Definitions for Bounds on quantum nonlocality via partial transposition

QAlgorithms.Papers.Independent.Intakethird100.P09.BellCoefficients

For any Bell scenario $S=(n_x,n_y,n_a,n_b)$, Bell coefficients are an arbitrary real-valued array $s_{xyab}$, indexed by $0\le x<n_x$, $0\le y<n_y$, $0\le a<n_a$, and $0\le b<n_b$. No sign, normalization, or other constraint is imposed on this array.

QAlgorithms.Papers.Independent.Intakethird100.P09.BellExperiment

For natural numbers $a,b$, possibly zero, this structure consists of positive output dimensions $c,d$, a scenario with positive natural numbers $n_x,n_y,n_a,n_b$ of Alice and Bob inputs and outcomes, a function $\Phi$ from complex matrices on $\mathbb C^a\otimes\mathbb C^b$ to matrices on $\mathbb C^c\otimes\mathbb C^d$, and local measurement matrices $A_{x,\alpha}$ on $\mathbb C^c$ and $B_{y,\beta}$ on $\mathbb C^d$. For each input $x$, all $A_{x,\alpha}$ must be positive semidefinite and satisfy $\sum_\alpha A_{x,\alpha}=I_c$; the analogous conditions hold for every $y$ and its $B_{y,\beta}$. The function $\Phi$ must lie in the pointwise topological closure of maps recursively constructed from bipartite identities and finite local Kraus instruments: each Alice or Bob instrument has local Kraus matrices whose adjoint products sum to the local identity, each outcome is followed by a recursively constructed continuation, and the branch outputs are summed. The structure contains no input state and imposes no additional relation among its output dimensions, scenario counts, and measurements.

QAlgorithms.Papers.Independent.Intakethird100.P09.BellScenario

A Bell scenario consists of four strictly positive integers $n_x,n_y,n_a,n_b$. These give the numbers of Alice’s inputs, Bob’s inputs, Alice’s outputs, and Bob’s outputs, respectively; none may be zero.

QAlgorithms.Papers.Independent.Intakethird100.P09.HasFiniteLocalModel

Let $S$ specify positive natural numbers $n_x,n_y,n_a,n_b$, giving Alice’s and Bob’s input and output index sets, and let $P(x,y,a,b)$ be any real-valued array on these sets. This predicate means that there exist a natural number $k$, real weights $w_i\geq0$ for $0\leq i<k$ with $\sum_iw_i=1$, and deterministic response functions $f_i:\{0,\ldots,n_x-1\}\to\{0,\ldots,n_a-1\}$ and $g_i:\{0,\ldots,n_y-1\}\to\{0,\ldots,n_b-1\}$, such that, for every input and output quadruple,

$$P(x,y,a,b)=\sum_{i=0}^{k-1}w_i\,\mathbf1_{\{f_i(x)=a,\ g_i(y)=b\}}.$$

Zero weights are permitted; $k=0$ cannot satisfy the requirement that the weights sum to $1$. No separate correlation or no-signaling hypothesis is required of $P$.

QAlgorithms.Papers.Independent.Intakethird100.P09.IsOperationalLOCC

For any natural numbers $a,b,c,d$, including zero, and any function $L$ from complex matrices on $\mathbb C^a\otimes\mathbb C^b$ to complex matrices on $\mathbb C^c\otimes\mathbb C^d$, this predicate means that $L$ belongs to the topological closure of the following recursively generated set of maps: identity maps on bipartite systems are included; an Alice step chooses finitely many rectangular matrices $A_i$ with $\sum_iA_i^\dagger A_i=I$, applies $X\mapsto(A_i\otimes I)X(A_i\otimes I)^\dagger$, follows each outcome $i$ by another recursively generated map $L_i$, and sums the resulting outputs; a Bob step has the same form using $I\otimes B_i$ with $\sum_iB_i^\dagger B_i=I$. Local dimensions may change, and each continuation may depend on its outcome. The closure uses the topology on functions given by pointwise convergence of output matrices. No separate linearity or channel condition is included in this predicate beyond membership in that closure.

QAlgorithms.Papers.Independent.Intakethird100.P09.LocalFilter

For any natural numbers $a,b$, a local filter consists of positive integers $r_A,r_B$, families of complex square matrices $A_i\in\mathbb C^{a\times a}$ for $0\le i<r_A$ and $B_j\in\mathbb C^{b\times b}$ for $0\le j<r_B$, satisfying

$$\sum_{i=0}^{r_A-1}A_i^\dagger A_i=I_a, \qquad \sum_{j=0}^{r_B-1}B_j^\dagger B_j=I_b,$$

together with selected indices $i_*\in\{0,\ldots,r_A-1\}$ and $j_*\in\{0,\ldots,r_B-1\}$. The local dimensions $a,b$ may be zero. The selected matrices may be zero; no positive probability or nonzero action is required of the selected branch.

QAlgorithms.Papers.Independent.Intakethird100.P09.Measurements

For any Bell scenario $S=(n_x,n_y,n_a,n_b)$ and finite types $I,J$, each equipped with decidable equality, a measurement specification consists of complex matrices $A_{x,a}$ indexed by $I\times I$ and $B_{y,b}$ indexed by $J\times J$. For every $x<n_x$, all $A_{x,a}$ are positive semidefinite and $\sum_{a<n_a}A_{x,a}=1_I$; for every $y<n_y$, all $B_{y,b}$ are positive semidefinite and $\sum_{b<n_b}B_{y,b}=1_J$. The types $I,J$ are allowed to be empty; on an empty index type, the identity and all matrices are the unique empty matrix, so the corresponding requirements are satisfied.

