Definitions for The statistical strength of nonlocality proofs

QAlgorithms.Papers.Independent.Intakethird100.P80.AdmissiblePrior

For every Bell experiment $S$ with a positive number $k$ of parties, positive setting counts $m_i$, and positive outcome counts $o_{ia}$, let $V=\prod_i\prod_{a=0}^{m_i-1}\{0,\ldots,o_{ia}-1\}$, let $\Delta_V$ be the probability simplex on $V$, and let $\nu$ be the $(|V|-1)$-dimensional Hausdorff measure on $\mathbb R^V$ restricted to $\Delta_V$. For every function $w:\mathbb R^V\to\mathbb R$, this predicate means that $w$ is continuous on $\Delta_V$, that there exists a real constant $c>0$ such that $c<w(p)$ for every $p\in\Delta_V$, and that

$$\int_{\mathbb R^V}w(p)\,d\nu(p)=1.$$

The lower bound is strict, and the continuity and lower-bound conditions impose no requirements outside the simplex.

QAlgorithms.Papers.Independent.Intakethird100.P80.BellExperiment

A Bell experiment consists of a natural number $n>0$ of parties; a natural number $m_i>0$ of settings for each party $i\in\{0,\ldots,n-1\}$; and a natural number $r_{i,a}>0$ of outcomes for each party $i$ and each setting $a\in\{0,\ldots,m_i-1\}$. The data include proofs of all these positivity conditions. Outcome counts may depend on both the party and its setting.

QAlgorithms.Papers.Independent.Intakethird100.P80.BellScenario

This denotes the Bell experiment with exactly two parties, indexed by $\{0,1\}$, exactly two settings at each party, indexed by $\{0,1\}$, and exactly two outcomes at every setting, indexed by $\{0,1\}$. All required positivity conditions are supplied by the inequality $0<2$.

QAlgorithms.Papers.Independent.Intakethird100.P80.BellTrial

For a Bell experiment $S$ with $k>0$ parties, positive setting counts $m_i$, and positive outcome counts $o_{i,a}$, this is the type of pairs $(a,x)$, where $a=(a_i)_i$ is a joint setting with $0\leq a_i<m_i$, and $x=(x_i)_i$ is a joint outcome satisfying $0\leq x_i<o_{i,a_i}$ for every party. The allowed outcome tuples therefore depend on the chosen setting tuple. This type is finite and nonempty.

QAlgorithms.Papers.Independent.Intakethird100.P80.DominatesQuantum

Let $S$ consist of a positive finite number of parties, positive finite setting counts, and positive finite outcome counts at every local setting. Let $\mathcal A$ be the joint settings, $\mathcal X_a$ the joint outcomes at $a$, and $\mathcal V$ the assignments specifying an outcome for every local setting of every party; write $v(a)$ for the outcome selected at $a$. For any real table $Q(a,x)$ and any real function $p:\mathcal V\to\mathbb R$, the domination predicate is

$$\forall a\in\mathcal A\ \forall x\in\mathcal X_a,\qquad Q(a,x)>0\ \Longrightarrow\ \sum_{\substack{v\in\mathcal V\\v(a)=x}}p(v)>0.$$

No positivity or normalization assumption is imposed on $Q$ or $p$, and no condition is imposed at entries where $Q(a,x)\le0$. In particular, the predicate holds vacuously if $Q$ has no positive entries.

QAlgorithms.Papers.Independent.Intakethird100.P80.Equalizer

Let $S$ specify a positive finite number of parties, positive finite setting counts, and positive finite outcome counts for each local setting. Write $\mathcal A$ for the joint settings, $\mathcal X_a$ for the joint outcomes at $a$, and $\mathcal V$ for potential assignments of outcomes to every local setting, with $v(a)$ the selected outcome. For any real table $Q(a,x)$ and any real function $p:\mathcal V\to\mathbb R$, the equalizer predicate means that there exists $K\in[0,\infty]$ such that

$$D_{Q,s}(p)=K \quad\text{for every }s:\mathcal A\to\mathbb R \text{ satisfying }s\ge0\text{ and }\sum_as(a)=1.$$

Here $L_p(a,x)=\sum_{v:\,v(a)=x}p(v)$. With $q(a,x)=s(a)Q(a,x)$ and $r(a,x)=s(a)L_p(a,x)$, the payoff is $+\infty$ if some trial has $q>0$ and $r=0$; otherwise it is the nonnegative truncation of $\sum_{a,x}q(a,x)\log_2(q(a,x)/r(a,x))$, with zero terms when $q=0$. Division by zero is zero, and the total real logarithm is zero at zero and uses absolute values at negative arguments. The constant $K$ may be infinite. No positivity or normalization is required of $Q$ or $p$, and the universal quantifier includes setting distributions with zero coordinates.

QAlgorithms.Papers.Independent.Intakethird100.P80.GHZScenario

This denotes the Bell experiment with exactly three parties, indexed by $\{0,1,2\}$, exactly two settings at each party, indexed by $\{0,1\}$, and exactly two outcomes at every setting, indexed by $\{0,1\}$. The required positivity conditions are supplied by $0<3$ for the party count and $0<2$ for the setting and outcome counts.

QAlgorithms.Papers.Independent.Intakethird100.P80.IsPrefixCode

For any type $Z$ and any assignment $C$ of a finite Boolean word to each element of $Z$, the predicate $\operatorname{IsPrefixCode}(C)$ means that, for every $x,y\in Z$, if $C(x)$ is a prefix of $C(y)$, including the possibility that the two words are equal, then $x=y$. This includes injectivity of $C$. The condition is vacuous when $Z$ is empty and permits the empty word when $Z$ has only one element.

