Definitions for On Mutual Information in Multipartite Quantum States and Equality in Strong Subadditivity of Entropy

QAlgorithms.Papers.Independent.Intakethird100.P63.BinaryTree.Valid

For every natural number $n$, this definition assigns a validity proposition to binary trees whose leaves are labelled by $\{0,\ldots,n-1\}$. Every leaf is valid. A node with children $l,r$ is valid exactly when both children are valid and their sets of leaf labels are disjoint, where a leaf's label set is its singleton and a node's label set is the union of its children's label sets. Thus validity requires that no label occur in two leaves, but does not require that every possible label occur. For $n=0$, there are no trees to which the definition applies.

QAlgorithms.Papers.Independent.Intakethird100.P63.BinaryTree.information

For every natural number $n$, every complex kernel $\rho$ indexed by pairs of configurations in $\mathbb N^n$, and every binary tree $t$ with leaves labelled by $\{0,\ldots,n-1\}$, this definition assigns a real number $J_\rho(t)$ recursively by $J_\rho(\operatorname{leaf}(i))=0$ and

$$J_\rho(\operatorname{split}(l,r)) =S_\rho(A_l)+S_\rho(A_r)-S_\rho(A_l\cup A_r)+J_\rho(l)+J_\rho(r),$$

where $A_t$ is the set of labels occurring in $t$. Here, for a finite coordinate set $C$, define the reduced kernel by $R_C(x,y)=\sum_{z\in\mathbb N^{C^c}}\rho(x\oplus_C z,y\oplus_C z)$, where $x\oplus_C z$ agrees with $x$ on $C$ and with $z$ on its complement, and put $S_\rho(C)=H(R_C).\mathrm{toReal}$. For a kernel $K$ on a label set $I$, $H(K)$ is the infimum of $\sum_{k=0}^\infty\max\{-w_k\log w_k,0\}$, interpreted in $[0,\infty]$, over all representations $K(i,j)=\sum_{k=0}^\infty w_k u_k(i)\overline{u_k(j)}$, with nonnegative summable real weights summing to $1$, vectors $u_k\in\ell^2(I;\mathbb C)$ of norm $1$ whenever $w_k>0$, and $\langle u_k,u_l\rangle=0$ for distinct indices of positive weight. The infimum is $+\infty$ if there is no representation, and conversion to a real number sends $+\infty$ to $0$; kernel sums use the total infinite-sum convention, which returns $0$ for a nonsummable family. No density, finite-entropy, tree-validity, or coverage assumption is required, so the children's label sets may overlap.

QAlgorithms.Papers.Independent.Intakethird100.P63.BinaryTree

For each natural number $n$, this defines the type of finite trees generated by two constructors: a leaf carrying an index $i\in\{0,\ldots,n-1\}$, and a split carrying an ordered pair of trees of the same type, designated left and right. Every tree is built using these constructors. Leaf labels may repeat, and there is no requirement that every index occur or that the labels in the two children be disjoint. There is no constructor for an empty tree; consequently, when $n=0$, this type has no elements.

QAlgorithms.Papers.Independent.Intakethird100.P63.BinaryTree.parties

For every natural number $n$, a binary tree has leaves labelled by elements of $\{0,\ldots,n-1\}$ and internal nodes with two children. This definition assigns a finite set of labels to each tree: a leaf labelled $i$ is assigned $\{i\}$, and a node with children $l,r$ is assigned the union of their assigned sets. Repeated occurrences of a label contribute only one element to this set. No validity or coverage condition is imposed. When $n=0$, there are no such trees.