QAlgorithms.Papers.Independent.Intakethird100.P09.asFin

For any finite types $I,J$, possibly empty, and any complex matrix $\rho$ indexed by $I\times J$, choose the bijections $e_I:I\to\{0,\ldots,|I|-1\}$ and $e_J:J\to\{0,\ldots,|J|-1\}$ supplied by their finite-type structures. This defines a matrix indexed by $\{0,\ldots,|I|-1\}\times\{0,\ldots,|J|-1\}$ through

$$(\operatorname{asFin}\rho)_{(a,b),(a',b')} =\rho_{(e_I^{-1}(a),e_J^{-1}(b)),(e_I^{-1}(a'),e_J^{-1}(b'))}.$$

Thus the definition reindexes both rows and columns using those bijections. It requires no nonemptiness, decidable-equality, positivity, or normalization hypotheses.

QAlgorithms.Papers.Independent.Intakethird100.P09.asymptoticR

For natural numbers $a,b$, a Boolean flag $u$, and any complex matrix $\rho$ on $\{0,\ldots,a-1\}\times\{0,\ldots,b-1\}$, this defines the extended nonnegative real number

$$\limsup_{m\to\infty}\frac{\mathcal R_u(\rho^{[m]})}{m}.$$

Here $\rho^{[m]}$ is the matrix with entries $\prod_{t=0}^{m-1}\rho_{(x_t,y_t),(x'_t,y'_t)}$, relabelled onto local index sets of sizes $a^m,b^m$ using the specified finite-set bijections. For a matrix $\omega$, $\mathcal R_u(\omega)$ is the supremum over experiments consisting of a map $\Lambda$ in the closure of finite LOCC maps, positive finite output dimensions, a Bell scenario with positive finite input and outcome counts, and local POVMs $(A_{x,\alpha})_\alpha,(B_{y,\beta})_\beta$, of the following quantity for $P(\alpha,\beta\mid x,y)=\operatorname{Re}\operatorname{tr}[(A_{x,\alpha}\otimes B_{y,\beta})\Lambda(\omega)]$:

$$\begin{cases} \displaystyle\inf_{Q\text{ finite local}}D_{\mathrm{unif}}(P\|Q),&u=\mathrm{true},\\[2mm] \displaystyle\sup_{\pi\text{ input probability law}}\inf_{Q\text{ finite local}}D_\pi(P\|Q),&u=\mathrm{false}. \end{cases}$$

Each POVM consists of positive semidefinite matrices summing to the identity. A finite local box is a finite convex combination of deterministic boxes $\mathbf1_{\alpha=f(x),\,\beta=g(y)}$. The divergence is $D_\pi(P\|Q)=\sum_{x,y}\max(\pi_{xy},0)\,K(P_{xy},Q_{xy})$, interpreted in $[0,\infty]$, where $K(p,q)=\infty$ if some $p_i>0$ has $q_i=0$, and otherwise

$$K(p,q)=\max\!\left(0,\sum_{i:p_i\ne0}p_i\frac{\log(p_i/q_i)}{\log2}\right).$$

The uniform input law assigns $1/(\#x\,\#y)$ to each pair. Finite LOCC maps are generated by identity maps and finite, outcome-dependent local Kraus operations, summing over branches, with each local Kraus family satisfying $\sum_j E_j^\dagger E_j=I$. The limiting sequence uses only $m=n+1\ge1$, so its denominator never vanishes. Neither positivity nor trace one is required of $\rho$, and $a$ or $b$ may be zero; all expressions retain their literal total-function meanings, including division by zero and $\log0=0$. Suprema and the limit superior take values in $[0,\infty]$, with an empty supremum equal to zero.

QAlgorithms.Papers.Independent.Intakethird100.P09.asymptoticRH

For natural numbers $a,b$, a Boolean flag $u$, and any complex matrix $\rho$ on local index sets of sizes $a,b$, this defines

$$\limsup_{m\to\infty}\frac{\mathcal H_u(\rho^{[m]})}{m}\in[0,\infty],$$

where $\rho^{[m]}$ has entries $\prod_{t=0}^{m-1}\rho_{(i_t,j_t),(i'_t,j'_t)}$, relabelled onto local index sets of sizes $a^m,b^m$ by the specified bijections. For any matrix $\omega$, $\mathcal H_u(\omega)$ is the supremum over a map $\Lambda$ in the closure of finite LOCC maps, positive finite output dimensions, local output filters with complete nonempty finite Kraus families $C_i,D_j$ and selected indices $i_*,j_*$, and Bell measurements with positive finite input and outcome counts. Write

$$\tau=(C_{i_*}\otimes D_{j_*})\Lambda(\omega)(C_{i_*}\otimes D_{j_*})^\dagger, \qquad p=\operatorname{Re}\operatorname{tr}\tau.$$