QAlgorithms.Papers.Independent.Intakethird100.P80.MarginalConsistency

For any Bell experiment $S$ with $n>0$ parties, positive setting counts $m_i$, and positive outcome counts $r_{i,a}$, and any real table $Q(s,x)$ on its setting and outcome tuples, this property requires the following: for every subset $J\subseteq\{0,\ldots,n-1\}$, every two setting tuples $s,t$ satisfying $s_i=t_i$ for every $i\in J$, and every potential assignment $v$ specifying a permitted outcome $v_{i,a}$ for every party and setting,

$$\sum_{\substack{x:\ \forall i,\ 0\le x_i<r_{i,s_i}\\ \forall i\in J,\ x_i=v_{i,s_i}}}Q(s,x) = \sum_{\substack{x:\ \forall i,\ 0\le x_i<r_{i,t_i}\\ \forall i\in J,\ x_i=v_{i,t_i}}}Q(t,x).$$

The coordinate equalities compare the underlying integer outcome labels. The subset $J$ may be empty, in which case the requirement equates the total sums at any two settings. No nonnegativity or normalization of $Q$ is assumed by this property.

QAlgorithms.Papers.Independent.Intakethird100.P80.MeasuredOutcomes

For a Bell experiment $S$ with $n>0$ parties, positive setting counts $m_i$, and positive outcome counts $r_{i,a}$, and for any setting tuple $s$ with $0\le s_i<m_i$, the measured outcomes at $s$ are all functions $x$ assigning to every party $i$ an outcome $x_i\in\{0,\ldots,r_{i,s_i}-1\}$. The permitted outcome range for each party uses that party’s chosen setting.

QAlgorithms.Papers.Independent.Intakethird100.P80.MerminScenario

This denotes the Bell experiment with exactly two parties, indexed by $\{0,1\}$, exactly three settings at each party, indexed by $\{0,1,2\}$, and exactly two outcomes at every setting, indexed by $\{0,1\}$. The required positivity conditions are supplied by $0<2$ for the party and outcome counts and $0<3$ for the setting counts.

QAlgorithms.Papers.Independent.Intakethird100.P80.NonlocalityProof

For a Bell experiment $S$, let $k>0$ be its number of parties, let $m_i>0$ be the number of settings for party $i$, and let $o_{i,a}>0$ be its number of outcomes at setting $a$. Write $A=\prod_{i=0}^{k-1}\{0,\ldots,m_i-1\}$ for the joint settings, $X_a=\prod_i\{0,\ldots,o_{i,a_i}-1\}$ for the outcomes at $a\in A$, and $V=\prod_i\prod_{b=0}^{m_i-1}\{0,\ldots,o_{i,b}-1\}$ for the assignments of an outcome to every party and setting. For any real table $Q(a,x)$, the predicate holds precisely when all three following conditions hold: each $Q(a,x)$ is nonnegative and $\sum_{x\in X_a}Q(a,x)=1$; for every subset $J$ of parties, every $a,b\in A$ agreeing on $J$, and every $v\in V$, $\sum_{x\in X_a:\,\forall i\in J,\ x_i=v_{i,a_i}}Q(a,x)=\sum_{y\in X_b:\,\forall i\in J,\ y_i=v_{i,b_i}}Q(b,y)$; and there exist positive integers $d_i$, a positive semidefinite complex matrix $\rho$ indexed by $D=\prod_i\{0,\ldots,d_i-1\}$ with trace $1$, and positive semidefinite matrices $M_{i,b,c}$ of size $d_i\times d_i$, with $\sum_{c=0}^{o_{i,b}-1}M_{i,b,c}=I_{d_i}$ for every $i,b$, such that $Q(a,x)=\operatorname{Re}\operatorname{tr}(E_{a,x}\rho)$ for every $a,x$, where $E_{a,x}(u,w)=\prod_i M_{i,a_i,x_i}(u_i,w_i)$. The marginal condition includes the empty subset $J$.

QAlgorithms.Papers.Independent.Intakethird100.P80.NontrivialTable

For a Bell experiment $S$ with a positive finite number of parties, positive finite setting counts, and positive finite outcome counts for every local setting, and an arbitrary real table $Q$ assigning $Q(a,x)$ to each joint setting $a$ and each compatible joint outcome $x$, the nontrivial-table predicate means

$$\exists a\ \exists x,\qquad 0<Q(a,x)<1.$$

It requires only one entry strictly between zero and one; no positivity, normalization, or other restriction is imposed on the remaining entries.

QAlgorithms.Papers.Independent.Intakethird100.P80.PiLocal

For a Bell experiment $S$ with $n>0$ parties, positive setting counts $m_i$, and positive outcome counts $r_{i,a}$, $\Pi_{\mathrm{Local}}(S)$ is the standard real simplex on the finite set of potential assignments $v$, where $v_{i,a}\in\{0,\ldots,r_{i,a}-1\}$ for every party and setting. Its elements are real-valued functions $p$ on these assignments satisfying $p(v)\ge0$ for every $v$ and $\sum_vp(v)=1$; coordinates may be zero.

QAlgorithms.Papers.Independent.Intakethird100.P80.PositiveCoordinates

For any type $I$ and any function $p:I\to\mathbb R$, this predicate says that $p(i)>0$ for every $i\in I$. There is no finiteness or normalization assumption; when $I$ is empty, the condition holds vacuously.

QAlgorithms.Papers.Independent.Intakethird100.P80.Potential

For a Bell experiment $S$ with $n>0$ parties, positive setting counts $m_i$, and positive outcome counts $r_{i,a}$, a potential assignment is a function $v$ specifying an outcome $v_{i,a}\in\{0,\ldots,r_{i,a}-1\}$ for every party $i\in\{0,\ldots,n-1\}$ and every one of its settings $a\in\{0,\ldots,m_i-1\}$. It specifies outcomes for all settings simultaneously.