QAlgorithms.Papers.Independent.Intakethird100.P63.Biorthogonal

For every natural number $n$, let $L$ be the configuration set $\{0,\ldots,n-1\}\to\mathbb N$, and let $e$ consist of real weights $w_k$ and complex matrices $\rho_k$ indexed by $L$, for $k\in\mathbb N$, with nonnegative summable weights satisfying $\sum'_k w_k=1$. At every positive weight, $\rho_k$ is required to have a representation $\rho_k(x,y)=\sum'_r u_r(x)\overline{v_r(y)}$ with $u_r,v_r\in\ell^2(L;\mathbb C)$ and $\sum'_r\|u_r\|\|v_r\|<\infty$, to have every finite principal quadratic form $\sum_{x,y\in s}\overline{a_x}\rho_k(x,y)a_y$ real and nonnegative for every finite $s\subseteq L$ and every choice of complex coefficients, and to have summable real diagonal entries with $\sum'_x\operatorname{Re}\rho_k(x,x)=1$; matrices at zero weights have no such requirement. For any subsets $A,B\subseteq\{0,\ldots,n-1\}$, the defined property means that for every ordered pair $k\ne l$ with $w_k>0$ and $w_l>0$, and for each $D\in\{A,B\}$, the matrix whose $(a,b)$ entry is $\sum'_{z:D\to\mathbb N}R_{k,D}(a,z)R_{l,D}(z,b)$ is zero, where $R_{k,D}(a,b)=\sum'_{t:D^c\to\mathbb N}\rho_k(a\sqcup t,b\sqcup t)$ and $\sqcup$ joins assignments on complementary subsets. These complex sums are assigned $0$ when nonsummable. The subsets need not be nonempty, disjoint, or covering, and $n=0$ is allowed. Only distinct positive-weight components are tested; if there is just one positive-weight component, the property is vacuous.

QAlgorithms.Papers.Independent.Intakethird100.P63.Cluster

For every natural number $n$, a cluster is a finite subset of the index set $\{0,\ldots,n-1\}$. The empty subset is allowed, and when $n=0$ it is the only cluster.

QAlgorithms.Papers.Independent.Intakethird100.P63.Coordinates

For any natural number $n$ and any finite subset $C\subseteq\{0,\ldots,n-1\}$, the coordinates on $C$ are all functions $C\to\mathbb N$, assigning an arbitrary natural number to each member of $C$. If $C$ is empty, there is exactly one such function, the empty assignment.

QAlgorithms.Papers.Independent.Intakethird100.P63.DiscreteDensityEnsemble

For any type $I$, a discrete density ensemble consists of real weights $w_k$ and complex matrices $X_k:I\times I\to\mathbb C$, indexed by $k\in\mathbb N$, with $w_k\ge0$, summable weights, and $\sum_k w_k=1$. For every $k$ with $w_k>0$, $X_k$ must satisfy all of the following: there exist sequences $a_r,b_r\in\ell^2(I;\mathbb C)$ with $\sum_r\|a_r\|\|b_r\|<\infty$ and $X_k(i,j)=\sum_r a_r(i)\overline{b_r(j)}$ for every $i,j$; for every finite $s\subseteq I$ and every $z:I\to\mathbb C$, the number $\sum_{i\in s}\sum_{j\in s}\overline{z(i)}X_k(i,j)z(j)$ has imaginary part zero and nonnegative real part; and the real diagonal family is summable with $\sum_{i\in I}\operatorname{Re}X_k(i,i)=1$. Components at zero-weight indices are arbitrary matrices. No countability assumption is imposed on $I$; if $I$ is empty, no ensemble exists because a positive-weight component would have to have diagonal sum $1$.