The optimized value is zero for $p=0$, and otherwise is $\max(p,0)\mathcal N_u(P)$, where $P(\alpha,\beta\mid x,y)=\operatorname{Re}\operatorname{tr}[(A_{x,\alpha}\otimes B_{y,\beta})p^{-1}\tau]$, and each local measurement is a family of positive semidefinite effects summing to the identity. The quantity $\mathcal N_u(P)$ is $\inf_{Q\text{ finite local}}D_{\mathrm{unif}}(P\|Q)$ for $u=\mathrm{true}$, and $\sup_{\pi\text{ input probability law}}\inf_{Q\text{ finite local}}D_\pi(P\|Q)$ otherwise. Finite local boxes are finite convex combinations of deterministic product responses. Here $D_\pi(P\|Q)=\sum_{x,y}\max(\pi_{xy},0)K(P_{xy},Q_{xy})$, where $K(p,q)=\infty$ if some $p_i>0$ has $q_i=0$, and otherwise $K(p,q)=\max(0,\sum_{i:p_i\ne0}p_i\log(p_i/q_i)/\log2)$; the uniform input law weights input pairs equally. Finite LOCC maps are generated by identities and finite, outcome-dependent complete local Kraus operations, summing over branches. The limit uses only $m=n+1\ge1$. No density-matrix hypothesis is imposed on $\rho$, and zero local dimensions are allowed; negative branch traces are converted to zero in $[0,\infty]$, where $0\cdot\infty=0$, and empty suprema are zero.

QAlgorithms.Papers.Independent.Intakethird100.P09.asymptoticT

For natural numbers $a,b$ whose index sets $\{0,\ldots,a-1\}$ and $\{0,\ldots,b-1\}$ are nonempty, hence $a,b>0$, a Boolean flag $u$, and any complex matrix $\rho$ on their product, this defines

$$\limsup_{m\to\infty}\frac{\mathcal T_u(\rho,m)}{m}\in[0,\infty].$$

For each $m$, $\mathcal T_u(\rho,m)$ is the supremum over Bell experiments on local dimensions $a^m,b^m$. Such an experiment comprises a map $\Lambda$ in the closure of finite LOCC maps, positive finite output dimensions, positive finite numbers of inputs and outcomes, and local POVMs $A_{x,\alpha},B_{y,\beta}$. Set

$$P_\rho(\alpha,\beta\mid x,y) =\operatorname{Re}\operatorname{tr}[(A_{x,\alpha}\otimes B_{y,\beta})\Lambda(\rho^{[m]})],$$

where $\rho^{[m]}$ has tensor-product entries $\prod_{t=0}^{m-1}\rho_{(i_t,j_t),(i'_t,j'_t)}$, relabelled by the specified bijections onto index sets of sizes $a^m,b^m$. The experiment contributes

$$\begin{cases} \displaystyle\inf_{\sigma\text{ separable density matrix}}D_{\mathrm{unif}}(P_\rho\|P_\sigma),&u=\mathrm{true},\\[2mm] \displaystyle\sup_{\pi\text{ input probability law}}\inf_{\sigma\text{ separable density matrix}}D_\pi(P_\rho\|P_\sigma),&u=\mathrm{false}. \end{cases}$$

Here $\sigma$ is a single matrix on the original $a\times b$ system, and $P_\sigma$ uses $\sigma^{[m]}$ with the same experiment. “Separable density matrix” means a finite sum $\sum_j w_j S_j\otimes T_j$, where $w_j\ge0$, $\sum_jw_j=1$, and $S_j,T_j$ are positive semidefinite of trace one. Each POVM consists of positive semidefinite matrices summing to the identity. Finite LOCC maps are generated by identities and finite, outcome-dependent local Kraus operations with complete Kraus families, summing over branches. The divergence is $D_\pi(P\|Q)=\sum_{x,y}\max(\pi_{xy},0)K(P_{xy},Q_{xy})$, where $K(p,q)=\infty$ if some $p_i>0$ has $q_i=0$, and otherwise $K(p,q)=\max(0,\sum_{i:p_i\ne0}p_i\log(p_i/q_i)/\log2)$; the uniform input law gives every input pair equal weight. The limit uses $m=n+1\ge1$. No density-matrix assumption is imposed on $\rho$; the real division and logarithms are total, and all optimization and limiting operations shown here are in $[0,\infty]$.

QAlgorithms.Papers.Independent.Intakethird100.P09.bellOperator

For any Bell scenario $S=(n_x,n_y,n_a,n_b)$, finite types $I,J$ with decidable equality, real coefficient array $s_{xyab}$, and measurement specification $M$, this defines the complex matrix

$$\sum_{x<n_x}\sum_{y<n_y}\sum_{a<n_a}\sum_{b<n_b} s_{xyab}\,(A_{x,a}\otimes B_{y,b})$$

on $I\times J$. For every $x$, the matrices $A_{x,a}$ are positive semidefinite and sum over $a$ to $1_I$, and for every $y$, the matrices $B_{y,b}$ are positive semidefinite and sum over $b$ to $1_J$; tensor-product entries are $A_{ii'}B_{jj'}$. The coefficients may have either sign. The types $I,J$ may be empty, yielding an empty matrix.

QAlgorithms.Papers.Independent.Intakethird100.P09.bellOperatorNorm

For any finite type $I$ equipped with decidable equality and any complex matrix $X$ indexed by $I\times I$, this defines a real number equal to the ambient matrix norm $\|X\|$. No Hermiticity, positivity, or normalization assumption is imposed on $X$. The index type may be empty, in which case this norm is zero.

QAlgorithms.Papers.Independent.Intakethird100.P09.bellValue

For any Bell scenario $S=(n_x,n_y,n_a,n_b)$ and arbitrary real arrays $s_{xyab}$ and $P_{xyab}$ on its input and output indices, the Bell value is

$$\sum_{x<n_x}\sum_{y<n_y}\sum_{a<n_a}\sum_{b<n_b}s_{xyab}P_{xyab}.$$

Neither array is required to be nonnegative or normalized.