QAlgorithms.Papers.Independent.Intakethird100.P80.Proper

For a Bell experiment $S$ with $k>0$ parties, positive setting counts $m_i$, and positive outcome counts $o_{i,a}$, let $A=\prod_i\{0,\ldots,m_i-1\}$, $X_a=\prod_i\{0,\ldots,o_{i,a_i}-1\}$, and $V=\prod_i\prod_{b=0}^{m_i-1}\{0,\ldots,o_{i,b}-1\}$. For any real table $Q(a,x)$, this predicate says that there is no function $p:V\to\mathbb R$ satisfying $p(v)\geq0$ for every $v$, $\sum_{v\in V}p(v)=1$, and $Q(a,x)=\sum_{v\in V:\,\forall i,\ v_{i,a_i}=x_i}p(v)$ for every $a\in A$ and $x\in X_a$. No nonnegativity or normalization assumption on $Q$ is imposed.

QAlgorithms.Papers.Independent.Intakethird100.P80.QuantumGenerated

For any Bell experiment $S$ with $n>0$ parties, positive setting counts $m_i$, and positive outcome counts $r_{i,a}$, and any real table $Q(s,x)$ on its setting and outcome tuples, this property means that there exist natural numbers $d_i>0$, a complex matrix $\rho$ indexed by $\prod_i\{0,\ldots,d_i-1\}$, and complex $d_i\times d_i$ matrices $M_{i,a,b}$ for every party $i$, setting $a$, and permitted outcome $b$, such that $\rho$ is positive semidefinite with $\operatorname{tr}(\rho)=1$, every $M_{i,a,b}$ is positive semidefinite, and $\sum_{b=0}^{r_{i,a}-1}M_{i,a,b}=I_{d_i}$ for every $i,a$. For all setting tuples $s$ and permitted outcome tuples $x$, these data must satisfy

$$Q(s,x)=\operatorname{Re}\operatorname{tr}(E_{s,x}\rho), \qquad E_{s,x}(u,v)=\prod_{i=0}^{n-1}M_{i,s_i,x_i}(u_i,v_i).$$

QAlgorithms.Papers.Independent.Intakethird100.P80.Saddle

Let $S$ specify a positive finite number of parties, positive finite setting counts, and positive finite outcome counts at every local setting. Write $\mathcal A$ for the joint settings, $\mathcal X_a$ for their joint outcomes, and $\mathcal V$ for the assignments of outcomes to every local setting, with $v(a)$ the selected outcome. Given any real table $Q(a,x)$, set $A\subseteq(\mathcal A\to\mathbb R)$, and functions $s:\mathcal A\to\mathbb R$ and $p:\mathcal V\to\mathbb R$, the saddle predicate means precisely

$$s\in A,\qquad p\ge0,\qquad \sum_vp(v)=1,\qquad \forall t\in A,\ D_{Q,t}(p)\le D_{Q,s}(p),\qquad \forall r\ge0\text{ with }\sum_vr(v)=1,\ D_{Q,s}(p)\le D_{Q,s}(r).$$

For any setting function $u$ and potential function $h$, put $L_h(a,x)=\sum_{v:\,v(a)=x}h(v)$, $q(a,x)=u(a)Q(a,x)$, and $\ell(a,x)=u(a)L_h(a,x)$. The payoff $D_{Q,u}(h)$ is $+\infty$ if some trial satisfies $q>0$ and $\ell=0$; otherwise it is the nonnegative truncation of $\sum_{a,x}q(a,x)\log_2(q(a,x)/\ell(a,x))$, with each term defined as zero when $q(a,x)=0$. Division by zero is zero, and the total real logarithm is zero at zero and uses absolute values at negative arguments. The inequalities are non-strict inequalities of extended nonnegative reals, so infinite payoffs are allowed. No positivity or normalization is imposed on $Q$ or on members of $A$; if $A$ is empty, no pair satisfies this predicate.

QAlgorithms.Papers.Independent.Intakethird100.P80.Settings

For a Bell experiment $S$ with $n>0$ parties, positive setting counts $m_i$, and positive outcome counts $r_{i,a}$, its set of settings consists of all functions $s$ assigning to every party $i\in\{0,\ldots,n-1\}$ a setting $s_i\in\{0,\ldots,m_i-1\}$. Thus a setting is a simultaneous choice of one setting for each party.

QAlgorithms.Papers.Independent.Intakethird100.P80.Sigma

For a Bell experiment $S$ with $n>0$ parties, positive setting counts $m_i$, and positive outcome counts $r_{i,a}$, $\Sigma(S)$ is the standard real simplex on the finite set of setting tuples: its elements are functions $\sigma$ assigning a real number to every tuple $s$ with $0\le s_i<m_i$, subject to $\sigma(s)\ge0$ for every $s$ and $\sum_s\sigma(s)=1$. Coordinates may be zero.

QAlgorithms.Papers.Independent.Intakethird100.P80.Table

For a Bell experiment $S$ with $n>0$ parties, positive setting counts $m_i$, and positive outcome counts $r_{i,a}$, a table is an arbitrary real-valued function $Q(s,x)$, defined for every setting tuple $s$ satisfying $0\le s_i<m_i$ and every outcome tuple $x$ satisfying $0\le x_i<r_{i,s_i}$. This definition imposes no nonnegativity, normalization, or consistency conditions on those real values.

QAlgorithms.Papers.Independent.Intakethird100.P80.TwoByTwo

For a Bell experiment $S$, which already specifies a positive number of parties, a positive number of settings per party, and a positive number of outcomes at each local setting, the two-by-two predicate means that there are exactly two parties and that each party has exactly two settings:

$$S.\mathrm{parties}=2 \quad\text{and}\quad \forall i\in\{0,\ldots,S.\mathrm{parties}-1\},\ S.\mathrm{settings}(i)=2.$$

It places no further restriction on the positive outcome counts.

QAlgorithms.Papers.Independent.Intakethird100.P80.Uncorrelated

For a Bell experiment $S$ with $k>0$ parties, positive setting counts $m_i$, and positive outcome counts at every party and setting, let $A=\prod_i\{0,\ldots,m_i-1\}$. For an arbitrary function $s:A\to\mathbb R$, this predicate says that there exist functions $p_i:\{0,\ldots,m_i-1\}\to\mathbb R$, one for each party, such that $p_i(b)\geq0$, $\sum_{b=0}^{m_i-1}p_i(b)=1$, and $s(a)=\prod_i p_i(a_i)$ for every $a\in A$. The coordinates of the $p_i$, and consequently some coordinates of $s$, may be zero.