QAlgorithms.Papers.Independent.Intakethird100.P63.FiniteSubsystemEntropies

For every natural number $n$ and every complex matrix $\rho$ indexed by configurations $\{0,\ldots,n-1\}\to\mathbb N$, this property means that $H(R_C)\ne\infty$ for every subset $C\subseteq\{0,\ldots,n-1\}$, including the empty and full subsets. Here $R_C(x,y)=\sum'_{z:C^c\to\mathbb N}\rho(x\sqcup z,y\sqcup z)$ for assignments $x,y:C\to\mathbb N$, where $x\sqcup z$ uses $x$ on $C$ and $z$ on its complement. For a matrix $R$ indexed by a label set $L$, $H(R)$ is the infimum in $[0,\infty]$ of $\sum'_k\max\{0,-w_k\log w_k\}$ over all representations $R(a,b)=\sum'_k w_k u_k(a)\overline{u_k(b)}$, where $k\in\mathbb N$, $u_k\in\ell^2(L;\mathbb C)$, the real weights are nonnegative and summable with sum $1$, and the vectors at positive weights are unit vectors and mutually orthogonal for distinct indices. Vectors at zero weights are unrestricted within $\ell^2$. The logarithm is natural, with $\log 0=0$, and an empty infimum is $\infty$. No density or convergence assumption on $\rho$ is part of this property; nonsummable complex sums are assigned $0$. When $n=0$, the property still tests the unique subset, namely $\varnothing$.

QAlgorithms.Papers.Independent.Intakethird100.P63.IsPartition

For every natural number $n$ and every finite set $P$ of subsets of $\{0,\ldots,n-1\}$, this property means that every $C\in P$ is nonempty, any two distinct members $C,D\in P$ are disjoint, and every $i\in\{0,\ldots,n-1\}$ belongs to some $C\in P$. Thus the members cover the entire index set and are pairwise disjoint. The case $n=0$ is included: then the property holds exactly when $P=\varnothing$.

QAlgorithms.Papers.Independent.Intakethird100.P63.Monoorthogonal

For every natural number $n$, let $L$ be the configuration set $\{0,\ldots,n-1\}\to\mathbb N$, and let $e$ consist of real weights $w_k$ and complex matrices $\rho_k$ indexed by $L$, for $k\in\mathbb N$, with nonnegative summable weights satisfying $\sum'_k w_k=1$. For every $k$ with $w_k>0$, the matrix $\rho_k$ must admit a representation $\rho_k(x,y)=\sum'_r u_r(x)\overline{v_r(y)}$ with $u_r,v_r\in\ell^2(L;\mathbb C)$ and summable products $\|u_r\|\|v_r\|$, every finite principal quadratic form $\sum_{x,y\in s}\overline{a_x}\rho_k(x,y)a_y$ must be real and nonnegative for every finite $s\subseteq L$ and every choice of complex coefficients, and its real diagonal entries must be summable with sum $1$; zero-weight components have no such requirement. For any subset $B\subseteq\{0,\ldots,n-1\}$, this property means that for all distinct $k,l$ with $w_k>0$ and $w_l>0$, $\sum'_{z:B\to\mathbb N}R_{k,B}(a,z)R_{l,B}(z,b)=0$ for every pair of assignments $a,b:B\to\mathbb N$, where $R_{k,B}(a,b)=\sum'_{t:B^c\to\mathbb N}\rho_k(a\sqcup t,b\sqcup t)$ and $\sqcup$ joins assignments on complementary subsets. Nonsummable complex sums are assigned $0$. The empty subset and $n=0$ are included. No condition is tested between a component and itself or on zero-weight components; with just one positive-weight component, the property is vacuous.

QAlgorithms.Papers.Independent.Intakethird100.P63.SpectralExpansion

For any type $I$, a spectral expansion consists of real weights $w_k$ and vectors $u_k\in\ell^2(I;\mathbb C)$, indexed by $k\in\mathbb N$, such that $w_k\ge 0$ for every $k$, the sequence of weights is summable, $\sum_{k=0}^{\infty}w_k=1$, every vector with $w_k>0$ has norm $1$, and $\langle u_k,u_l\rangle=0$ whenever $k\ne l$ and both weights are positive. Vectors at zero-weight indices are arbitrary elements of $\ell^2(I;\mathbb C)$; neither normalization nor orthogonality is required of them. There is no countability assumption on $I$. If $I$ is empty, no such expansion exists, since the mass condition requires a positive weight, whereas $\ell^2(I;\mathbb C)$ has no unit vector.

