Definitions for Nonlocality free wirings and the distinguishability between Bell boxes

QAlgorithms.Papers.Independent.Intakethird100.P29.AfterAlice

For Bell scenarios $s,t$, whose input and output alphabets $X_s,Y_s,A_s,B_s$ and $X_t,Y_t,A_t,B_t$ are finite and nonempty, an object of this type has exactly one of two forms: a wait object containing functions $i_B:Y_t\to Y_s$, $o_A:X_t\to A_t$, and $o_B:Y_t\times B_s\to B_t$; or a measure object containing an element $y\in Y_s$ and functions $o_A:B_s\times X_t\to A_t$ and $o_B:B_s\times Y_t\to B_t$. These are arbitrary functions of the indicated arguments.

QAlgorithms.Papers.Independent.Intakethird100.P29.AfterBob

For Bell scenarios $s,t$, whose input and output alphabets $X_s,Y_s,A_s,B_s$ and $X_t,Y_t,A_t,B_t$ are finite and nonempty, an object of this type has exactly one of two forms: a wait object containing functions $i_A:X_t\to X_s$, $o_A:X_t\times A_s\to A_t$, and $o_B:Y_t\to B_t$; or a measure object containing an element $x\in X_s$ and functions $o_A:A_s\times X_t\to A_t$ and $o_B:A_s\times Y_t\to B_t$. These are arbitrary functions of the indicated arguments.

QAlgorithms.Papers.Independent.Intakethird100.P29.AgreeOnNS

For Bell scenarios $s,t$ with finite nonempty alphabets $X_s,Y_s,A_s,B_s$ and $X_t,Y_t,A_t,B_t$, let $w,v$ be any two functions taking real-valued arrays on $X_s\times Y_s\times A_s\times B_s$ to real-valued arrays on $X_t\times Y_t\times A_t\times B_t$. They agree on no-signaling correlations precisely when $w[p]=v[p]$ for every array $p$ such that $p(x,y,a,b)\ge0$ for all indices, $\sum_{a,b}p(x,y,a,b)=1$ for every $x,y$, and

$$\sum_{a\in A_s}p(x,y,a,b)=\sum_{a\in A_s}p(z,y,a,b) \quad(x,z\in X_s,\ y\in Y_s,\ b\in B_s),$$ $$\sum_{b\in B_s}p(x,y,a,b)=\sum_{b\in B_s}p(x,z,a,b) \quad(y,z\in Y_s,\ x\in X_s,\ a\in A_s).$$

The equality is equality of the entire output arrays, hence holds at every $\chi\in X_t,\psi\in Y_t,\alpha\in A_t,\beta\in B_t$; the property imposes no equality on other input arrays.

QAlgorithms.Papers.Independent.Intakethird100.P29.Behavior

For any Bell scenario $s$, a behavior is an arbitrary real-valued function $p:X_s\times Y_s\times A_s\times B_s\to\mathbb R$, written $p(x,y,a,b)$, where $X_s,Y_s,A_s,B_s$ are the integer index sets of sizes $s.n_X,s.n_Y,s.n_A,s.n_B$, respectively. This definition imposes no nonnegativity, normalization, or no-signaling conditions.

QAlgorithms.Papers.Independent.Intakethird100.P29.BellOperation

For any Bell scenarios $s,t$, a Bell operation is an arbitrary function from the real-valued functions on $X_s\times Y_s\times A_s\times B_s$ to the real-valued functions on $X_t\times Y_t\times A_t\times B_t$, where each index set consists of the integers starting at zero and has the corresponding positive size specified by its scenario. No linearity or preservation of nonnegativity, normalization, or no-signaling is required.

QAlgorithms.Papers.Independent.Intakethird100.P29.BellScenario.A

For any Bell scenario $s$, its set $A_s$ is the finite set $\{0,\ldots,s.n_A-1\}$. The positivity requirement on $s.n_A$ makes this set nonempty.

QAlgorithms.Papers.Independent.Intakethird100.P29.BellScenario.B

For any Bell scenario $s$, its set $B_s$ is the finite set $\{0,\ldots,s.n_B-1\}$. The positivity requirement on $s.n_B$ makes this set nonempty.

QAlgorithms.Papers.Independent.Intakethird100.P29.BellScenario.X

For any Bell scenario $s$, its set $X_s$ is the finite set $\{0,\ldots,s.n_X-1\}$. The positivity requirement on $s.n_X$ makes this set nonempty.

QAlgorithms.Papers.Independent.Intakethird100.P29.BellScenario.Y

For any Bell scenario $s$, its set $Y_s$ is the finite set $\{0,\ldots,s.n_Y-1\}$. The positivity requirement on $s.n_Y$ makes this set nonempty.

QAlgorithms.Papers.Independent.Intakethird100.P29.BellScenario

A Bell scenario consists of four natural numbers $n_X,n_Y,n_A,n_B$ together with the requirements $0<n_X$, $0<n_Y$, $0<n_A$, and $0<n_B$. Thus all four numbers are positive; zero is excluded.

QAlgorithms.Papers.Independent.Intakethird100.P29.GlobalWiring.apply

For any Bell scenarios $s,t$ and any global wiring $g$ with input distributions $I_g(\chi,\psi,x,y)$ on $X_s\times Y_s$ and output distributions $O_g(a,b,x,y,\chi,\psi,\alpha,\beta)$ on $A_t\times B_t$, each nonnegative and of total mass one for every fixed choice of its conditioning arguments, the associated operation sends every arbitrary real-valued behavior $p$ to

$$(gp)(\chi,\psi,\alpha,\beta)= \sum_{a\in A_s}\sum_{b\in B_s}\sum_{x\in X_s}\sum_{y\in Y_s} O_g(a,b,x,y,\chi,\psi,\alpha,\beta)\,p(x,y,a,b)\,I_g(\chi,\psi,x,y).$$

This holds for every $\chi\in X_t,\psi\in Y_t,\alpha\in A_t,\beta\in B_t$; all scenario index sets are the integer index sets of their specified positive sizes, and no conditions on $p$ are required.