QAlgorithms.Papers.Independent.Intakethird100.P80.bayesMarginal

For every Bell experiment $S$ with a positive number $k$ of parties, positive setting counts $m_i$, and positive outcome counts $o_{ia}$, let $A=\prod_i\{0,\ldots,m_i-1\}$, $X_a=\prod_i\{0,\ldots,o_{i,a_i}-1\}$, $Z=\coprod_{a\in A}X_a$, and $V=\prod_i\prod_{a=0}^{m_i-1}\{0,\ldots,o_{ia}-1\}$. Let $\nu$ be the $(|V|-1)$-dimensional Hausdorff measure on $\mathbb R^V$ restricted to the probability simplex $\Delta_V$. For every real-valued function $s$ on $A$, every function $w:\mathbb R^V\to\mathbb R$, every natural number $n$, and every sample $z:\{0,\ldots,n-1\}\to Z$, write $z_j=(a_j,x_j)$ and define $L_p(a,x)=\sum_{v:\,\forall i,\ v_{i,a_i}=x_i}p(v)$. This definition assigns the real integral

$$B(s,w,z)=\int_{\mathbb R^V} w(p)\prod_{j=0}^{n-1}\bigl(s(a_j)L_p(a_j,x_j)\bigr)\,d\nu(p).$$

No probability, admissibility, or integrability hypotheses are supplied for $s$ or $w$; the integral uses the total real-valued integral convention, which returns $0$ for a nonintegrable integrand. For $n=0$, the product is $1$, so the value is $\int w\,d\nu$.

QAlgorithms.Papers.Independent.Intakethird100.P80.binaryScenario

For every pair of natural numbers $k,m$, together with hypotheses $0<k$ and $0<m$, this defines a Bell experiment with exactly $k$ parties indexed by $\{0,\ldots,k-1\}$, exactly $m$ settings at each party indexed by $\{0,\ldots,m-1\}$, and exactly two outcomes, indexed by $\{0,1\}$, at every setting of every party. The supplied inequalities establish positivity of the party and setting counts, and the outcome count is positive because it is $2$; the cases $k=0$ and $m=0$ cannot satisfy the required hypotheses.

QAlgorithms.Papers.Independent.Intakethird100.P80.chshLocal

For every assignment $v$ specifying a binary outcome $v_{i,a}\in\{0,1\}$ for each party $i\in\{0,1\}$ and each setting $a\in\{0,1\}$, this defines a real-valued function equal to $1/8$ when

$$(v_{0,0},v_{0,1},v_{1,0},v_{1,1}) \in \{(0,0,0,0),(1,1,1,1),(0,0,0,1),(1,1,1,0), (0,0,1,0),(1,1,0,1),(1,0,0,1),(0,1,1,0)\},$$

and equal to $0$ for every other assignment.

QAlgorithms.Papers.Independent.Intakethird100.P80.chshTable

For two parties indexed by $0,1$, each with settings and outcomes in $\{0,1\}$, this defines a real-valued table for every setting pair $s=(s_0,s_1)$ and outcome pair $x=(x_0,x_1)$ by

$$Q(s,x)= \begin{cases} \frac14+\frac{\sqrt2}{8}, & (x_0\ne x_1)\ \Longleftrightarrow\ (s_0=s_1=1),\\ \frac14-\frac{\sqrt2}{8},&\text{otherwise}. \end{cases}$$

QAlgorithms.Papers.Independent.Intakethird100.P80.correlatedStrength

Let $S$ specify $k>0$ parties, a positive number of settings for each party, and a positive number of outcomes for each local setting. Write $\mathcal A$ for the set of joint settings, $\mathcal X_a$ for the joint outcomes at setting $a$, and $\mathcal V$ for the potential assignments that specify an outcome for every local setting of every party; write $v(a)$ for the joint outcome selected from $v$ at $a$. For an arbitrary real table $Q(a,x)$, the correlated strength is

$$\sup_{\substack{s:\mathcal A\to\mathbb R\\s\ge0,\ \sum_a s(a)=1}} \ \inf_{\substack{p:\mathcal V\to\mathbb R\\p\ge0,\ \sum_v p(v)=1}} D_{Q,s}(p)\ \in[0,\infty].$$

Here $L_p(a,x)=\sum_{v:\,v(a)=x}p(v)$, and $D_{Q,s}(p)$ is $+\infty$ if some $(a,x)$ satisfies $s(a)Q(a,x)>0$ and $s(a)L_p(a,x)=0$; otherwise it is

$$\max\!\left\{0,\ \sum_{a\in\mathcal A}\sum_{x\in\mathcal X_a} \begin{cases} 0,&s(a)Q(a,x)=0,\\ s(a)Q(a,x)\log_2\!\dfrac{s(a)Q(a,x)}{s(a)L_p(a,x)},&s(a)Q(a,x)\ne0. \end{cases}\right\}.$$

The maximum is regarded as an extended nonnegative real. Division by zero is defined as zero, and the total real logarithm satisfies $\log_2 0=0$ and $\log_2 u=\log_2|u|$ for $u<0$. No positivity or normalization hypothesis is imposed on $Q$, and neither extremum is asserted to be attained.

QAlgorithms.Papers.Independent.Intakethird100.P80.ghzAllowed

For three parties indexed by $0,1,2$, each with settings and outcomes in $\{0,1\}$, this defines a predicate on every setting triple $s=(s_0,s_1,s_2)$ and outcome triple $x=(x_0,x_1,x_2)$ by

$$(x_0+x_1+x_2)\bmod 2= \begin{cases} 1,&s_0=s_1=s_2=0,\\ 0,&\text{otherwise}. \end{cases}$$

Thus the required outcome parity is odd at the all-zero setting and even at every other setting, including settings whose sum is $1$ or $3$.