QAlgorithms.Papers.Independent.Intakethird100.P63.amongCorrelation

For every natural number $n$, every complex matrix $\rho$ indexed by configurations $\{0,\ldots,n-1\}\to\mathbb N$, and every finite set $P$ of subsets of $\{0,\ldots,n-1\}$, this defines $\sum_{C\in P}S_\rho(C)-S_\rho(\{0,\ldots,n-1\})$. Here $S_\rho(D)=H(R_D).\mathrm{toReal}$, where $R_D(x,y)=\sum'_{z:D^c\to\mathbb N}\rho(x\sqcup z,y\sqcup z)$ for assignments $x,y:D\to\mathbb N$, with $\sqcup$ joining assignments on complementary subsets. For a matrix $R$ indexed by $L$, $H(R)$ is the infimum in $[0,\infty]$ of $\sum'_k\max\{0,-w_k\log w_k\}$ over representations $R(a,b)=\sum'_k w_k u_k(a)\overline{u_k(b)}$, where $u_k\in\ell^2(L;\mathbb C)$ and the nonnegative summable real weights have sum $1$, with unit vectors at positive weights and orthogonality between distinct positive-weight vectors; zero-weight vectors are unrestricted within $\ell^2$. The logarithm is natural, with $\log 0=0$. An empty infimum is $\infty$, and conversion to a real number sends $\infty$ to $0$. Neither a density or finite-entropy assumption on $\rho$ nor a partition condition on $P$ is required; nonsummable complex sums are assigned $0$. In particular, $P=\varnothing$ gives the negative of the full-subset marginal entropy, and $n=0$ is allowed.

QAlgorithms.Papers.Independent.Intakethird100.P63.attachThird

For any complex kernel $\sigma$ indexed by pairs of configurations in $\mathbb N^2$ and any complex kernel $\tau$ indexed by pairs of natural numbers, this definition produces a kernel on $\mathbb N^3$ whose entries are

$$\operatorname{attachThird}(\sigma,\tau)(x,y) =\sigma\bigl((x_0,x_1),(y_0,y_1)\bigr)\tau(x_2,y_2).$$

There are no positivity, normalization, summability, or density assumptions on either input.

QAlgorithms.Papers.Independent.Intakethird100.P63.clusterInformation

For every natural number $n$, every complex matrix $\rho$ indexed by configurations $\{0,\ldots,n-1\}\to\mathbb N$, and every subset $C\subseteq\{0,\ldots,n-1\}$, this defines $\sum_{i\in C}S_\rho(\{i\})-S_\rho(C)$. For any subset $D$, put $R_D(x,y)=\sum'_{z:D^c\to\mathbb N}\rho(x\sqcup z,y\sqcup z)$ for assignments $x,y:D\to\mathbb N$, with $\sqcup$ joining assignments on complementary subsets, and let $S_\rho(D)=H(R_D).\mathrm{toReal}$. For a matrix $R$ indexed by $L$, $H(R)$ is the infimum in $[0,\infty]$ of $\sum'_k\max\{0,-w_k\log w_k\}$ over representations $R(a,b)=\sum'_k w_k u_k(a)\overline{u_k(b)}$ by vectors $u_k\in\ell^2(L;\mathbb C)$ and nonnegative summable real weights of sum $1$, with unit vectors at positive weights and orthogonality between distinct positive-weight vectors; zero-weight vectors have no such restrictions. The logarithm is natural and $\log 0=0$. An empty infimum is $\infty$, and conversion to a real number sends $\infty$ to $0$. No density, finiteness-of-entropy, or convergence assumption on $\rho$ is imposed; nonsummable complex sums are assigned $0$. In particular, $C=\varnothing$ gives $-S_\rho(\varnothing)$, and $n=0$ is allowed.