QAlgorithms.Papers.Independent.Intakethird100.P29.GlobalWiring

For any Bell scenarios $s,t$, a global wiring consists of real-valued functions $I(\chi,\psi,x,y)$ and $O(a,b,x,y,\chi,\psi,\alpha,\beta)$, with $x\in X_s,y\in Y_s,a\in A_s,b\in B_s,\chi\in X_t,\psi\in Y_t,\alpha\in A_t,\beta\in B_t$, where each index set is the integer index set of its scenario’s specified positive size. For every $\chi,\psi$, all values of $I(\chi,\psi,\cdot,\cdot)$ are nonnegative and $\sum_{x,y}I(\chi,\psi,x,y)=1$; for every $a,b,x,y,\chi,\psi$, all values of $O(a,b,x,y,\chi,\psi,\cdot,\cdot)$ are nonnegative and $\sum_{\alpha,\beta}O(a,b,x,y,\chi,\psi,\alpha,\beta)=1$. No factorization or no-signaling requirement is imposed on these distributions.

QAlgorithms.Papers.Independent.Intakethird100.P29.HasGWRestriction

For Bell scenarios $s,t$ with finite nonempty alphabets $X_s,Y_s,A_s,B_s$ and $X_t,Y_t,A_t,B_t$, let $w$ be any function taking real-valued arrays on $X_s\times Y_s\times A_s\times B_s$ to real-valued arrays on $X_t\times Y_t\times A_t\times B_t$. It has the stated restriction precisely when there exist real-valued functions $I(\chi,\psi,x,y)$ and $O(a,b,x,y,\chi,\psi,\alpha,\beta)$ such that, for each $\chi,\psi$, $I$ is nonnegative and sums to $1$ over $(x,y)\in X_s\times Y_s$, and, for each $a,b,x,y,\chi,\psi$, $O$ is nonnegative and sums to $1$ over $(\alpha,\beta)\in A_t\times B_t$, with

$$w[p](\chi,\psi,\alpha,\beta) =\sum_{a\in A_s}\sum_{b\in B_s}\sum_{x\in X_s}\sum_{y\in Y_s} O(a,b,x,y,\chi,\psi,\alpha,\beta)\,p(x,y,a,b)\,I(\chi,\psi,x,y)$$

for every $\chi,\psi,\alpha,\beta$ and every nonnegative array $p$ normalized by $\sum_{a,b}p(x,y,a,b)=1$ for all $x,y$ and satisfying $\sum_a p(x,y,a,b)=\sum_a p(z,y,a,b)$ for all $x,z\in X_s,y\in Y_s,b\in B_s$, and $\sum_b p(x,y,a,b)=\sum_b p(x,z,a,b)$ for all $y,z\in Y_s,x\in X_s,a\in A_s$. This equality is required only for those arrays $p$.

QAlgorithms.Papers.Independent.Intakethird100.P29.HasLOSRRestriction

For Bell scenarios $s,t$, each consisting of four finite input and output alphabets of strictly positive sizes, and any map $w$ from real tables on $s$ to real tables on $t$, this property means that there is another such map $v$ with the following form. There are arbitrary types $L_1,L_2$, probability mass functions $\mu$ on $L_1$ and $\nu$ on $L_2$, and real kernels $A_{\mathrm{in}}(\ell,\chi,x)$, $B_{\mathrm{in}}(\ell,\psi,y)$, $A_{\mathrm{out}}(k,a,x,\chi,\alpha)$, and $B_{\mathrm{out}}(k,b,y,\psi,\beta)$, each nonnegative and summing to $1$ over its final argument for every choice of its other arguments, such that, for every real table $r$,

$$(vr)(\chi,\psi,\alpha,\beta)= \sum_{a,b,x,y} \left(\sum_{k\in L_1}\mu(k)A_{\mathrm{out}}(k,a,x,\chi,\alpha)B_{\mathrm{out}}(k,b,y,\psi,\beta)\right) r(x,y,a,b) \left(\sum_{\ell\in L_2}\nu(\ell)A_{\mathrm{in}}(\ell,\chi,x)B_{\mathrm{in}}(\ell,\psi,y)\right).$$

Here the potentially infinite sums are the specified sums of real values of the probability masses. Moreover, $wr=vr$ for every $r$ whose entries are nonnegative, whose output entries sum to $1$ at every input pair, and which satisfies

$$\sum_a r(x,y,a,b)=\sum_a r(x',y,a,b),\qquad \sum_b r(x,y,a,b)=\sum_b r(x,y',a,b)$$

for all indicated inputs and outputs. Agreement is required only on these tables, whereas the displayed formula for $v$ is required on every real table.

QAlgorithms.Papers.Independent.Intakethird100.P29.IsBellMonotone

For a function $f(s,p)\in[0,\infty]$, this predicate is the conjunction of the following two monotonicity requirements, each including finiteness on nonsignaling correlations and value zero on local correlations. Scenarios have four positive integer sizes and finite nonempty input and output index sets; a nonsignaling correlation is a nonnegative array $p_{xy}^{ab}$ with $\sum_{a,b}p_{xy}^{ab}=1$, $\sum_a p_{xy}^{ab}$ independent of $x$, and $\sum_b p_{xy}^{ab}$ independent of $y$. A local correlation additionally admits $p_{xy}^{ab}=\sum_\lambda\mu(\lambda)A_\lambda(a\mid x)B_\lambda(b\mid y)$, with a probability mass function $\mu$ on an arbitrary type and probability vectors $A_\lambda,B_\lambda$. The first requirement is $f(t,w(p))\le f(s,p)$ whenever $w$, on all real arrays, is obtained by a local input preprocessing mixture and a separate local output postprocessing mixture: the input distribution is $\sum_\eta\nu(\eta)I^A_\eta(x\mid\chi)I^B_\eta(y\mid\psi)$, the output distribution is $\sum_\lambda\mu(\lambda)O^A_\lambda(\alpha\mid a,x,\chi)O^B_\lambda(\beta\mid b,y,\psi)$, and their product with $p_{xy}^{ab}$ is summed over $a,b,x,y$; all kernels are probability vectors and both mixing types may be infinite. The second requirement is the same inequality whenever $w$ agrees on all nonsignaling correlations with a probability mixture of deterministic preparations of these forms: both source inputs depend separately on the target inputs and each final output depends on its own target input and source output; or Alice’s source input is fixed first and, depending on her outcome, Bob’s source input either depends on his target input while Alice’s final output ignores Bob’s outcome, or is fixed while both final outputs may depend on Bob’s outcome; or the symmetric construction with Bob first. In each latter construction final outputs depend only on the stated available outcomes and their own target inputs, and probabilities are obtained by summing source entries with the corresponding output indicators. No agreement away from nonsignaling correlations is required for the second class.