QAlgorithms.Papers.Independent.Intakethird100.P80.ghzConstrained

For three parties indexed by $0,1,2$, each with settings in $\{0,1\}$, this defines a predicate on every setting triple $s=(s_0,s_1,s_2)$: it holds exactly when the ordinary integer sum $s_0+s_1+s_2$ equals $0$ or $2$.

QAlgorithms.Papers.Independent.Intakethird100.P80.ghzCorrelatedSettings

For three parties indexed by $0,1,2$, each with settings in $\{0,1\}$, this defines a real-valued function on every setting triple $s=(s_0,s_1,s_2)$, equal to $1/4$ when $s_0+s_1+s_2$ is $0$ or $2$, and equal to $0$ when that sum is $1$ or $3$.

QAlgorithms.Papers.Independent.Intakethird100.P80.ghzLocalTable

For three parties indexed by $0,1,2$, each with settings and outcomes in $\{0,1\}$, this defines a real-valued table for every setting triple $s$ and outcome triple $x$. Write $r=s_0+s_1+s_2$ and $e=(x_0+x_1+x_2)\bmod 2$; then

$$Q(s,x)= \begin{cases} \frac{3}{16},&(r=0\text{ and }e=1)\text{ or }(r=2\text{ and }e=0),\\ \frac{1}{16},&(r=0\text{ and }e=0)\text{ or }(r=2\text{ and }e=1),\\ \frac18,&r=1\text{ or }r=3. \end{cases}$$

QAlgorithms.Papers.Independent.Intakethird100.P80.ghzTable

For three parties indexed by $0,1,2$, each with settings and outcomes in $\{0,1\}$, this defines a real-valued table for every setting triple $s$ and outcome triple $x$. Write $r=s_0+s_1+s_2$ and $e=(x_0+x_1+x_2)\bmod 2$; then

$$Q(s,x)= \begin{cases} \frac14,&(r=0\text{ and }e=1)\text{ or }(r=2\text{ and }e=0),\\ 0,&(r=0\text{ and }e=0)\text{ or }(r=2\text{ and }e=1),\\ \frac18,&r=1\text{ or }r=3. \end{cases}$$

QAlgorithms.Papers.Independent.Intakethird100.P80.hardy

For two parties indexed by $0,1$, each with settings and outcomes in $\{0,1\}$, this defines a real-valued table for every setting pair $s=(s_0,s_1)$ and outcome pair $x=(x_0,x_1)$ as follows: let $\alpha=\sqrt{2+2\sqrt{-13+6\sqrt5}}/2$, $\beta=\sqrt{1-\alpha^2}$, and

$$(u_0,v_0)= \left(\sqrt{\frac{\beta}{\alpha+\beta}}, \sqrt{\frac{\alpha}{\alpha+\beta}}\right), \qquad (u_1,v_1)= \left(-\sqrt{\frac{\beta^3}{\alpha^3+\beta^3}}, \sqrt{\frac{\alpha^3}{\alpha^3+\beta^3}}\right).$$

For each $a\in\{0,1\}$, set $(w(a,1,0),w(a,1,1))=(u_a,v_a)$ and $(w(a,0,0),w(a,0,1))=(-v_a,u_a)$; then the table is

$$Q(s,x)= \left( \alpha\,w(s_0,x_0,0)w(s_1,x_1,0) -\beta\,w(s_0,x_0,1)w(s_1,x_1,1) \right)^2.$$

QAlgorithms.Papers.Independent.Intakethird100.P80.hardyAlpha

This defines the real number

$$\alpha=\frac{\sqrt{\,2+2\sqrt{-13+6\sqrt5}\,}}{2}.$$

QAlgorithms.Papers.Independent.Intakethird100.P80.hardyBeta

This defines the real number

$$\beta=\sqrt{1-\alpha^2}, \qquad \alpha=\frac{\sqrt{\,2+2\sqrt{-13+6\sqrt5}\,}}{2}.$$

QAlgorithms.Papers.Independent.Intakethird100.P80.hardyVector

For every $a,x,j\in\{0,1\}$, this defines a real number $w(a,x,j)$ as follows: put $\alpha=\sqrt{2+2\sqrt{-13+6\sqrt5}}/2$ and $\beta=\sqrt{1-\alpha^2}$, and define

$$u_a= \begin{cases} \sqrt{\frac{\beta}{\alpha+\beta}},&a=0,\\ -\sqrt{\frac{\beta^3}{\alpha^3+\beta^3}},&a=1, \end{cases} \qquad v_a= \begin{cases} \sqrt{\frac{\alpha}{\alpha+\beta}},&a=0,\\ \sqrt{\frac{\alpha^3}{\alpha^3+\beta^3}},&a=1. \end{cases}$$

The values for both coordinate indices are specified by

$$\bigl(w(a,1,0),w(a,1,1)\bigr)=(u_a,v_a), \qquad \bigl(w(a,0,0),w(a,0,1)\bigr)=(-v_a,u_a).$$

QAlgorithms.Papers.Independent.Intakethird100.P80.idealLength

For any type $Z$, any real-valued function $p:Z\to\mathbb R$, any natural number $n$, and any sample $z:\{0,\ldots,n-1\}\to Z$, put $L_p(z)=\prod_{i=0}^{n-1}p(z(i))$. The extended nonnegative real number called the ideal length is

$$\operatorname{idealLength}(p,z)= \begin{cases} +\infty,&L_p(z)=0,\\ \max\{0,-\log_2 L_p(z)\},&L_p(z)\ne0. \end{cases}$$

Here the finite value is embedded in $[0,+\infty]$, and the real logarithm is total: for a nonzero negative argument, $\log_2 t=\log|t|/\log2$. No nonnegativity or normalization assumption is imposed on $p$. When $n=0$, the empty product is $1$, so the ideal length is $0$.