QAlgorithms.Papers.Independent.Intakethird100.P09.boxD

For a scenario $S$ with positive numbers $n_x,n_y,n_a,n_b$ of inputs and outputs, any real input array $p(x,y)$, and any real arrays $P(x,y,a,b)$ and $Q(x,y,a,b)$, this defines

$$\sum_{x=0}^{n_x-1}\sum_{y=0}^{n_y-1} [p(x,y)]_+\,D\bigl(P(x,y,\cdot,\cdot),Q(x,y,\cdot,\cdot)\bigr) \quad\text{in }[0,+\infty],$$

where $[r]_+=\max(r,0)$, embedded into the extended nonnegative reals. For arrays $u,v$ on the output pairs, $D(u,v)=+\infty$ if some output pair has $u>0$ and $v=0$; otherwise it is the nonnegative part of $\sum_{a,b:u(a,b)\neq0}u(a,b)\log(u(a,b)/v(a,b))/\log2$. These expressions use total real division and logarithms, with division by zero and $\log0$ equal to zero and $\log t=\log|t|$ for negative $t$. No probability, normalization, or no-signaling assumptions are imposed on $p,P,Q$. Extended multiplication uses $0\cdot(+\infty)=0$, so input pairs with nonpositive $p$ contribute zero even when their divergence is infinite.

QAlgorithms.Papers.Independent.Intakethird100.P09.boxNonlocality

For a scenario $S$ with positive input and output counts and any real array $P(x,y,a,b)$, this defines, when the Boolean flag is true, the infimum over finite local models $Q$ of the input-weighted divergence at the uniform input law; when the flag is false, it defines

$$\sup_{\substack{p(x,y)\geq0\\\sum_{x,y}p(x,y)=1}} \ \inf_{Q\text{ a finite local model}} \sum_{x,y}[p(x,y)]_+D(P_{xy},Q_{xy}).$$

All extrema and sums here take values in $[0,+\infty]$. A finite local model has the form $Q(x,y,a,b)=\sum_{i=0}^{k-1}w_i\mathbf1_{\{f_i(x)=a,\ g_i(y)=b\}}$, for a natural number $k$, nonnegative real weights summing to $1$, and deterministic local response functions with the scenario’s input and output sets. For output arrays $u,v$, $D(u,v)=+\infty$ if some coordinate has $u>0,v=0$; otherwise it is the nonnegative part of $\sum_{a,b:u(a,b)\neq0}u(a,b)\log(u(a,b)/v(a,b))/\log2$. Division and logarithms are total, with division by zero and $\log0$ equal to zero and negative logarithm arguments interpreted through their absolute values; extended multiplication satisfies $0\cdot(+\infty)=0$. In the uniform case $p(x,y)=1/(n_xn_y)$. The infimizing local model may depend on the chosen input law, and $P$ itself is not assumed to be a correlation.

QAlgorithms.Papers.Independent.Intakethird100.P09.chshCoefficients

For inputs $x,y\in\{0,1\}$ and outcomes $\alpha,\beta\in\{0,1\}$, this defines the real coefficient

$$s_{xy\alpha\beta} = \begin{cases}-1,&x=y=1,\\1,&\text{otherwise}\end{cases} \begin{cases}1,&\alpha=\beta,\\-1,&\alpha\ne\beta.\end{cases}$$

Thus equal outcomes receive coefficient $1$ and unequal outcomes coefficient $-1$, with both signs reversed for the input pair $(1,1)$.

QAlgorithms.Papers.Independent.Intakethird100.P09.chshScenario

This defines a Bell scenario with two inputs for Alice, two inputs for Bob, two outcomes for Alice, and two outcomes for Bob. Each input and outcome index therefore ranges over $\{0,1\}$.

QAlgorithms.Papers.Independent.Intakethird100.P09.classicalBound

For any Bell scenario $S=(n_x,n_y,n_a,n_b)$ and arbitrary real coefficient array $s_{xyab}$, the classical bound is the real supremum of the values

$$\sum_{x<n_x}\sum_{y<n_y}s_{xy\,f(x)\,g(y)}$$

over all functions $f:\{0,\ldots,n_x-1\}\to\{0,\ldots,n_a-1\}$ and $g:\{0,\ldots,n_y-1\}\to\{0,\ldots,n_b-1\}$. This is a finite nonempty collection because all four scenario sizes are strictly positive.

QAlgorithms.Papers.Independent.Intakethird100.P09.classicalKLBits

For any finite type $I$, possibly empty, and arbitrary real-valued functions $p,q:I\to\mathbb R$, this defines an extended nonnegative real number by

$$D(p,q)= \begin{cases} +\infty,&\exists i,\ p_i>0\ \text{and}\ q_i=0,\\[2mm] \displaystyle\max\!\left\{0,\sum_{i\in I} \begin{cases} 0,&p_i=0,\\ p_i\,\dfrac{\log(p_i/q_i)}{\log2},&p_i\neq0 \end{cases}\right\},&\text{otherwise}. \end{cases}$$

The finite value in the second branch is embedded into the extended nonnegative reals. Neither function is assumed nonnegative or normalized. Division and logarithms are total: division by zero gives zero, $\log0=0$, and the real logarithm at a negative argument is the logarithm of its absolute value. The exceptional infinite branch tests strictly positive $p_i$, not merely nonzero $p_i$. If $I$ is empty, the value is zero.