QAlgorithms.Papers.Independent.Intakethird100.P29.IsGW

For any Bell scenarios $s,t$ and any operation $w$ between their arbitrary real-valued behaviors, this predicate holds exactly when there exist nonnegative input distributions $I(\chi,\psi,\cdot,\cdot)$ of total mass one on $X_s\times Y_s$ and nonnegative output distributions $O(a,b,x,y,\chi,\psi,\cdot,\cdot)$ of total mass one on $A_t\times B_t$, for every choice of conditioning arguments, such that for every real-valued behavior $p$ and every target index $\chi,\psi,\alpha,\beta$,

$$(wp)(\chi,\psi,\alpha,\beta)= \sum_{a\in A_s}\sum_{b\in B_s}\sum_{x\in X_s}\sum_{y\in Y_s} O(a,b,x,y,\chi,\psi,\alpha,\beta)\,p(x,y,a,b)\,I(\chi,\psi,x,y).$$

Here each index set is the integer index set of the corresponding positive size in $s$ or $t$. The same distributions must give this identity for all $p$, including negative, unnormalized, and signaling functions.

QAlgorithms.Papers.Independent.Intakethird100.P29.IsLOSR

For any Bell scenarios $s,t$ and any operation $w$ between their spaces of arbitrary real-valued behaviors, this predicate holds exactly when there exist two types $L_1,L_2$, probability mass functions $\mu_1$ on $L_1$ and $\mu_2$ on $L_2$, and real-valued functions $u_A(\ell,\chi,x)$, $u_B(\ell,\psi,y)$, $v_A(j,a,x,\chi,\alpha)$, and $v_B(j,b,y,\psi,\beta)$ with the following properties: for every $\ell\in L_2$ and $\chi\in X_t$, $u_A(\ell,\chi,\cdot)$ is a nonnegative distribution of total mass one on $X_s$; for every $\ell,\psi\in Y_t$, $u_B(\ell,\psi,\cdot)$ is such a distribution on $Y_s$; for every $j\in L_1,a\in A_s,x\in X_s,\chi\in X_t$, $v_A(j,a,x,\chi,\cdot)$ is such a distribution on $A_t$; and for every $j,b\in B_s,y\in Y_s,\psi\in Y_t$, $v_B(j,b,y,\psi,\cdot)$ is such a distribution on $B_t$. For every real-valued behavior $p$ and every $\chi\in X_t,\psi\in Y_t,\alpha\in A_t,\beta\in B_t$, the required identity is

$$(wp)(\chi,\psi,\alpha,\beta)= \sum_{a\in A_s}\sum_{b\in B_s}\sum_{x\in X_s}\sum_{y\in Y_s} \left(\sum_{j\in L_1}\mu_1(j)v_A(j,a,x,\chi,\alpha)v_B(j,b,y,\psi,\beta)\right) p(x,y,a,b) \left(\sum_{\ell\in L_2}\mu_2(\ell)u_A(\ell,\chi,x)u_B(\ell,\psi,y)\right).$$

Here the scenario index sets are the integer index sets of their specified positive sizes, and $L_1,L_2$ need not be finite. The identity is required even for $p$ that is negative, unnormalized, or signaling.

QAlgorithms.Papers.Independent.Intakethird100.P29.IsLOSRMonotone

For a function $f(s,p)\in[0,\infty]$ on scenarios and real behavior arrays, this predicate requires finiteness on every nonnegative normalized array whose Alice marginal is independent of Bob’s input and whose Bob marginal is independent of Alice’s input, and value zero on every nonnegative normalized array of the form $p_{xy}^{ab}=\sum_\lambda\mu(\lambda)A_\lambda(a\mid x)B_\lambda(b\mid y)$, with a probability mass function $\mu$ on an arbitrary type and probability vectors $A_\lambda,B_\lambda$. It additionally requires $f(t,w(p))\le f(s,p)$ for every pair of scenarios $s,t$, every such nonsignaling correlation $p$ on $s$, and every operation $w$ with the following representation, valid on all real arrays $r$:

$$w(r)_{\chi\psi}^{\alpha\beta} =\sum_{a,b,x,y} \left(\sum_\lambda\mu(\lambda)O^A_\lambda(\alpha\mid a,x,\chi)O^B_\lambda(\beta\mid b,y,\psi)\right) r_{xy}^{ab} \left(\sum_\eta\nu(\eta)I^A_\eta(x\mid\chi)I^B_\eta(y\mid\psi)\right).$$

Here $x,y,a,b$ are source indices, $\chi,\psi,\alpha,\beta$ are target indices, $\mu,\nu$ are separate probability mass functions on arbitrary types, and every displayed $I$ or $O$ kernel is a nonnegative probability vector in its first argument for every choice of its conditioning arguments. All scenario index sets are finite and nonempty.

QAlgorithms.Papers.Independent.Intakethird100.P29.IsNS

For any Bell scenario $s$ and any real-valued behavior $p(x,y,a,b)$ on its finite, nonempty index sets $X_s,Y_s,A_s,B_s$ of sizes $s.n_X,s.n_Y,s.n_A,s.n_B$, respectively, the predicate holds exactly when $p(x,y,a,b)\ge0$ for every $x,y,a,b$, $\sum_{a\in A_s}\sum_{b\in B_s}p(x,y,a,b)=1$ for every $x,y$, and both marginal equalities hold:

$$\sum_{a\in A_s}p(x,y,a,b)=\sum_{a\in A_s}p(z,y,a,b) \quad\text{for every }x,z\in X_s,\ y\in Y_s,\ b\in B_s, \qquad \sum_{b\in B_s}p(x,y,a,b)=\sum_{b\in B_s}p(x,z,a,b) \quad\text{for every }y,z\in Y_s,\ x\in X_s,\ a\in A_s.$$

QAlgorithms.Papers.Independent.Intakethird100.P29.IsNormalizedQuantity

For a function $f$ assigning a value in $[0,\infty]$ to every scenario $s$ and every real array $p_{xy}^{ab}$ on that scenario, this predicate requires two properties. Each scenario has positive integer numbers of Alice inputs, Bob inputs, Alice outputs, and Bob outputs, so all four index sets are finite and nonempty. First, $f(s,p)<\infty$ whenever $p$ is nonnegative, $\sum_{a,b}p_{xy}^{ab}=1$ for every $x,y$, and its marginals satisfy $\sum_a p_{xy}^{ab}=\sum_a p_{x'y}^{ab}$ and $\sum_b p_{xy}^{ab}=\sum_b p_{xy'}^{ab}$ for all applicable indices. Second, $f(s,p)=0$ whenever $p$ is a nonnegative normalized correlation admitting a representation

$$p_{xy}^{ab}=\sum_{\lambda}\mu(\lambda)A_\lambda(a\mid x)B_\lambda(b\mid y),$$

where $\mu$ is a probability mass function on an arbitrary, potentially infinite type, and each $A_\lambda(\cdot\mid x)$ and $B_\lambda(\cdot\mid y)$ is a nonnegative real probability vector. The series uses the real values of the probability masses. No condition is imposed here on other arrays.