QAlgorithms.Papers.Independent.Intakethird100.P80.iidLaw

For every Bell experiment $S$ with a positive number $k$ of parties, positive setting counts $m_i$, and positive outcome counts $o_{ia}$, let $A=\prod_i\{0,\ldots,m_i-1\}$, $X_a=\prod_i\{0,\ldots,o_{i,a_i}-1\}$, and $Z=\coprod_{a\in A}X_a$, with every subset of $Z$ measurable. For every real-valued table $Q(a,x)$ and real-valued setting function $s(a)$, this definition applies the countable infinite-product measure construction to the constant family of measures

$$\mu_{Q,s}=\sum_{(a,x)\in Z}\max\{s(a)Q(a,x),0\}\,\delta_{(a,x)},$$

producing a measure on sequences $Z^{\mathbb N}$. No hypothesis requires $Q$, $s$, or $\mu_{Q,s}$ to be normalized probability data.

QAlgorithms.Papers.Independent.Intakethird100.P80.initialTrials

For any type $Z$, any sequence $z:\mathbb N\to Z$, and any natural number $n$, $\operatorname{initialTrials}(z,n)$ is the function on $\{0,\ldots,n-1\}$ given by $i\mapsto z(i)$. Thus it records the first $n$ entries, starting at index $0$; when $n=0$, it is the unique function from the empty set to $Z$.

QAlgorithms.Papers.Independent.Intakethird100.P80.kl

For any finite type $I$, equipped with a finite enumeration, and arbitrary functions $q,p:I\to\mathbb R$, this definition gives a value in $[0,\infty]$: it is $+\infty$ if there exists $x\in I$ with $q(x)>0$ and $p(x)=0$; otherwise it is $\max\{0,\sum_{x\in I}h(x)\}$, regarded as a finite nonnegative extended real, where $h(x)=0$ when $q(x)=0$ and $h(x)=q(x)\log_2(q(x)/p(x))$ otherwise. Neither function is assumed nonnegative or normalized. Division by zero is defined to be zero, and the real logarithm used here satisfies $\log_2(0)=0$ and $\log_2(r)=\log|r|/\log2$ for $r\ne0$. Thus a zero denominator triggers infinity only when the corresponding numerator is strictly positive. For empty $I$, the value is zero.

QAlgorithms.Papers.Independent.Intakethird100.P80.localTable

For any Bell experiment $S$ with $n>0$ parties, positive setting counts $m_i$, and positive outcome counts $r_{i,a}$, and any real-valued function $p$ on all potential assignments $v$, where $0\le v_{i,a}<r_{i,a}$, the associated table is defined, for every setting tuple $s$ and permitted outcome tuple $x$, by

$$L_p(s,x)=\sum_{\substack{v\\ \forall i,\ v_{i,s_i}=x_i}}p(v).$$

The sum ranges over all potential assignments whose selected outcomes equal $x$ at every party. There is no assumption that $p$ is nonnegative or sums to one; an empty sum is zero.

QAlgorithms.Papers.Independent.Intakethird100.P80.localTensorEffect

For any Bell experiment $S$ with $n>0$ parties, positive setting counts $m_i$, and positive outcome counts $r_{i,a}$, choose arbitrary natural numbers $d_i$ and, for every party $i$, setting $a$, and outcome $b\in\{0,\ldots,r_{i,a}-1\}$, an arbitrary complex $d_i\times d_i$ matrix $M_{i,a,b}$. For any setting tuple $s$ and permitted outcome tuple $x$, define a complex matrix $E_{s,x}$ indexed by tuples $u,v\in\prod_i\{0,\ldots,d_i-1\}$ through

$$E_{s,x}(u,v)=\prod_{i=0}^{n-1}M_{i,s_i,x_i}(u_i,v_i).$$

No positivity, normalization, or positive-dimension assumptions are imposed. In particular, if any $d_i=0$, the matrix’s indexing set is empty.

QAlgorithms.Papers.Independent.Intakethird100.P80.lowerValue

For a Bell experiment $S$ with $k>0$ parties, positive setting counts $m_i$, and positive outcome counts $o_{i,a}$, let $A=\prod_i\{0,\ldots,m_i-1\}$, $X_a=\prod_i\{0,\ldots,o_{i,a_i}-1\}$, and $V=\prod_i\prod_{b=0}^{m_i-1}\{0,\ldots,o_{i,b}-1\}$. Given an arbitrary real table $Q(a,x)$ and arbitrary $s:A\to\mathbb R$, this definition is the infimum in $[0,\infty]$, over all $p:V\to\mathbb R$ with $p(v)\geq0$ and $\sum_vp(v)=1$, of the following quantity: set $q_{a,x}=s(a)Q(a,x)$ and $r_{a,x}=s(a)\sum_{v:\,\forall i,\ v_{i,a_i}=x_i}p(v)$; the quantity is $+\infty$ if some $q_{a,x}>0$ has $r_{a,x}=0$, and otherwise is $\max\{0,\sum_{a,x:\,q_{a,x}\ne0}q_{a,x}\log_2(q_{a,x}/r_{a,x})\}$. Division by zero is zero, $\log_2(0)=0$, and $\log_2(t)=\log|t|/\log2$ for $t\ne0$. Neither $Q$ nor $s$ is required to be nonnegative or normalized, and no attainment of the infimum is asserted.

QAlgorithms.Papers.Independent.Intakethird100.P80.mermin

For two parties indexed by $0,1$, each with settings in $\{0,1,2\}$ and outcomes in $\{0,1\}$, this defines a real-valued table for every setting pair $s=(s_0,s_1)$ and outcome pair $x=(x_0,x_1)$ by

$$Q(s,x)= \begin{cases} \frac12,&s_0=s_1\text{ and }x_0=x_1,\\ 0,&s_0=s_1\text{ and }x_0\ne x_1,\\ \frac18,&s_0\ne s_1\text{ and }x_0=x_1,\\ \frac38,&s_0\ne s_1\text{ and }x_0\ne x_1. \end{cases}$$