QAlgorithms.Papers.Independent.Intakethird100.P63.clusterMutualInformation

For every natural number $n$, every complex matrix $\rho$ indexed by configurations $\{0,\ldots,n-1\}\to\mathbb N$, and any two subsets $A,B\subseteq\{0,\ldots,n-1\}$, this defines $S_\rho(A)+S_\rho(B)-S_\rho(A\cup B)$. Here $S_\rho(D)=H(R_D).\mathrm{toReal}$, where $R_D(x,y)=\sum'_{z:D^c\to\mathbb N}\rho(x\sqcup z,y\sqcup z)$ for assignments $x,y:D\to\mathbb N$, and $\sqcup$ joins assignments on complementary subsets. For a matrix $R$ indexed by $L$, $H(R)$ is the infimum in $[0,\infty]$ of $\sum'_k\max\{0,-w_k\log w_k\}$ over representations $R(a,b)=\sum'_k w_k u_k(a)\overline{u_k(b)}$ with $u_k\in\ell^2(L;\mathbb C)$ and nonnegative summable real weights of sum $1$, such that every positive-weight vector is unit and vectors at distinct positive weights are orthogonal; zero-weight vectors need not be unit or orthogonal. The logarithm is natural, with $\log 0=0$; an empty infimum is $\infty$, whose conversion to a real number is $0$. There is no density, finite-entropy, or convergence hypothesis on $\rho$, and nonsummable complex sums are assigned $0$. The subsets may be empty, overlap, or coincide, and $n=0$ is included.

QAlgorithms.Papers.Independent.Intakethird100.P63.frameCompression

For any type $I$, any natural number $m$, any complex matrix $\rho:I\times I\to\mathbb C$, and any family $v_0,\ldots,v_{m-1}\in\ell^2(I;\mathbb C)$, the defined compression is the $m\times m$ matrix

$$C_{\rho,v}(a,b) = \sum_{i\in I}\overline{v_a(i)} \left(\sum_{j\in I}\rho(i,j)v_b(j)\right) \qquad(0\le a,b<m).$$

There is no assumption that the family is orthonormal, that $\rho$ is a density matrix, or that either displayed sum is summable. These are nested total infinite sums in the displayed order; a nonsummable complex family has total sum zero. The index type $I$ need not be countable. The value $m=0$ is allowed and gives the matrix with no entries.

QAlgorithms.Papers.Independent.Intakethird100.P63.herbutEntropy

For any type $I$ and any complex matrix $\rho:I\times I\to\mathbb C$, its defined entropy is the extended nonnegative number

$$\inf\left\{ \sum_{k=0}^{\infty}\max\{-w_k\log w_k,0\}: \rho(i,j)=\sum_{k=0}^{\infty}w_k u_k(i)\overline{u_k(j)} \text{ for every }i,j\in I \right\},$$

where the infimum ranges over all summable sequences $w_k\ge0$ with $\sum_k w_k=1$ and all sequences $u_k\in\ell^2(I;\mathbb C)$ satisfying $\|u_k\|=1$ when $w_k>0$ and $\langle u_k,u_l\rangle=0$ for distinct indices with positive weights. Each nonnegative summand is regarded as an element of $[0,\infty]$, and $\log$ is the natural logarithm. Zero weights contribute zero, using the total real logarithm with $\log 0=0$, and impose no normalization or orthogonality requirements on their vectors. The sum may be infinite. No density hypothesis is imposed on $\rho$; if it has no such expansion, the infimum is over the empty set and equals $+\infty$.