QAlgorithms.Papers.Independent.Intakethird100.P29.IsWPICC

For any Bell scenarios $s,t$ with positive finite input and output cardinalities, and any function $w$ from real arrays $p:s.X\times s.Y\times s.A\times s.B\to\mathbb R$ to real arrays on $t.X\times t.Y\times t.A\times t.B$, this predicate asserts the existence of a real $\lambda\in[0,1]$, an operation $v$, a type $L$, a probability mass function $\mu$ on $L$, and preparations $r_l$ for every $l\in L$, such that every $r_l$ is an Alice first or Bob first preparation, and, for every nonnegative array $p$ satisfying $\sum_{a,b}p(x,y,a,b)=1$, with $\sum_a p(x,y,a,b)$ independent of $x$ and $\sum_b p(x,y,a,b)$ independent of $y$, and every $\chi\in t.X,\psi\in t.Y,\alpha\in t.A,\beta\in t.B$, one has

$$(wp)(\chi,\psi,\alpha,\beta) =\lambda\sum_{l\in L}\mu(l)\,R_l(p;\chi,\psi,\alpha,\beta) +(1-\lambda)(vp)(\chi,\psi,\alpha,\beta).$$

Here the sum over $L$ is the possibly infinite sum of real terms, with each mass interpreted as a real number. Writing $\mathbf1[u=v]$ for the equality indicator, an Alice first preparation has a fixed $x_0\in s.X$ and contributes

$$R_l(p;\chi,\psi,\alpha,\beta) =\sum_{a,b}p(x_0,Y_a(\psi),a,b)\, \mathbf1[\alpha=A_a(b,\chi)]\, \mathbf1[\beta=B_a(b,\psi)].$$

For each $a$, its continuation independently chooses either the waiting form, in which $Y_a:t.Y\to s.Y$ is arbitrary, $A_a(b,\chi)$ depends only on $\chi$, and $B_a(b,\psi)$ is arbitrary, or the measuring form, in which $Y_a$ is constant and both output maps are arbitrary functions of the displayed arguments. A Bob first preparation instead has a fixed $y_0\in s.Y$ and contributes

$$R_l(p;\chi,\psi,\alpha,\beta) =\sum_{a,b}p(X_b(\chi),y_0,a,b)\, \mathbf1[\alpha=A_b(a,\chi)]\, \mathbf1[\beta=B_b(a,\psi)].$$

For each $b$, its waiting form permits arbitrary $X_b:t.X\to s.X$ and $A_b$, while $B_b(a,\psi)$ depends only on $\psi$; its measuring form makes $X_b$ constant and permits arbitrary displayed output maps. The required condition on $v$ means that there exist types $L_1,L_2$, probability mass functions $\nu_1,\nu_2$, input response laws $i_A(h,\chi,x)$, $i_B(h,\psi,y)$, and output response laws $o_A(k,a,x,\chi,\alpha)$, $o_B(k,b,y,\psi,\beta)$, each nonnegative and summing to $1$ over its final argument for every choice of its other arguments, such that for every real array $p$,

$$(vp)(\chi,\psi,\alpha,\beta) =\sum_{a,b,x,y} \left(\sum_{k\in L_1}\nu_1(k)o_A(k,a,x,\chi,\alpha)o_B(k,b,y,\psi,\beta)\right) p(x,y,a,b) \left(\sum_{h\in L_2}\nu_2(h)i_A(h,\chi,x)i_B(h,\psi,y)\right).$$

The representation of $w$ is required only on the normalized nonsignaling arrays specified above. Both endpoints $\lambda=0,1$ are allowed, but all existential witnesses and the requirement that every $r_l$ be prepared remain required, including for indices of zero $\mu$-mass.

QAlgorithms.Papers.Independent.Intakethird100.P29.IsWPICCMonotone

For a function $f(s,p)\in[0,\infty]$ on scenarios and real behavior arrays, this predicate requires finiteness on every nonnegative normalized correlation with each party’s marginal independent of the other party’s input, and value zero on every nonnegative normalized correlation $p_{xy}^{ab}=\sum_\lambda\mu(\lambda)A_\lambda(a\mid x)B_\lambda(b\mid y)$, where $\mu$ is a probability mass function on an arbitrary type and $A_\lambda,B_\lambda$ are probability vectors. It also requires $f(t,w(p))\le f(s,p)$ for every pair of scenarios, every nonsignaling correlation $p$ on the source, and every operation $w$ that, on every nonsignaling source correlation, agrees with a probability mixture of the following deterministic transformations. One may choose inputs $x=i_A(\chi)$, $y=i_B(\psi)$ and output $\alpha=o_A(\chi,a)$, $\beta=o_B(\psi,b)$; alternatively, one may fix Alice’s input $x$ and, separately for each resulting $a$, either choose Bob’s input as a function of $\psi$, Alice’s output as a function of $\chi$, and Bob’s output as a function of $\psi,b$, or fix Bob’s input and make both final outputs functions of $b$ and their respective target inputs. Symmetrically, one may fix Bob’s input $y$ and, separately for each resulting $b$, either choose Alice’s input as a function of $\chi$, Alice’s output as a function of $\chi,a$, and Bob’s output as a function of $\psi$, or fix Alice’s input and make both outputs functions of $a$ and their respective target inputs. A deterministic transformation sums $p_{xy}^{ab}$ over source outcomes with indicators enforcing these output functions; the mixture uses a probability mass function on an arbitrary type. The operation is unrestricted on arrays that are not nonsignaling correlations. All scenario index sets are finite and nonempty.

