i^\dagger\right) + H\!\left(\frac13\sum_{i=0}^2 (a_i\otimes b_i)(a_i\otimes b_i)^\dagger\right) \right].$$

Here $(a_i\otimes b_i)(a,b)=a_i(a)b_i(b)$, an outer product $xx^\dagger$ has entries $x_r\overline{x_s}$, and $H(M)=-\operatorname{Re}\operatorname{tr}(f(M))$, with $f(x)=x\log_2x$ applied through continuous functional calculus and $f(0)=0$. Both parameters are unrestricted, and negative real inputs to the square-root function give $0$; no trace-one conditions are imposed on these averaged matrices.

QAlgorithms.Papers.Independent.Intakethird100.P11.hiddenJoint

For every pair of real numbers $\alpha,\beta$ and every $i\in\{0,1,2\}$, this defines a vector $j_i\in\mathbb C^{\{0,1\}\times\{0,1,2\}}$ by $j_i(a,b)=a_i(a)b_i(b)$, where

$$(a_0,a_1,a_2)= \left(e_0,\sqrt\alpha\,e_0+\sqrt{1-\alpha}\,e_1,e_1\right)$$

in $\mathbb C^2$, and

$$(b_0,b_1,b_2)= \left(\sqrt{1-\beta}\,e_0+\sqrt\beta\,e_1,\ e_1,\ \sqrt{1-\beta}\,e_2+\sqrt\beta\,e_0\right)$$

in $\mathbb C^3$. Thus $j_i=a_i\otimes b_i$, with the tensor product specified by the coordinate multiplication above. All $e_k$ are coordinate unit vectors; both parameters are unrestricted, and real square roots of negative inputs are $0$.

QAlgorithms.Papers.Independent.Intakethird100.P11.hiddenRateA

For every real number $\alpha$, let $u=\sqrt\alpha\,e_0+\sqrt{1-\alpha}\,e_1\in\mathbb C^2$. This defines

$$H\!\left(\frac{e_0e_0^\dagger+uu^\dagger+e_1e_1^\dagger}{3}\right), \qquad H(M)=-\operatorname{Re}\operatorname{tr}\!\bigl(f(M)\bigr), \quad f(x)=x\log_2x,$$

where $f(M)$ is formed by continuous functional calculus, $e_0,e_1$ are coordinate unit vectors, and $uu^\dagger$ has entries $u_a\overline{u_b}$. The scalar convention gives $f(0)=0$. The parameter $\alpha$ is unrestricted, real square roots of negative inputs are $0$, and the matrix inside $H$ is not required by this definition to have trace $1$.

QAlgorithms.Papers.Independent.Intakethird100.P11.hiddenRateB

For every real number $\beta$, let $u=\sqrt{1-\beta}\,e_0+\sqrt\beta\,e_1\in\mathbb C^3$. This defines

$$H\!\left(\frac{uu^\dagger+e_1e_1^\dagger+e_0e_0^\dagger}{3}\right), \qquad H(M)=-\operatorname{Re}\operatorname{tr}\!\bigl(f(M)\bigr), \quad f(x)=x\log_2x,$$

where $f(M)$ is defined by continuous functional calculus, $e_0,e_1,e_2$ are coordinate unit vectors, and $uu^\dagger$ has entries $u_a\overline{u_b}$. Here $f(0)=0$. There is no restriction on $\beta$; square roots of negative real inputs are $0$, and the definition does not require the matrix inside $H$ to have trace $1$.

QAlgorithms.Papers.Independent.Intakethird100.P11.hybridChi

For arbitrary Boolean-indexed real weights $p_i$ and complex coefficients $\alpha_i,\beta_i$, define $w_{ji}=p_i|\alpha_i|^2$ for $j=\mathrm{false}$ and $w_{ji}=p_i|\beta_i|^2$ for $j=\mathrm{true}$, and set $q_j=\sum_iw_{ji}$. Let $C_j=|\mathrm{false}\rangle\langle\mathrm{false}|$ when $q_j=0$, and otherwise let $C_j=q_j^{-1}\sum_iw_{ji}|j\mathbin{\oplus}i\rangle\langle j\mathbin{\oplus}i|$, with $|b\rangle$ the coordinate vector at $b$. This defines

$$\chi=S\!\left(\sum_jq_jC_j\right)-\sum_jq_jS(C_j), \qquad S(X)=-\operatorname{Re}\operatorname{Tr}\!\left[ \operatorname{cfc}\bigl(t\mapsto t\log_2t\bigr)(X)\right],$$

where $\operatorname{cfc}$ denotes continuous functional calculus and the real logarithm is the total real logarithm, with value $0$ at $0$. No distribution, positivity, or normalization hypotheses are required.