QAlgorithms.Papers.Independent.Intakethird100.P80.minimax

Let $S$ specify a positive finite number of parties, positive finite setting counts, and positive finite outcome counts for every local setting. Write $\mathcal A$ for the joint settings, $\mathcal X_a$ for the outcomes at $a$, and $\mathcal V$ for the assignments of outcomes to all local settings; let $v(a)$ be the outcome selected at $a$. For any real table $Q(a,x)$ and any set $A\subseteq(\mathcal A\to\mathbb R)$, define

$$\operatorname{minimax}(Q,A)= \inf_{\substack{p:\mathcal V\to\mathbb R\\p\ge0,\ \sum_vp(v)=1}} \ \sup_{s\in A}D_{Q,s}(p)\ \in[0,\infty].$$

Here $L_p(a,x)=\sum_{v:\,v(a)=x}p(v)$. With $q(a,x)=s(a)Q(a,x)$ and $r(a,x)=s(a)L_p(a,x)$, the payoff is $+\infty$ if some $(a,x)$ has $q(a,x)>0$ and $r(a,x)=0$; otherwise it equals

$$\max\!\left\{0,\ \sum_{a,x} \begin{cases} 0,&q(a,x)=0,\\ q(a,x)\log_2\!\bigl(q(a,x)/r(a,x)\bigr),&q(a,x)\ne0. \end{cases}\right\}.$$

Division by zero is zero, and the total real logarithm is zero at zero and equals the logarithm of the absolute value at negative arguments. No normalization or positivity is required of $Q$ or of the functions in $A$. The set $A$ may be empty, giving minimax value zero; attainment of the infimum or supremum is not part of the definition.

QAlgorithms.Papers.Independent.Intakethird100.P80.minimizers

Let $S$ consist of a positive finite number of parties, each with positive finite setting counts and positive finite outcome counts at each setting. Let $\mathcal A$ denote the joint settings, $\mathcal X_a$ their joint outcomes, and $\mathcal V$ the assignments of outcomes to all local settings, with $v(a)$ the selected outcome. For any real table $Q(a,x)$ and set $A\subseteq(\mathcal A\to\mathbb R)$, the minimizers are exactly the functions $p:\mathcal V\to\mathbb R$ satisfying

$$p\ge0,\qquad \sum_vp(v)=1,\qquad \sup_{s\in A}D_{Q,s}(p)= \inf_{\substack{r:\mathcal V\to\mathbb R\\r\ge0,\ \sum_vr(v)=1}} \sup_{s\in A}D_{Q,s}(r).$$

Here $L_h(a,x)=\sum_{v:\,v(a)=x}h(v)$; for $q(a,x)=s(a)Q(a,x)$ and $\ell(a,x)=s(a)L_h(a,x)$, $D_{Q,s}(h)$ is $+\infty$ if some trial has $q>0$ and $\ell=0$, and otherwise is the nonnegative truncation of $\sum_{a,x}q(a,x)\log_2(q(a,x)/\ell(a,x))$, with a zero term whenever $q(a,x)=0$. Division by zero is zero, and the total real logarithm is zero at zero and uses absolute values at negative arguments. Neither $Q$ nor the elements of $A$ are required to be probability distributions. The definition does not assert existence or uniqueness of a minimizer; equality at $+\infty$ is permitted. If $A$ is empty, every probability distribution on $\mathcal V$ belongs to this set.

QAlgorithms.Papers.Independent.Intakethird100.P80.optimizedBell

In the experiment with two parties, two settings per party, and two outcomes per setting, this defines a real table for every setting pair $(a,b)\in\{0,1\}^2$ and outcome pair $(x,y)\in\{0,1\}^2$ by

$$Q_{(a,b)}(x,y)= \begin{cases} \frac12,&(a,b)=(0,0)\ \text{and}\ x=y,\\ 0,&(a,b)=(0,0)\ \text{and}\ x\ne y,\\ \frac18,&(a,b)=(0,1)\ \text{and}\ x=y,\\ \frac38,&(a,b)=(0,1)\ \text{and}\ x\ne y,\\ \frac38,&a=1\ \text{and}\ x=y,\\ \frac18,&a=1\ \text{and}\ x\ne y. \end{cases}$$

Each listed value is assigned to each individual outcome pair satisfying its case.

QAlgorithms.Papers.Independent.Intakethird100.P80.optimizerScore

Let $S$ specify a positive finite number of parties, positive finite setting counts, and positive finite outcome counts at each local setting. Write $\mathcal A$ for the joint settings, $\mathcal X_a$ for their joint outcomes, and $\mathcal V$ for the assignments of outcomes to every local setting, with $v(a)\in\mathcal X_a$ the outcome selected from $v$. Given any real table $Q(a,x)$, any functions $s:\mathcal A\to\mathbb R$ and $p:\mathcal V\to\mathbb R$, and any $v\in\mathcal V$, the optimizer score is the real number

$$\sum_{a\in\mathcal A} \begin{cases} 0,&Q(a,v(a))=0,\\[2pt] \dfrac{s(a)Q(a,v(a))} {\displaystyle\sum_{\substack{u\in\mathcal V\\u(a)=v(a)}}p(u)}, &Q(a,v(a))\ne0. \end{cases}$$

No probability, normalization, positivity, domination, or optimality assumptions are imposed. The second branch also applies when $Q(a,v(a))<0$; if its denominator is zero, that quotient is defined to be zero.

QAlgorithms.Papers.Independent.Intakethird100.P80.originalBell

In the experiment with two parties, two settings per party, and two outcomes per setting, this defines a real table on every setting pair $(a,b)\in\{0,1\}^2$ and outcome pair $(x,y)\in\{0,1\}^2$ by setting $\alpha_0=0$, $\alpha_1=\pi/8$, $\beta_0=\pi/8$, and $\beta_1=\pi/4$, and then putting

$$Q_{(a,b)}(x,y)= \begin{cases} \dfrac{\cos^2(\alpha_a-\beta_b)}{2},&x=y,\\[2pt] \dfrac{\sin^2(\alpha_a-\beta_b)}{2},&x\ne y. \end{cases}$$

Each expression specifies the value for an individual outcome pair.