QAlgorithms.Papers.Independent.Intakethird100.P29.OptimaAttained

For a Bell scenario $s$, with finite nonempty input alphabets $X,Y$ and output alphabets $A,B$ of strictly positive sizes, and any real table $p$, define $\mathcal L$ to be the nonnegative output-normalized tables $q$ expressible as $q(x,y,a,b)=\sum_{\ell\in L}\mu(\ell)u(\ell,x,a)v(\ell,y,b)$, for some arbitrary type $L$, probability mass function $\mu$, and nonnegative response functions normalized over their respective outputs for every hidden value and input. Put $D_{x,y}(p,q)=+\infty$ if some output satisfies $p(x,y,a,b)>0=q(x,y,a,b)$, and otherwise put $D_{x,y}(p,q)=\max\{0,\sum_{a,b}p(x,y,a,b)\log(p(x,y,a,b)/q(x,y,a,b))\}$; total division by zero and $\log 0$ are $0$, and logarithms of negative nonzero arguments use absolute values. For any real input weights $d$, write

$$B(q)=\sup_{x,y}D_{x,y}(p,q),\qquad E_d(q)=\sum_{x,y}\max\{d(x,y),0\}D_{x,y}(p,q),\qquad L(d)=\inf_{q\in\mathcal L}E_d(q),$$

using extended nonnegative arithmetic, including $0\cdot(+\infty)=0$, and let $u(x,y)=1/(|X||Y|)$. A law means a nonnegative real function summing to $1$. This property asserts that all four numbers

$$S=\inf_{q\in\mathcal L}B(q),\qquad U=L(u),\qquad C=\sup_{d_X,d_Y\text{ laws}}L(d_Xd_Y),\qquad J=\sup_{d\text{ a joint law}}L(d)$$

are different from $+\infty$, where $(d_Xd_Y)(x,y)=d_X(x)d_Y(y)$; that some $q\in\mathcal L$ satisfies $B(q)=S$; that for every joint input law $d$, some $q\in\mathcal L$ satisfies $E_d(q)=L(d)$; that some $q\in\mathcal L$ satisfies $E_u(q)=U$; that some pair of input laws $d_X,d_Y$ satisfies $L(d_Xd_Y)=C$; and that some joint input law $d$ satisfies $L(d)=J$. These witnesses may differ, no uniqueness is required, and $p$ itself is not assumed to be a correlation or no-signaling table.

QAlgorithms.Papers.Independent.Intakethird100.P29.Preparation.apply

For Bell scenarios $s,t$ with finite nonempty alphabets $X_s,Y_s,A_s,B_s$ and $X_t,Y_t,A_t,B_t$, a preparation $r$ defines a transformation of every real-valued array $p:X_s\times Y_s\times A_s\times B_s\to\mathbb R$. At $(\chi,\psi,\alpha,\beta)\in X_t\times Y_t\times A_t\times B_t$, a none preparation with functions $i_A,i_B,o_A,o_B$ has value

$$\sum_{a\in A_s}\sum_{b\in B_s} p(i_A(\chi),i_B(\psi),a,b)\, \mathbf1_{\alpha=o_A(\chi,a)}\mathbf1_{\beta=o_B(\psi,b)}.$$

For an aliceFirst preparation with fixed $x\in X_s$ and continuation chosen separately for each $a\in A_s$, the value is $\sum_{a\in A_s}C_a$: a wait continuation with functions $i_B:Y_t\to Y_s,\ o_A:X_t\to A_t,\ o_B:Y_t\times B_s\to B_t$ gives $C_a=\sum_b p(x,i_B(\psi),a,b)\mathbf1_{\alpha=o_A(\chi)}\mathbf1_{\beta=o_B(\psi,b)}$, whereas a measure continuation with $y\in Y_s,\ o_A:B_s\times X_t\to A_t,\ o_B:B_s\times Y_t\to B_t$ gives $C_a=\sum_b p(x,y,a,b)\mathbf1_{\alpha=o_A(b,\chi)}\mathbf1_{\beta=o_B(b,\psi)}$. For a bobFirst preparation with fixed $y\in Y_s$ and continuation chosen separately for each $b\in B_s$, the value is $\sum_{b\in B_s}D_b$: a wait continuation with $i_A:X_t\to X_s,\ o_A:X_t\times A_s\to A_t,\ o_B:Y_t\to B_t$ gives $D_b=\sum_a p(i_A(\chi),y,a,b)\mathbf1_{\alpha=o_A(\chi,a)}\mathbf1_{\beta=o_B(\psi)}$, whereas a measure continuation with $x\in X_s,\ o_A:A_s\times X_t\to A_t,\ o_B:A_s\times Y_t\to B_t$ gives $D_b=\sum_a p(x,y,a,b)\mathbf1_{\alpha=o_A(a,\chi)}\mathbf1_{\beta=o_B(a,\psi)}$. All continuation data may depend on the outcome indexing that continuation; all indicators take values $1$ or $0$ according to their displayed equalities. No normalization or no-signaling assumption on $p$ is required.

QAlgorithms.Papers.Independent.Intakethird100.P29.Preparation.isPrepared

For any two Bell scenarios $s,t$, each having positive finite numbers of Alice inputs, Bob inputs, Alice outputs, and Bob outputs, and any preparation $r$ from $s$ to $t$, this predicate is false exactly when $r$ has the “none” constructor and true when it has either the “Alice first” or “Bob first” constructor. The “none” constructor contains deterministic input maps $t.X\to s.X$, $t.Y\to s.Y$ and deterministic output maps $t.X\times s.A\to t.A$, $t.Y\times s.B\to t.B$. An “Alice first” preparation contains a fixed input $x\in s.X$ and, for each outcome $a\in s.A$, a continuation of one of two forms: a waiting continuation with maps $t.Y\to s.Y$, $t.X\to t.A$, and $t.Y\times s.B\to t.B$; or a measuring continuation with a fixed $y\in s.Y$ and maps $s.B\times t.X\to t.A$, $s.B\times t.Y\to t.B$. A “Bob first” preparation contains a fixed input $y\in s.Y$ and, for each outcome $b\in s.B$, either a waiting continuation with maps $t.X\to s.X$, $t.X\times s.A\to t.A$, and $t.Y\to t.B$, or a measuring continuation with a fixed $x\in s.X$ and maps $s.A\times t.X\to t.A$, $s.A\times t.Y\to t.B$. The predicate imposes no further condition on any of these inputs, maps, or continuations.