QAlgorithms.Papers.Independent.Intakethird100.P63.herbutMarginalEntropy

For every natural number $n$, every complex matrix $\rho$ indexed by pairs of configurations in $\mathbb N^{\{0,\ldots,n-1\}}$, and every subset $C\subseteq\{0,\ldots,n-1\}$, this defines a real number as follows. Write $C^c$ for the complement in $\{0,\ldots,n-1\}$ and define the matrix $R_C$, indexed by assignments $x,y:C\to\mathbb N$, by $R_C(x,y)=\sum'_{z:C^c\to\mathbb N}\rho(x\sqcup z,y\sqcup z)$, where $x\sqcup z$ agrees with $x$ on $C$ and with $z$ on $C^c$. For any matrix $R$ with label set $L$, define $H(R)$ to be the infimum, in $[0,\infty]$, of $\sum'_{k\in\mathbb N}\max\{0,-w_k\log w_k\}$ over all representations $R(a,b)=\sum'_k w_k u_k(a)\overline{u_k(b)}$ with $u_k\in\ell^2(L;\mathbb C)$, nonnegative summable real weights satisfying $\sum'_k w_k=1$, unit vectors at every positive weight, and orthogonality between vectors at distinct positive weights; vectors at zero weights have no normalization or orthogonality requirement. The defined marginal entropy is $H(R_C).\mathrm{toReal}$: its finite value if $H(R_C)<\infty$, and $0$ if $H(R_C)=\infty$, including when there is no such representation and the infimum is over the empty set. Here $\log$ is the natural logarithm, with $\log 0=0$. No density, positivity, normalization, or convergence hypothesis on $\rho$ is imposed; a nonsummable complex sum $\sum'$ has value $0$. Both $C=\varnothing$ and $n=0$ are included.

QAlgorithms.Papers.Independent.Intakethird100.P63.herbutMixture

For any type $I$ and a discrete density ensemble $e$ with summable nonnegative weights $w_k$ totaling $1$, the mixture is the complex matrix

$$M_e(i,j)=\sum_{k=0}^{\infty}w_kX_k(i,j) \qquad(i,j\in I).$$

Here, for each positive-weight index $k$, the component $X_k$ has a representation $X_k(i,j)=\sum_r a_r(i)\overline{b_r(j)}$ with $a_r,b_r\in\ell^2(I;\mathbb C)$ and summable $\|a_r\|\|b_r\|$; every finite quadratic form $\sum_{i,j\in s}\overline{z(i)}X_k(i,j)z(j)$, for finite $s\subseteq I$ and arbitrary $z:I\to\mathbb C$, is real and nonnegative; and its real diagonal is summable with total $1$. Zero-weight components are unrestricted and contribute zero. The displayed entrywise series uses the total complex sum, whose value is zero if its summands are not summable; no separate summability hypothesis on this series is supplied.

QAlgorithms.Papers.Independent.Intakethird100.P63.herbutReduction

For any natural number $n$, any arbitrary complex matrix $\rho$ indexed in both entries by assignments $\{0,\ldots,n-1\}\to\mathbb N$, and any finite subset $C\subseteq\{0,\ldots,n-1\}$, the reduction is the matrix indexed by assignments $x,y:C\to\mathbb N$ defined by

$$R_C(x,y)= \sum_{z:C^{\mathrm c}\to\mathbb N} \rho\bigl(J_C(x,z),J_C(y,z)\bigr),$$

where $C^{\mathrm c}$ is the complement within $\{0,\ldots,n-1\}$, and $J_C(x,z)$ equals $x$ on $C$ and $z$ on its complement. No positivity, density, trace-class, or summability hypothesis is imposed on $\rho$; the displayed sum is the total complex sum and equals zero if its summands are not summable. If $C$ is empty, the reduction has one entry, given by the sum of all diagonal entries of $\rho$. If $C$ is the full index set, there is one complement assignment, so the reduction returns $\rho$ under the natural identification of coordinates; these conventions also cover $n=0$.

QAlgorithms.Papers.Independent.Intakethird100.P63.joinCoordinate

For any natural number $n$, any finite subset $C\subseteq\{0,\ldots,n-1\}$, and assignments $x:C\to\mathbb N$ and $z:C^{\mathrm c}\to\mathbb N$, where the complement is taken within $\{0,\ldots,n-1\}$, the joined assignment $J_C(x,z):\{0,\ldots,n-1\}\to\mathbb N$ is defined by

$$J_C(x,z)(i)= \begin{cases} x(i),&i\in C,\\ z(i),&i\notin C. \end{cases}$$

Empty clusters and empty complements are allowed; in particular, for $n=0$ the result is the unique empty assignment.