QAlgorithms.Papers.Independent.Intakethird100.P09.deterministicBox

For any Bell scenario $S=(n_x,n_y,n_a,n_b)$ and functions $f:\{0,\ldots,n_x-1\}\to\{0,\ldots,n_a-1\}$ and $g:\{0,\ldots,n_y-1\}\to\{0,\ldots,n_b-1\}$, this defines the real array $P_{xyab}$ to equal $1$ when both $f(x)=a$ and $g(y)=b$, and $0$ otherwise.

QAlgorithms.Papers.Independent.Intakethird100.P09.flatProbability

For every positive integer $d$, including $d=1$, this defines the real number

$$p=\frac{1}{\sqrt d+1}.$$

QAlgorithms.Papers.Independent.Intakethird100.P09.flatX

For every positive integer $d$ and every complex $d\times d$ unitary matrix $U$, this defines a complex matrix $X$ indexed by pairs from $\{0,\ldots,d-1\}$, with

$$X_{(i,j),(k,\ell)} = \begin{cases} \dfrac{U_{ij}}{d\sqrt d},&(k,\ell)=(j,i),\\ 0,&\text{otherwise}. \end{cases}$$

The case $d=1$ is included. The assumption on $U$ is unitarity; no equal-magnitude condition on its entries is imposed.

QAlgorithms.Papers.Independent.Intakethird100.P09.flatY

For every positive integer $d$ and every complex $d\times d$ unitary matrix $U$, this defines $Y=\sqrt d\,X^{T_2}$, where

$$X_{(i,j),(k,\ell)} =\begin{cases}U_{ij}/(d\sqrt d),&(k,\ell)=(j,i),\\0,&\text{otherwise},\end{cases} \qquad (X^{T_2})_{(i,j),(k,\ell)}=X_{(i,\ell),(k,j)}.$$

Equivalently,

$$Y_{(i,j),(k,\ell)} = \begin{cases} U_{ik}/d,&i=j\ \text{and}\ k=\ell,\\ 0,&\text{otherwise}. \end{cases}$$

All four indices range over $\{0,\ldots,d-1\}$, including when $d=1$; only unitarity is assumed of $U$.

QAlgorithms.Papers.Independent.Intakethird100.P09.hiddenBlock

For natural numbers $a,b$, a Boolean flag $u$, and any complex matrix $\rho$ on the product of local index sets of sizes $a,b$, this defines an extended nonnegative supremum over Bell experiments and selected local-filter branches. An experiment consists of a map $\Lambda$ in the closure of finite LOCC maps, positive finite output dimensions, positive finite numbers of inputs and outcomes, and local POVMs $A_{x,\alpha},B_{y,\beta}$. A filter on the output consists of nonempty finite families $C_i,D_j$, satisfying $\sum_iC_i^\dagger C_i=I$, $\sum_jD_j^\dagger D_j=I$, and selected indices $i_*,j_*$. Put

$$K=C_{i_*}\otimes D_{j_*},\qquad \tau=K\Lambda(\rho)K^\dagger,\qquad p=\operatorname{Re}\operatorname{tr}\tau.$$

The value included in the supremum is zero when $p=0$, and otherwise is

$$\max(p,0)\,\mathcal N_u(P),\qquad P(\alpha,\beta\mid x,y) =\operatorname{Re}\operatorname{tr}[(A_{x,\alpha}\otimes B_{y,\beta})(p^{-1}\tau)].$$

Here $\mathcal N_u(P)$ is $\inf_{Q\text{ finite local}}D_{\mathrm{unif}}(P\|Q)$ when $u=\mathrm{true}$, and $\sup_{\pi\text{ input probability law}}\inf_{Q\text{ finite local}}D_\pi(P\|Q)$ otherwise. A finite local box is a finite convex combination of $\mathbf1_{\alpha=f(x),\,\beta=g(y)}$. The divergence is $D_\pi(P\|Q)=\sum_{x,y}\max(\pi_{xy},0)K(P_{xy},Q_{xy})$, with $K(p,q)=\infty$ if some $p_i>0$ has $q_i=0$, and otherwise

$$K(p,q)=\max\!\left(0,\sum_{i:p_i\ne0}p_i\frac{\log(p_i/q_i)}{\log2}\right).$$

The uniform input law assigns equal weight to every input pair. POVM effects are positive semidefinite and sum to the identity; finite LOCC maps are generated by identities and finite, outcome-dependent complete local Kraus operations, summing over branches. No positivity or normalization hypothesis is imposed on $\rho$, and $a,b$ may be zero. In particular, $p$ need not be positive; for $p<0$, its extended nonnegative conversion is zero, and multiplication uses the convention $0\cdot\infty=0$. An empty supremum is zero.