QAlgorithms.Papers.Independent.Intakethird100.P29.Preparation

For Bell scenarios $s,t$ with finite nonempty alphabets $X_s,Y_s,A_s,B_s$ and $X_t,Y_t,A_t,B_t$, a preparation is one of three forms. A none preparation contains functions $i_A:X_t\to X_s$, $i_B:Y_t\to Y_s$, $o_A:X_t\times A_s\to A_t$, and $o_B:Y_t\times B_s\to B_t$. An aliceFirst preparation contains $x\in X_s$ and, for every $a\in A_s$, a choice between wait data $i_B:Y_t\to Y_s,\ o_A:X_t\to A_t,\ o_B:Y_t\times B_s\to B_t$, and measure data $y\in Y_s,\ o_A:B_s\times X_t\to A_t,\ o_B:B_s\times Y_t\to B_t$. A bobFirst preparation contains $y\in Y_s$ and, for every $b\in B_s$, a choice between wait data $i_A:X_t\to X_s,\ o_A:X_t\times A_s\to A_t,\ o_B:Y_t\to B_t$, and measure data $x\in X_s,\ o_A:A_s\times X_t\to A_t,\ o_B:A_s\times Y_t\to B_t$. All these functions and continuation choices are arbitrary.

QAlgorithms.Papers.Independent.Intakethird100.P29.Quantity

This abbreviates the type of functions that assign an extended nonnegative real number in $[0,+\infty]$ to every pair consisting of a Bell scenario $s$ and an arbitrary real table $p:X_s\times Y_s\times A_s\times B_s\to\mathbb R$. A scenario specifies four strictly positive natural alphabet sizes, with each alphabet the corresponding finite index set. Such a function is defined on all these tables, without any positivity, normalization, or no-signaling requirement on them, and its values may be $+\infty$.

QAlgorithms.Papers.Independent.Intakethird100.P29.UncorrelatedWiring.apply

For Bell scenarios $s,t$ with finite nonempty input and output alphabets $X_s,Y_s,A_s,B_s$ and $X_t,Y_t,A_t,B_t$, let $u$ consist of conditional probability distributions $I_A(\chi,x)$, $I_B(\psi,y)$, $O_A(a,x,\chi,\alpha)$, and $O_B(b,y,\psi,\beta)$, normalized over $x,y,\alpha,\beta$, respectively, for each fixed choice of their other arguments. Its application is the function taking any real-valued array $p:X_s\times Y_s\times A_s\times B_s\to\mathbb R$ to the array defined, for every $\chi\in X_t,\psi\in Y_t,\alpha\in A_t,\beta\in B_t$, by

$$(u[p])(\chi,\psi,\alpha,\beta) =\sum_{a\in A_s}\sum_{b\in B_s}\sum_{x\in X_s}\sum_{y\in Y_s} O_A(a,x,\chi,\alpha)\,O_B(b,y,\psi,\beta)\, p(x,y,a,b)\,I_A(\chi,x)\,I_B(\psi,y).$$

There is no hypothesis here that $p$ is nonnegative, normalized, or no-signaling.

QAlgorithms.Papers.Independent.Intakethird100.P29.UncorrelatedWiring

For Bell scenarios $s,t$, let $X_s,Y_s,A_s,B_s$ and $X_t,Y_t,A_t,B_t$ denote their respective input and output alphabets, each a finite set of indices of a specified positive natural size. An uncorrelated wiring from $s$ to $t$ consists of four real-valued functions

$$I_A:X_t\times X_s\to\mathbb R,\quad I_B:Y_t\times Y_s\to\mathbb R,\quad O_A:A_s\times X_s\times X_t\times A_t\to\mathbb R,\quad O_B:B_s\times Y_s\times Y_t\times B_t\to\mathbb R.$$

For every $\chi\in X_t$, the numbers $I_A(\chi,x)$, indexed by $x\in X_s$, are nonnegative and sum to $1$; for every $\psi\in Y_t$, the numbers $I_B(\psi,y)$, indexed by $y\in Y_s$, are nonnegative and sum to $1$. For every $a\in A_s,x\in X_s,\chi\in X_t$, the numbers $O_A(a,x,\chi,\alpha)$, indexed by $\alpha\in A_t$, are nonnegative and sum to $1$; for every $b\in B_s,y\in Y_s,\psi\in Y_t$, the numbers $O_B(b,y,\psi,\beta)$, indexed by $\beta\in B_t$, are nonnegative and sum to $1$.

QAlgorithms.Papers.Independent.Intakethird100.P29.aliceContribution

For Bell scenarios $s,t$ with finite nonempty alphabets $X_s,Y_s,A_s,B_s$ and $X_t,Y_t,A_t,B_t$, take any real-valued array $p:X_s\times Y_s\times A_s\times B_s\to\mathbb R$, elements $x\in X_s,a\in A_s,\chi\in X_t,\psi\in Y_t,\alpha\in A_t,\beta\in B_t$, and continuation data $c$ of either of the following forms. If $c$ contains wait functions $i_B:Y_t\to Y_s,\ o_A:X_t\to A_t,\ o_B:Y_t\times B_s\to B_t$, its Alice contribution is

$$\sum_{b\in B_s}p(x,i_B(\psi),a,b)\, \mathbf1_{\alpha=o_A(\chi)}\mathbf1_{\beta=o_B(\psi,b)}.$$

If $c$ contains measure data $y\in Y_s,\ o_A:B_s\times X_t\to A_t,\ o_B:B_s\times Y_t\to B_t$, the contribution is

$$\sum_{b\in B_s}p(x,y,a,b)\, \mathbf1_{\alpha=o_A(b,\chi)}\mathbf1_{\beta=o_B(b,\psi)}.$$

Here every indicator is $1$ when its displayed equality holds and $0$ otherwise. No probability or no-signaling hypothesis is imposed on $p$.