QAlgorithms.Papers.Independent.Intakethird100.P63.kernelProduct

For any type $I$ and any two complex matrices $\rho,\sigma$ indexed by $I\times I$, this defines the matrix $K(i,j)=\sum'_{z\in I}\rho(i,z)\sigma(z,j)$ for every $i,j\in I$. There is no finiteness, countability, positivity, or summability assumption on $I$ or the matrices. The sum is the unconditional sum over $I$, assigned the value $0$ when its summands are not summable. The empty label set is also allowed.

QAlgorithms.Papers.Independent.Intakethird100.P63.productAcross

For every natural number $n$, every complex matrix $\rho$ indexed by configurations $\{0,\ldots,n-1\}\to\mathbb N$, and every subset $C\subseteq\{0,\ldots,n-1\}$, this defines a matrix on the same configurations by $P_C(x,y)=R_C(x|_C,y|_C)\,R_{C^c}(x|_{C^c},y|_{C^c})$. Here the complement is taken in $\{0,\ldots,n-1\}$, restrictions retain the indicated coordinates, and, for each subset $D$, $R_D(a,b)=\sum'_{z:D^c\to\mathbb N}\rho(a\sqcup z,b\sqcup z)$ for assignments $a,b:D\to\mathbb N$, with $\sqcup$ joining assignments on complementary subsets. No density, positivity, normalization, or convergence hypothesis on $\rho$ is imposed; nonsummable complex sums are assigned $0$. Empty and full choices of $C$ are permitted, as is $n=0$; assignments on an empty subset form a singleton.

QAlgorithms.Papers.Independent.Intakethird100.P63.productMixture

Let $e$ consist of real weights $w_k$ and complex kernels $K_k$ on $\mathbb N^2$, indexed by $k\in\mathbb N$, with $w_k\ge0$, the weights summable, and $\sum_k w_k=1$. Whenever $w_k>0$, $K_k$ must have a representation $K_k(i,j)=\sum_r a_r(i)\overline{b_r(j)}$ with $a_r,b_r\in\ell^2(\mathbb N^2;\mathbb C)$ and $\sum_r\|a_r\|\|b_r\|<\infty$; every finite quadratic form $\sum_{i,j\in F}\overline{z_i}K_k(i,j)z_j$, for finite $F\subseteq\mathbb N^2$ and arbitrary $z:\mathbb N^2\to\mathbb C$, must be real and nonnegative; and its real diagonal must be summable with sum $1$. For every family of arbitrary complex kernels $\tau_k$ on $\mathbb N$, the definition produces the kernel

$$\operatorname{productMixture}(e,\tau)(x,y) =\sum_{k=0}^\infty w_k K_k\bigl((x_0,x_1),(y_0,y_1)\bigr)\tau_k(x_2,y_2), \qquad x,y\in\mathbb N^3.$$

The components $K_k$ with zero weight have no density requirement, and there are no assumptions on the $\tau_k$. Each displayed series is the total complex infinite sum, equal to $0$ if that entry's series is not summable; the definition itself asserts no density property of the resulting kernel.

QAlgorithms.Papers.Independent.Intakethird100.P63.pureExpansion

For every label type $I$, every vector $v\in\ell^2(I;\mathbb C)$, and an assumption $\|v\|=1$, this definition constructs a spectral expansion with weights $w_0=1$ and $w_k=0$ for every $k\ne0$, and with vector $v$ at every index. The resulting weights are nonnegative and summable with sum $1$; its vectors have norm $1$ at every positive-weight index; and vectors at distinct positive-weight indices are required to be orthogonal. That last requirement is vacuous here because $0$ is the only positive-weight index. No countability or nonemptiness assumption is imposed on $I$; if $I$ is empty, the required unit vector cannot exist.