QAlgorithms.Papers.Independent.Intakethird100.P09.hidingResource

For every natural number $m$, this defines a complex matrix $H_m$ on two parties, each indexed by $\{0,1\}\times\mathcal A_m$, where $\mathcal A_m=\{0,\ldots,m^2-1\}^{\{0,\ldots,m-1\}^2}$. To specify its entries, put $d=m^2$, let $I_d$ be the identity on $\mathbb C^d\otimes\mathbb C^d$, let $F_d$ have entries $(F_d)_{(i,j),(k,l)}=1$ when $i=l$ and $j=k$, and $0$ otherwise, and define $W_+=(I_d+F_d)/(d^2+d)$ and $W_-=(I_d-F_d)/(d^2-d)$. On pairs of length-$m$ strings over $\{0,\ldots,d-1\}$, define $T_1$ and $T_2$ by $(T_1)_{(u,v),(u',v')}=\prod_{s=0}^{m-1}((W_++W_-)/2)_{(u_s,v_s),(u'_s,v'_s)}$ and $(T_2)_{(u,v),(u',v')}=\prod_{s=0}^{m-1}(W_+)_{(u_s,v_s),(u'_s,v'_s)}$. Put $D=(T_1+T_2)/6$, $O=(T_1-T_2)/6$, $J=T_2/6$, and $\mathcal T_m(Z)_{(A,B),(A',B')}=\prod_{t=0}^{m-1}Z_{(A_t,B_t),(A'_t,B'_t)}$. Writing $a=(i,j)$, $b=(i',j')$, and $c_m=(2(1/3)^m+2(1/6)^m)^{-1}$, the entry of $H_m$ with row $((i,A),(j,B))$ and column $((i',A'),(j',B'))$ is $c_m\mathcal T_m(D)_{(A,B),(A',B')}$ if $a=b=(0,0)$ or $a=b=(1,1)$, is $c_m\mathcal T_m(O)_{(A,B),(A',B')}$ if $(a,b)=((0,0),(1,1))$ or $((1,1),(0,0))$, is $c_m\mathcal T_m(J)_{(A,B),(A',B')}$ if $a=b=(0,1)$ or $a=b=(1,0)$, and is $0$ otherwise. This definition includes $m=0$, with empty products equal to $1$, and $m=1$; scalar inverses at zero are defined to be zero, including those in the formulas for $W_\pm$. The declaration itself imposes no positivity or trace condition on $H_m$.

QAlgorithms.Papers.Independent.Intakethird100.P09.inputOptimization

For a Boolean flag called “uniform,” a scenario $S$ with positive input and output counts $n_x,n_y,n_a,n_b$, and any function $f$ from real arrays on the input pairs to $[0,+\infty]$, this defines

$$\begin{cases} f\bigl((x,y)\mapsto1/(n_xn_y)\bigr),&\text{if the flag is true},\\[1mm] \displaystyle\sup_{\substack{p(x,y)\geq0\\\sum_{x,y}p(x,y)=1}}f(p),&\text{if the flag is false}. \end{cases}$$

The supremum is taken in the extended nonnegative reals and may be infinite. The function $f$ is unrestricted, and the optimized input laws may correlate Alice’s and Bob’s inputs and may assign zero weight to some input pairs.

QAlgorithms.Papers.Independent.Intakethird100.P09.measuredBox

For any Bell scenario $S=(n_x,n_y,n_a,n_b)$, finite types $I,J$ with decidable equality, measurement specification $M$, and arbitrary complex matrix $\rho$ indexed by $(I\times J)\times(I\times J)$, this defines the real array

$$P_{xyab}=\operatorname{Re}\operatorname{tr}\bigl((A_{x,a}\otimes B_{y,b})\rho\bigr).$$

Here, for every $x$, the matrices $A_{x,a}$ are positive semidefinite and sum over $a<n_a$ to $1_I$, and for every $y$, the matrices $B_{y,b}$ are positive semidefinite and sum over $b<n_b$ to $1_J$; the tensor product has entries $(A\otimes B)_{(i,j),(i',j')}=A_{ii'}B_{jj'}$. No density-matrix hypothesis is imposed on $\rho$, so the definition alone imposes no probability constraints on $P$. Either $I$ or $J$ may be empty, in which case every entry of $P$ is zero.

QAlgorithms.Papers.Independent.Intakethird100.P09.nearSep

For finite, nonempty types $I,J$ with decidable equality and any real number $\epsilon$, this defines the set of complex matrices $\rho$ on $I\times J$ that are positive semidefinite with trace $1$ and for which there exists a separable density matrix $\sigma$ satisfying

$$\|\rho^{T_J}-\sigma^{T_J}\|_1\leq\epsilon.$$

Here $(X^{T_J})_{(i,j),(i',j')}=X_{(i,j'),(i',j)}$, and $\|X\|_1=\operatorname{Re}\operatorname{tr}\sqrt{X^\dagger X}$, using continuous functional calculus. The required separability of $\sigma$ means that, for some natural number $q$, it has a decomposition $\sigma=\sum_{\ell=0}^{q-1}w_\ell A_\ell\otimes B_\ell$, where the real weights are nonnegative and sum to $1$, and every $A_\ell,B_\ell$ is positive semidefinite with trace $1$ on its respective subsystem. The definition allows every real $\epsilon$; for $\epsilon<0$, the set is empty.

QAlgorithms.Papers.Independent.Intakethird100.P09.paired

For arbitrary types $I,J,K,L$, and arbitrary complex matrices $\rho$ indexed by $I\times J$ and $\sigma$ indexed by $K\times L$, this defines a matrix on $(I\times K)\times(J\times L)$ by

$$\operatorname{paired}(\rho,\sigma)_{((i,k),(j,l)),((i',k'),(j',l'))} =\rho_{(i,j),(i',j')}\,\sigma_{(k,l),(k',l')}.$$

The types need not be finite or nonempty, and neither matrix is assumed positive, Hermitian, or normalized.