QAlgorithms.Papers.Independent.Intakethird100.P11.hybridConditional

For arbitrary real weights $p_i$, arbitrary complex coefficients $\alpha_i,\beta_i$, all indexed by Booleans, and a Boolean $j$, put $w_{ji}=p_i|\alpha_i|^2$ if $j=\mathrm{false}$, put $w_{ji}=p_i|\beta_i|^2$ if $j=\mathrm{true}$, and let $q_j=\sum_iw_{ji}$. This defines the Boolean-indexed complex matrix

$$C_j= \begin{cases} |\mathrm{false}\rangle\langle\mathrm{false}|,&q_j=0,\\[1mm] \displaystyle q_j^{-1}\sum_iw_{ji}|j\mathbin{\oplus}i\rangle \langle j\mathbin{\oplus}i|,&q_j\ne0, \end{cases}$$

where $|b\rangle\langle b|$ has its only nonzero entry, equal to $1$, at $(b,b)$. The zero case uses this fixed matrix even when $q_j=0$ results from cancellation of nonzero signed weights. No positivity or normalization assumptions are imposed on the inputs.

QAlgorithms.Papers.Independent.Intakethird100.P11.hybridPiggyback

For arbitrary Boolean-indexed real weights $p_i$ and complex coefficient functions $\alpha_i,\beta_i$, let $v_i=\alpha_i|\mathrm{false},i\rangle+\beta_i|\mathrm{true},\neg i\rangle$, where the kets are coordinate vectors, and let $\rho=\sum_i p_i|v_i\rangle\langle v_i|$, so $\rho_{ab}= \sum_i p_i v_i(a)\overline{v_i(b)}$. This defines the real number

$$B=1+S(\operatorname{Tr}_{\mathrm{first}}\rho)-S(\rho), \qquad (\operatorname{Tr}_{\mathrm{first}}\rho)_{b,b'} =\sum_{a\in\{\mathrm{false},\mathrm{true}\}}\rho_{(a,b),(a,b')},$$

where $S(X)=-\operatorname{Re}\operatorname{Tr}[\operatorname{cfc}(t\mapsto t\log_2t)(X)]$, using continuous functional calculus and the total real logarithm with $\ln0=0$. The inputs need not be probabilities or normalized coefficients, and $\rho$ is not assumed to be a density matrix.

QAlgorithms.Papers.Independent.Intakethird100.P11.hybridQ

For arbitrary functions $p:\{\mathrm{false},\mathrm{true}\}\to\mathbb R$ and $\alpha,\beta:\{\mathrm{false},\mathrm{true}\}\to\mathbb C$, and any Boolean $j$, this defines the real number

$$q_j= \begin{cases} \displaystyle\sum_{i\in\{\mathrm{false},\mathrm{true}\}}p_i|\alpha_i|^2, &j=\mathrm{false},\\[2mm] \displaystyle\sum_{i\in\{\mathrm{false},\mathrm{true}\}}p_i|\beta_i|^2, &j=\mathrm{true}. \end{cases}$$

The weights $p_i$ need not be nonnegative or sum to $1$, and the coefficients need not be normalized, so these numbers are not required to form a probability distribution.

QAlgorithms.Papers.Independent.Intakethird100.P11.hybridRateA

For arbitrary Boolean-indexed real weights $p_i$ and complex coefficients $\alpha_i,\beta_i$, set $v_i=\alpha_i|\mathrm{false},i\rangle+\beta_i|\mathrm{true},\neg i\rangle$ and $\rho=\sum_i p_i|v_i\rangle\langle v_i|$, with kets denoting coordinate vectors. Put $w_{ji}=p_i|\alpha_i|^2$ for $j=\mathrm{false}$, $w_{ji}=p_i|\beta_i|^2$ for $j=\mathrm{true}$, $q_j=\sum_iw_{ji}$, and

$$C_j= \begin{cases} |\mathrm{false}\rangle\langle\mathrm{false}|,&q_j=0,\\ q_j^{-1}\sum_iw_{ji}|j\mathbin{\oplus}i\rangle\langle j\mathbin{\oplus}i|,&q_j\ne0. \end{cases}$$