QAlgorithms.Papers.Independent.Intakethird100.P63.restrictCoordinate

For any natural number $n$, any finite subset $C\subseteq\{0,\ldots,n-1\}$, and any assignment $x:\{0,\ldots,n-1\}\to\mathbb N$, the restricted coordinate assignment is the function on $C$ sending each $i\in C$ to $x(i)$. This includes the unique empty assignment when $C=\varnothing$, and includes $n=0$.

QAlgorithms.Papers.Independent.Intakethird100.P63.spectralKernel

For any type $I$ and any spectral expansion $e$, consisting of summable nonnegative weights $w_k$ of total mass $1$ and vectors $u_k\in\ell^2(I;\mathbb C)$ that are unit vectors and pairwise orthogonal at distinct positive-weight indices, the associated matrix $K_e:I\times I\to\mathbb C$ is defined entrywise by

$$K_e(i,j)=\sum_{k=0}^{\infty}w_k\,u_k(i)\,\overline{u_k(j)} \qquad(i,j\in I).$$

Zero-weight vectors remain unrestricted apart from belonging to $\ell^2(I;\mathbb C)$, and their summands are zero.

QAlgorithms.Papers.Independent.Intakethird100.P63.tensorMarginals

For every natural number $n$ and every complex matrix $\rho$ indexed by configurations $\{0,\ldots,n-1\}\to\mathbb N$, this defines another matrix on the same configurations by $T(x,y)=\prod_{i=0}^{n-1}R_{\{i\}}(x|_{\{i\}},y|_{\{i\}})$. For any subset $D$, the matrix appearing here is $R_D(a,b)=\sum'_{z:D^c\to\mathbb N}\rho(a\sqcup z,b\sqcup z)$, where $a,b:D\to\mathbb N$ and the joined configuration uses the first assignment on $D$ and $z$ on its complement; thus the singleton assignments have values $x_i$ and $y_i$. No positivity, normalization, density, or summability assumption on $\rho$ is required, and a nonsummable complex sum is assigned $0$. When $n=0$, the product is empty and equals $1$, so the resulting matrix has its unique entry equal to $1$.

QAlgorithms.Papers.Independent.Intakethird100.P63.traceClassDivergence

For any type $I$ and arbitrary complex matrices $\rho,\sigma:I\times I\to\mathbb C$, the defined divergence is

$$\sup_{m\in\mathbb N}\ \sup_{\substack{v_0,\ldots,v_{m-1}\in\ell^2(I;\mathbb C)\\ \langle v_a,v_b\rangle=\delta_{ab}}} D_{\mathrm{fin}}(C_{\rho,v},C_{\sigma,v}),$$

with values in $[0,\infty]$, where

$$C_{\rho,v}(a,b)= \sum_{i\in I}\overline{v_a(i)} \left(\sum_{j\in I}\rho(i,j)v_b(j)\right),$$

and similarly for $\sigma$. For finite square matrices $X,Y$, $D_{\mathrm{fin}}(X,Y)=+\infty$ if $\ker Y\not\subseteq\ker X$; otherwise it is

$$\max\!\left\{ \operatorname{Re}\operatorname{tr} \bigl(f(X)-Xg(Y)-X+Y\bigr),\,0 \right\}, \qquad f(t)=t\log t,\quad g(t)=\log t,$$

regarded as an extended nonnegative number, with $f(X)$ and $g(Y)$ evaluated by the real continuous functional calculus used in the definition. No trace-class, positivity, normalization, or countability hypotheses are imposed on the input matrices or on $I$. The compression sums are total sums, taking value zero for nonsummable complex families. The supremum includes $m=0$, whose empty frame contributes zero; for any $m$ admitting no orthonormal frame, the corresponding inner supremum is zero.