QAlgorithms.Papers.Independent.Intakethird100.P09.pptPrivate

For every positive integer $d$ and every complex $d\times d$ unitary matrix $U$, this defines a complex matrix on $(\{0,1\}\times\{0,\ldots,d-1\})^2$ as follows. Define matrices on the two $d$-dimensional shield indices by

$$X_{(s,t),(s',t')}= \begin{cases} U_{st}/(d\sqrt d),&(s',t')=(t,s),\\ 0,&\text{otherwise}, \end{cases} \qquad Y_{(s,t),(s',t')}= \begin{cases} U_{ss'}/d,&s=t,\ s'=t',\\ 0,&\text{otherwise}, \end{cases} \qquad p=\frac1{\sqrt d+1}.$$

On key indices ordered as $00,01,10,11$, form the shield-valued block matrix

$$Z=\frac12 \begin{pmatrix} (1-p)\sqrt{XX^\dagger}&0&0&(1-p)X\\ 0&p\sqrt{YY^\dagger}&0&0\\ 0&0&p\sqrt{Y^\dagger Y}&0\\ (1-p)X^\dagger&0&0&(1-p)\sqrt{X^\dagger X} \end{pmatrix},$$

where the square roots are obtained by applying the real square-root function through continuous functional calculus. The returned matrix $W$ regroups key and shield indices by party:

$$W_{((i,s),(j,t)),\,((i',s'),(j',t'))} = Z_{((i,j),(s,t)),\,((i',j'),(s',t'))}.$$

The definition includes $d=1$ and assumes only unitarity of $U$, without any additional restriction on its entry magnitudes.

QAlgorithms.Papers.Independent.Intakethird100.P09.quantumBound

For any Bell scenario $S=(n_x,n_y,n_a,n_b)$ and arbitrary real coefficient array $s_{xyab}$, the quantum bound is the real supremum, over all strictly positive integers $d_A,d_B$, all positive semidefinite complex matrices $\rho$ of trace $1$ on $\{0,\ldots,d_A-1\}\times\{0,\ldots,d_B-1\}$, and all measurement families $A_{x,a},B_{y,b}$, of

$$\sum_{x<n_x}\sum_{y<n_y}\sum_{a<n_a}\sum_{b<n_b} s_{xyab}\operatorname{Re}\operatorname{tr}\bigl((A_{x,a}\otimes B_{y,b})\rho\bigr).$$

Every measurement matrix is positive semidefinite, with $\sum_{a<n_a}A_{x,a}=1_{d_A}$ for every $x$ and $\sum_{b<n_b}B_{y,b}=1_{d_B}$ for every $y$; tensor-product entries are $A_{ii'}B_{jj'}$. The local dimensions have no upper bound.

QAlgorithms.Papers.Independent.Intakethird100.P09.rBlock

For natural numbers $a,b$, a Boolean flag, and any complex matrix $\rho$ on $\mathbb C^a\otimes\mathbb C^b$, this defines the extended nonnegative supremum over experiments described as follows: choose positive output dimensions $c,d$, positive finite input and outcome counts, a map $\Phi$ in the pointwise closure of finite adaptive local Kraus protocols, and positive semidefinite local measurement matrices $A_{x,\alpha},B_{y,\beta}$ summing to their respective identities for every input. Such finite protocols are generated recursively from identities by complete local Kraus instruments, outcome-dependent continuations, and summation over branches. Form the real array

$$P_E(x,y,\alpha,\beta) =\operatorname{Re}\operatorname{tr}\!\left[(A_{x,\alpha}\otimes B_{y,\beta})\Phi(\rho)\right].$$

The value assigned to this experiment is $\inf_Q\sum_{x,y}p(x,y)D((P_E)_{xy},Q_{xy})$ at $p(x,y)=1/(n_xn_y)$ when the flag is true, and the supremum of that infimum over all nonnegative input laws $p$ summing to $1$ when the flag is false. Here $Q(x,y,\alpha,\beta)=\sum_iw_i\mathbf1_{\{f_i(x)=\alpha,\ g_i(y)=\beta\}}$ ranges over finite deterministic local mixtures with nonnegative weights summing to $1$. The output divergence $D(u,v)$ is infinite if some coordinate has $u>0,v=0$, and otherwise is $\max\{0,\sum_{u_i\neq0}u_i\log(u_i/v_i)/\log2\}$. All arithmetic follows the total real conventions for division and logarithms, and $0\cdot(+\infty)=0$. No density-matrix assumption is imposed on $\rho$, and $a$ or $b$ may be zero.

QAlgorithms.Papers.Independent.Intakethird100.P09.sepDistance

For any finite nonempty types $I,J$, each equipped with decidable equality, and arbitrary complex matrix $\rho$ on $I\times J$, this defines the real infimum of $\operatorname{Re}\operatorname{tr}\sqrt{\Delta^\dagger\Delta}$, where $\Delta=T_2(\rho)-T_2(\sigma)$ and $[T_2(X)]_{(i,j),(i',j')}=X_{(i,j'),(i',j)}$. The infimum ranges over matrices $\sigma$ admitting a finite decomposition $\sigma=\sum_{\ell<q}w_\ell(A_\ell\otimes B_\ell)$, where $q\in\mathbb N$, every $w_\ell\ge0$, $\sum_{\ell<q}w_\ell=1$, and every $A_\ell$ and $B_\ell$ is a positive semidefinite complex matrix of trace $1$ on $I$ and $J$, respectively. The sum-of-weights condition excludes $q=0$, but individual weights may vanish. The square root is the positive matrix square root, and $\dagger$ denotes conjugate transpose. No density-matrix condition is imposed on $\rho$.