QAlgorithms.Papers.Independent.Intakethird100.P29.bobContribution

For Bell scenarios $s,t$ with finite nonempty alphabets $X_s,Y_s,A_s,B_s$ and $X_t,Y_t,A_t,B_t$, take any real-valued array $p:X_s\times Y_s\times A_s\times B_s\to\mathbb R$, elements $y\in Y_s,b\in B_s,\chi\in X_t,\psi\in Y_t,\alpha\in A_t,\beta\in B_t$, and continuation data $c$ of either of the following forms. If $c$ contains wait functions $i_A:X_t\to X_s,\ o_A:X_t\times A_s\to A_t,\ o_B:Y_t\to B_t$, its Bob contribution is

$$\sum_{a\in A_s}p(i_A(\chi),y,a,b)\, \mathbf1_{\alpha=o_A(\chi,a)}\mathbf1_{\beta=o_B(\psi)}.$$

If $c$ contains measure data $x\in X_s,\ o_A:A_s\times X_t\to A_t,\ o_B:A_s\times Y_t\to B_t$, the contribution is

$$\sum_{a\in A_s}p(x,y,a,b)\, \mathbf1_{\alpha=o_A(a,\chi)}\mathbf1_{\beta=o_B(a,\psi)}.$$

Here every indicator is $1$ when its displayed equality holds and $0$ otherwise. No probability or no-signaling hypothesis is imposed on $p$.

QAlgorithms.Papers.Independent.Intakethird100.P29.classicalRelativeEntropyNats

For any finite type $I$, equipped with its finite enumeration, and arbitrary real functions $p,q:I\to\mathbb R$, this definition gives the extended nonnegative real value

$$\begin{cases} +\infty,&\text{if some }i\in I\text{ satisfies }p_i>0\text{ and }q_i=0,\\[2mm] \displaystyle\max\!\left\{0,\sum_{i\in I}p_i\log\!\left(\frac{p_i}{q_i}\right)\right\},&\text{otherwise}. \end{cases}$$

Neither function is assumed nonnegative or normalized. Division and logarithms are total: division by zero is $0$, $\log 0=0$, and for a negative nonzero argument the real logarithm is the logarithm of its absolute value. Thus a zero denominator forces infinity precisely when the corresponding numerator is strictly positive; otherwise the finite branch uses these total operations. If $I$ is empty, the value is $0$.

QAlgorithms.Papers.Independent.Intakethird100.P29.correlatedInputStrength

For a Bell scenario $s$, with finite nonempty input alphabets $X,Y$ and output alphabets $A,B$ of strictly positive sizes, and any real table $p$, this definition gives

$$\sup_{\substack{d:X\times Y\to\mathbb R,\ d\ge0\\ \sum_{x,y}d(x,y)=1}} \ \inf_{q\in\mathcal L} \sum_{x,y}d(x,y)\,D_{x,y}(p,q) \quad\text{in }[0,+\infty].$$

Here $\mathcal L$ consists of nonnegative tables normalized over outputs at every input pair that have a representation $q(x,y,a,b)=\sum_{\ell\in L}\mu(\ell)u(\ell,x,a)v(\ell,y,b)$, where $L$ is arbitrary, $\mu$ is a probability mass function, and the response functions are nonnegative and normalized over their respective outputs for every hidden value and input. The value $D_{x,y}(p,q)$ is $+\infty$ if some $a,b$ satisfy $p(x,y,a,b)>0=q(x,y,a,b)$, and otherwise equals $\max\{0,\sum_{a,b}p(x,y,a,b)\log(p(x,y,a,b)/q(x,y,a,b))\}$. No positivity, normalization, or no-signaling assumption is imposed on $p$. The joint law $d$ may have zeros, with $0\cdot(+\infty)=0$; division by zero and $\log 0$ equal $0$, and logarithms of negative nonzero arguments use their absolute values. No optimizer is asserted.

QAlgorithms.Papers.Independent.Intakethird100.P29.localDistance

For a Bell scenario $s$, with finite nonempty input alphabets $X,Y$ and output alphabets $A,B$ of strictly positive sizes, any real table $p$, and any real input weight function $d:X\times Y\to\mathbb R$, this definition gives

$$\inf_{q\in\mathcal L}\ \sum_{x,y}\max\{d(x,y),0\}\,D_{x,y}(p,q)\quad\text{in }[0,+\infty].$$

The set $\mathcal L$ consists of nonnegative tables normalized over $a,b$ at every $x,y$ that admit a representation $q(x,y,a,b)=\sum_{\ell\in L}\mu(\ell)u(\ell,x,a)v(\ell,y,b)$, where $L$ is an arbitrary type, $\mu$ is a probability mass function, and each response function is nonnegative and sums to $1$ over its output for every hidden value and input. Here $D_{x,y}(p,q)=+\infty$ if some $a,b$ satisfy $p(x,y,a,b)>0=q(x,y,a,b)$, and otherwise

$$D_{x,y}(p,q)=\max\!\left\{0,\sum_{a,b}p(x,y,a,b)\log\!\frac{p(x,y,a,b)}{q(x,y,a,b)}\right\}.$$

No distribution or no-signaling assumption is imposed on $p$, and $d$ need not be nonnegative or normalized. Negative weights become zero, $0\cdot(+\infty)=0$, division by zero and $\log 0$ are $0$, and logarithms of negative nonzero arguments use their absolute values. No minimizer is asserted.

QAlgorithms.Papers.Independent.Intakethird100.P29.responseIndicator

For any type $I$ and any $i,j\in I$, the response indicator is the real number

$$\mathbf 1_{i=j}= \begin{cases} 1,&i=j,\\ 0,&i\ne j. \end{cases}$$

No finiteness or nonemptiness assumption on $I$ is required; supplying $i,j$ already supplies elements of $I$.

QAlgorithms.Papers.Independent.Intakethird100.P29.sb

For a Bell scenario $s$, with finite nonempty input alphabets $X,Y$ and output alphabets $A,B$ of its specified strictly positive sizes, and arbitrary real tables $p,q:X\times Y\times A\times B\to\mathbb R$, this definition gives

$$\sup_{x\in X}\sup_{y\in Y} \begin{cases} +\infty,&\exists(a,b)\in A\times B:\ p(x,y,a,b)>0,\ q(x,y,a,b)=0,\\[1mm] \displaystyle\max\!\left\{0,\sum_{a,b}p(x,y,a,b) \log\!\frac{p(x,y,a,b)}{q(x,y,a,b)}\right\},&\text{otherwise}. \end{cases}$$

The supremum is taken in $[0,+\infty]$; because the input alphabets are finite and nonempty, it is a maximum. No nonnegativity, normalization, or no-signaling condition is imposed on either table. The finite branch uses total real division and logarithms, with division by zero and $\log 0$ equal to $0$, and the logarithm of a negative nonzero number equal to the logarithm of its absolute value.