With $S(X)=-\operatorname{Re}\operatorname{Tr}[\operatorname{cfc}(t\mapsto t\log_2t)(X)]$, define $H(q)=-\sum_jq_j\log_2q_j$, $\chi=S(\sum_jq_jC_j)-\sum_jq_jS(C_j)$, and $B=1+S(\operatorname{Tr}_{\mathrm{first}}\rho)-S(\rho)$, where $(\operatorname{Tr}_{\mathrm{first}}\rho)_{b,b'}=\sum_a\rho_{(a,b),(a,b')}$. The definition is

$$R_A=\frac{H(q)-\chi}{H(q)-\chi+B}.$$

No probability or normalization assumptions are imposed. Logarithms use the total real logarithm, including $\ln0=0$, and division by zero has value $0$, so $R_A=0$ whenever its denominator is zero.

QAlgorithms.Papers.Independent.Intakethird100.P11.hybridVector

For arbitrary functions $\alpha,\beta:\{\mathrm{false},\mathrm{true}\}\to\mathbb C$ and a Boolean label $i$, this defines a vector $v_i$ in the complex Euclidean space indexed by pairs of Booleans by

$$v_i=\alpha_i\,|\mathrm{false},i\rangle+ \beta_i\,|\mathrm{true},\neg i\rangle.$$

Here each displayed coordinate vector has entry $1$ at its indicated pair and $0$ elsewhere; equivalently, the amplitude at $(a,b)$ is zero unless $b=a\mathbin{\oplus}i$, and in that case it is $\alpha_i$ when $a=\mathrm{false}$ and $\beta_i$ when $a=\mathrm{true}$. No normalization or other restriction is imposed on either coefficient function.

QAlgorithms.Papers.Independent.Intakethird100.P11.letterFidelity

For an arbitrary label type $I$, finite types $A,B$ with decidable equality, any family of vectors $v_i\in\mathbb C^{A\times B}$, natural numbers $n,d_A,d_B$, a distributed block code $c$, and a word $w\in I^n$, define $V_w(a,b)=\prod_{t=0}^{n-1}v_{w(t)}(a(t),b(t))$ and the matrix $\rho_w$ by $(\rho_w)_{x,y}=V_w(x)\overline{V_w(y)}$. The letter fidelity is the real number

$$\operatorname{Re}\operatorname{tr}\!\left(\rho_w\,\mathcal C(\rho_w)\right), \qquad \mathcal C=\mathcal D\circ(\mathcal E_A\otimes\mathcal E_B),$$

where $c$ has positive length $n$, positive compressed dimensions $d_A,d_B$, and encoders $\mathcal E_A,\mathcal E_B$ and decoder $\mathcal D$, each admitting a finite representation $\Phi(X)=\sum_jK_jXK_j^\dagger$ with $\sum_jK_j^\dagger K_j=I$. The tensor action applies Alice’s encoder to the Alice matrix indices and Bob’s encoder to the Bob matrix indices. The vectors $v_i$ are not assumed normalized, so the definition does not itself require $\rho_w$ to be a density matrix or the resulting number to lie in $[0,1]$.

QAlgorithms.Papers.Independent.Intakethird100.P11.sourceMean

For a finite label type $I$, an arbitrary type $X$, arbitrary real weights $p_i$, and arbitrary vectors $v_i\in\mathbb C^X$, the source mean is the complex matrix indexed by $X$ whose entries are

$$(\operatorname{sourceMean}(p,v))_{x,y} = \sum_{i\in I}p_i\,v_i(x)\overline{v_i(y)}.$$

The definition obtains this sum by enumerating $I$ with the integers from $0$ to $|I|-1$ and summing the weighted matrices $v_iv_i^\dagger$. No finiteness assumption on $X$, probability conditions on the weights, or normalization conditions on the vectors are imposed. If $I$ is empty, this is the zero matrix.

QAlgorithms.Papers.Independent.Intakethird100.P11.sourceWordVector

For arbitrary types $I,A,B$, any family $v_i\in\mathbb C^{A\times B}$ of Euclidean-space vectors indexed by $i\in I$, any natural number $n$, and any word $w:\{0,\ldots,n-1\}\to I$, the source word vector is the vector indexed by $A^n\times B^n$ whose coordinates are

$$V_w(a,b)=\prod_{t=0}^{n-1}v_{w(t)}(a(t),b(t)).$$

No finiteness, decidable-equality, or normalization assumptions are imposed here. When $n=0$, the empty product is $1$, so the vector has value $1$ at the unique pair of empty functions.