QAlgorithms.Papers.Independent.Intakethird100.P09.sepDistanceUntransposed

For finite, nonempty types $I,J$ with decidable equality and any complex matrix $\rho$ indexed by $I\times J$, this defines

$$\inf_{\sigma\text{ separable density matrix}}\|\rho^{T_J}-\sigma\|_1,$$

where $(\rho^{T_J})_{(i,j),(i',j')}=\rho_{(i,j'),(i',j)}$ and $\|X\|_1=\operatorname{Re}\operatorname{tr}\sqrt{X^\dagger X}$, with the square root defined by continuous functional calculus. Here a separable density matrix is any matrix expressible as $\sigma=\sum_{\ell=0}^{q-1}w_\ell A_\ell\otimes B_\ell$, for some natural number $q$, real weights $w_\ell\geq0$ summing to $1$, and positive semidefinite matrices $A_\ell,B_\ell$ on $I,J$, respectively, each of trace $1$. Thus $q=0$ cannot satisfy the weight condition, although individual weights may vanish. Only $\rho$ is partially transposed in the distance expression; no positivity or normalization assumption is imposed on $\rho$.

QAlgorithms.Papers.Independent.Intakethird100.P09.stateValue

For any Bell scenario $S=(n_x,n_y,n_a,n_b)$, finite types $I,J$ with decidable equality, arbitrary real coefficient array $s_{xyab}$, and arbitrary complex matrix $\rho$ on $I\times J$, the state value is the real supremum of

$$\sum_{x<n_x}\sum_{y<n_y}\sum_{a<n_a}\sum_{b<n_b} s_{xyab}\operatorname{Re}\operatorname{tr}\bigl((A_{x,a}\otimes B_{y,b})\rho\bigr)$$

over all families of positive semidefinite matrices satisfying $\sum_{a<n_a}A_{x,a}=1_I$ for every $x$ and $\sum_{b<n_b}B_{y,b}=1_J$ for every $y$. Tensor-product entries are $A_{ii'}B_{jj'}$. No positivity or trace-one condition is required of $\rho$. Either local index type may be empty, in which case all values in this supremum are zero.

QAlgorithms.Papers.Independent.Intakethird100.P09.successfulFilterBranch

For any natural numbers $a,b$, any complex matrix $\rho$ on the product of the local index sets of sizes $a,b$, and a local filter consisting of nonempty finite matrix families $A_i,B_j$ with $\sum_iA_i^\dagger A_i=I_a$, $\sum_jB_j^\dagger B_j=I_b$, and selected indices $i_*,j_*$, this defines

$$(A_{i_*}\otimes B_{j_*})\,\rho\,(A_{i_*}\otimes B_{j_*})^\dagger.$$

No division by the trace is performed. Neither positivity nor trace one is required of $\rho$, the resulting matrix may be zero, and zero local dimensions are included.

QAlgorithms.Papers.Independent.Intakethird100.P09.tBlock

For natural numbers $a,b$ with nonempty index sets $\{0,\ldots,a-1\}$ and $\{0,\ldots,b-1\}$, hence $a,b>0$, any Boolean flag, any complex matrix $\rho$ on $\mathbb C^a\otimes\mathbb C^b$, and any natural number $n$, this defines the extended nonnegative supremum over experiments on input dimensions $a^n,b^n$ of the following value:

$$\begin{cases} \displaystyle\inf_{\sigma\text{ separable density matrix on }\mathbb C^a\otimes\mathbb C^b} \sum_{x,y}\frac{D(P^{E,\rho,n}_{xy},P^{E,\sigma,n}_{xy})}{n_xn_y}, &\text{if the flag is true},\\[3mm] \displaystyle\sup_{\substack{p(x,y)\geq0\\\sum_{x,y}p(x,y)=1}} \inf_{\sigma\text{ separable density matrix on }\mathbb C^a\otimes\mathbb C^b} \sum_{x,y}p(x,y)D(P^{E,\rho,n}_{xy},P^{E,\sigma,n}_{xy}), &\text{if the flag is false}. \end{cases}$$

An experiment chooses positive output dimensions, positive finite input and outcome counts, a map $\Phi_E$ in the pointwise closure of finite adaptive local Kraus protocols, and local positive semidefinite measurements $A_{x,\alpha},B_{y,\beta}$ summing to their identities for each input; the finite protocols are recursively generated from identities by complete local Kraus instruments, outcome-dependent continuations, and summation of branches. Its arrays are $P^{E,\omega,n}(x,y,\alpha,\beta)=\operatorname{Re}\operatorname{tr}[(A_{x,\alpha}\otimes B_{y,\beta})\Phi_E(\omega^{[n]})]$, where $\omega^{[n]}$ has entries $\prod_{t=0}^{n-1}\omega_{(i_t,j_t),(i'_t,j'_t)}$, reindexed by the chosen bijections from length-$n$ index sequences to indices of sizes $a^n,b^n$. A separable density matrix $\sigma$ is a finite sum $\sum_\ell w_\ell R_\ell\otimes S_\ell$ with nonnegative real weights summing to $1$ and positive semidefinite trace-one factors. The divergence $D(u,v)$ is infinite if some coordinate has $u>0,v=0$; otherwise it is the nonnegative part of $\sum_{u_i\neq0}u_i\log(u_i/v_i)/\log2$, using total real division and logarithms and $0\cdot(+\infty)=0$. The infimum chooses a single-copy $\sigma$ and compares its $n$-fold power using the same experiment; it may depend on the input law. No density assumption is imposed on $\rho$. The case $n=0$ is included, with both matrix powers equal to the one-dimensional matrix $1$ because their entry products are empty.