QAlgorithms.Papers.Independent.Intakethird100.P29.singletonScenario

This declaration specifies the scenario with exactly one Alice input, one Bob input, one Alice output, and one Bob output. Each index set is the singleton $\{0\}$, and all four required positivity conditions hold.

QAlgorithms.Papers.Independent.Intakethird100.P29.snl

For a Bell scenario $s$, with finite nonempty input alphabets $X,Y$ and output alphabets $A,B$ of strictly positive sizes, and any real table $p$, this definition gives

$$\inf_{q\in\mathcal L}\ \sup_{x,y}D_{x,y}(p,q)\quad\text{in }[0,+\infty].$$

Here $\mathcal L$ consists of nonnegative tables $q$ with $\sum_{a,b}q(x,y,a,b)=1$ for every $x,y$, for which there exist an arbitrary type $L$, a probability mass function $\mu$ on $L$, and response functions $u(\ell,x,a)$, $v(\ell,y,b)$, nonnegative and summing to $1$ over $a$ and $b$, respectively, for every $\ell,x,y$, such that

$$q(x,y,a,b)=\sum_{\ell\in L}\mu(\ell)u(\ell,x,a)v(\ell,y,b).$$

The value $D_{x,y}(p,q)$ is $+\infty$ if some $a,b$ have $p(x,y,a,b)>0=q(x,y,a,b)$, and otherwise is $\max\{0,\sum_{a,b}p(x,y,a,b)\log(p(x,y,a,b)/q(x,y,a,b))\}$. The table $p$ is unrestricted. Real division by zero and $\log 0$ equal $0$, and logarithms of negative nonzero arguments use their absolute values. The definition is an infimum and does not assert attainment.

QAlgorithms.Papers.Independent.Intakethird100.P29.su

For a Bell scenario $s$, with finite nonempty input alphabets $X,Y$ of sizes $n_X,n_Y$ and finite nonempty output alphabets $A,B$, and any real table $p$, this definition gives

$$\inf_{q\in\mathcal L}\ \sum_{x,y}\frac{1}{n_Xn_Y}\,D_{x,y}(p,q)\quad\text{in }[0,+\infty].$$

Here $\mathcal L$ consists of nonnegative tables normalized over outputs at every input pair and expressible as $q(x,y,a,b)=\sum_{\ell\in L}\mu(\ell)u(\ell,x,a)v(\ell,y,b)$, with an arbitrary hidden type $L$, a probability mass function $\mu$, and nonnegative response functions normalized over their respective outputs for every hidden value and input. The value $D_{x,y}(p,q)$ is $+\infty$ if some $a,b$ satisfy $p(x,y,a,b)>0=q(x,y,a,b)$, and otherwise equals $\max\{0,\sum_{a,b}p(x,y,a,b)\log(p(x,y,a,b)/q(x,y,a,b))\}$. The table $p$ has no normalization, positivity, or no-signaling requirement. Division by zero and $\log 0$ are $0$, logarithms of negative nonzero arguments use their absolute values, and no attainment of the infimum is asserted.

QAlgorithms.Papers.Independent.Intakethird100.P29.suc

For a Bell scenario $s$, with finite nonempty input alphabets $X,Y$ and output alphabets $A,B$ of strictly positive sizes, and any real table $p$, this definition gives

$$\sup_{\substack{d_X:X\to\mathbb R,\ d_X\ge0\\ \sum_xd_X(x)=1}} \ \sup_{\substack{d_Y:Y\to\mathbb R,\ d_Y\ge0\\ \sum_yd_Y(y)=1}} \ \inf_{q\in\mathcal L} \sum_{x,y}d_X(x)d_Y(y)\,D_{x,y}(p,q) \quad\text{in }[0,+\infty].$$

Here $\mathcal L$ consists of nonnegative output-normalized tables admitting $q(x,y,a,b)=\sum_{\ell\in L}\mu(\ell)u(\ell,x,a)v(\ell,y,b)$, with arbitrary hidden type $L$, probability mass function $\mu$, and nonnegative output-normalized response functions for every hidden value and input. The value $D_{x,y}(p,q)$ is $+\infty$ when some output has $p(x,y,a,b)>0=q(x,y,a,b)$, and otherwise equals $\max\{0,\sum_{a,b}p(x,y,a,b)\log(p(x,y,a,b)/q(x,y,a,b))\}$. The table $p$ is unrestricted, input laws may have zero probabilities, and $0\cdot(+\infty)=0$. Total real division and logarithms give division by zero and $\log 0$ the value $0$, and use absolute values for negative nonzero logarithm arguments. No supremum or infimum is asserted to be attained.

QAlgorithms.Papers.Independent.Intakethird100.P29.uniformInput

For any Bell scenario $s$, whose input sizes $n_X,n_Y$ and output sizes $n_A,n_B$ are natural numbers with all four strictly positive, this definition is the real function on input pairs $x\in\{0,\ldots,n_X-1\}$, $y\in\{0,\ldots,n_Y-1\}$ given by

$$(x,y)\longmapsto\frac{1}{n_Xn_Y}.$$

It is constant on all input pairs, and its denominator is positive by the scenario assumptions.

QAlgorithms.Papers.Independent.Intakethird100.P29.weightedEntropy

For a Bell scenario $s$, with finite nonempty input alphabets $X,Y$ and output alphabets $A,B$ of strictly positive sizes, arbitrary real tables $p,q$, and any real function $d:X\times Y\to\mathbb R$, this definition gives

$$\sum_{x,y}\max\{d(x,y),0\}\,D_{x,y}(p,q)\quad\text{in }[0,+\infty],$$

where $D_{x,y}(p,q)=+\infty$ if some output pair satisfies $p(x,y,a,b)>0$ and $q(x,y,a,b)=0$, and otherwise

$$D_{x,y}(p,q)= \max\!\left\{0,\sum_{a,b}p(x,y,a,b)\log\!\frac{p(x,y,a,b)}{q(x,y,a,b)}\right\}.$$

There are no distribution or no-signaling assumptions on $p,q,d$. Negative weights become zero, and extended nonnegative multiplication satisfies $0\cdot(+\infty)=0$. In the finite branch, division by zero and $\log 0$ are $0$, and the real logarithm at a negative nonzero argument is the logarithm of its absolute value.