QAlgorithms.Papers.Independent.Intakethird100.P80.posteriorOdds

For every Bell experiment $S$ with a positive number $k$ of parties, positive setting counts $m_i$, and positive outcome counts $o_{ia}$, let $A=\prod_i\{0,\ldots,m_i-1\}$, $X_a=\prod_i\{0,\ldots,o_{i,a_i}-1\}$, $Z=\coprod_{a\in A}X_a$, and $V=\prod_i\prod_{a=0}^{m_i-1}\{0,\ldots,o_{ia}-1\}$. Let $\nu$ be the $(|V|-1)$-dimensional Hausdorff measure on $\mathbb R^V$ restricted to the probability simplex on $V$. For every real-valued table $Q(a,x)$, real-valued function $s$ on $A$, function $w:\mathbb R^V\to\mathbb R$, real number $h$, natural number $n$, and sample $z:\{0,\ldots,n-1\}\to Z$, write $z_j=(a_j,x_j)$ and set $L_p(a,x)=\sum_{v:\,\forall i,\ v_{i,a_i}=x_i}p(v)$. This definition assigns

$$\frac{h}{1-h}\, \frac{\displaystyle\prod_{j=0}^{n-1}s(a_j)Q(a_j,x_j)} {\displaystyle\int_{\mathbb R^V} w(p)\prod_{j=0}^{n-1}s(a_j)L_p(a_j,x_j)\,d\nu(p)}.$$

There are no normalization, admissibility, integrability, or positivity hypotheses, and no restriction $0<h<1$. The integral is the total real-valued integral, returning $0$ for a nonintegrable integrand, and real division by $0$ is defined to be $0$; consequently the displayed value is $0$ if $h=1$ or if its integral denominator is $0$. When $n=0$, both sample products are $1$.

QAlgorithms.Papers.Independent.Intakethird100.P80.potentialAtSetting

For any Bell experiment $S$ with $n>0$ parties, positive setting counts $m_i$, and positive outcome counts $r_{i,a}$, any potential assignment $v$ specifying $v_{i,a}\in\{0,\ldots,r_{i,a}-1\}$ for every party and setting, and any setting tuple $s$ with $0\le s_i<m_i$, the outcome tuple selected from $v$ at $s$ is defined to have coordinate $v_{i,s_i}$ at each party $i$. Its $i$th coordinate therefore belongs to $\{0,\ldots,r_{i,s_i}-1\}$.

QAlgorithms.Papers.Independent.Intakethird100.P80.productSettings

For a Bell experiment $S$ with $k>0$ parties, positive setting counts $m_i$, and positive outcome counts for every local setting, let $\mathcal A=\prod_{i=0}^{k-1}\{0,\ldots,m_i-1\}$. The set of product settings consists exactly of the functions $s:\mathcal A\to\mathbb R$ such that $s(a)\ge0$ for every $a$, $\sum_a s(a)=1$, and there exist functions $p_i:\{0,\ldots,m_i-1\}\to\mathbb R$, one for each party, satisfying $p_i(b)\ge0$, $\sum_b p_i(b)=1$, and

$$s(a)=\prod_{i=0}^{k-1}p_i(a_i)\qquad\text{for every }a\in\mathcal A.$$

Zero coordinates and deterministic choices of settings are included; strict positivity is not required.

QAlgorithms.Papers.Independent.Intakethird100.P80.rotatedBellPairLaw

For every pair of real numbers $A,B$ and every pair of outcome labels $x,y\in\{0,1\}$, this defines the real number

$$\operatorname{rotatedBellPairLaw}(A,B,x,y)= \begin{cases} \dfrac{\cos^2(A-B)}{2},&x=y,\\[2pt] \dfrac{\sin^2(A-B)}{2},&x\ne y. \end{cases}$$

The real parameters $A,B$ are unrestricted, and the displayed value applies to each individual outcome pair.

QAlgorithms.Papers.Independent.Intakethird100.P80.sampleLikelihood

For every type $Z$, every real-valued function $p:Z\to\mathbb R$, every natural number $n$, and every indexed sample $z:\{0,\ldots,n-1\}\to Z$, this definition assigns

$$\operatorname{sampleLikelihood}(p,z)=\prod_{i=0}^{n-1}p(z_i).$$

The function $p$ need not be nonnegative or normalized, $Z$ need not be finite, and repeated sample values are permitted. For $n=0$, the product is $1$, including when $Z$ is empty.

QAlgorithms.Papers.Independent.Intakethird100.P80.shannonBefore

For any type $Z$ equipped with a finite enumeration, any function $p:Z\to\mathbb R$, and any $x,y\in Z$, let $e:Z\to\{0,\ldots,|Z|-1\}$ be the bijection supplied by that finite enumeration. The predicate $\operatorname{shannonBefore}(p,x,y)$ means

$$p(y)<p(x)\quad\text{or}\quad \bigl(p(x)=p(y)\ \text{and}\ e(x)<e(y)\bigr).$$

Thus larger values come first, with ties resolved by the supplied enumeration. No sign or normalization condition is imposed on $p$.

QAlgorithms.Papers.Independent.Intakethird100.P80.shannonCumulative

For any type $Z$ equipped with a finite enumeration, any function $p:Z\to\mathbb R$, and any $x\in Z$, let $e:Z\to\{0,\ldots,|Z|-1\}$ be the bijection supplied by that enumeration. The cumulative value is the real number

$$\operatorname{shannonCumulative}(p,x) = \sum_{\substack{y\in Z\\ p(x)<p(y)\ \text{or}\ (p(y)=p(x)\ \text{and}\ e(y)<e(x))}} p(y).$$

No nonnegativity or normalization is required, so this sum is not required to lie in $[0,1]$.

QAlgorithms.Papers.Independent.Intakethird100.P80.shannonSize

For any type $Z$, any function $p:Z\to\mathbb R$, and any $x\in Z$, the natural number $\operatorname{shannonSize}(p,x)$ is

$$\left\lceil-\log_2(p(x))\right\rceil_+,$$

where $\lceil r\rceil_+$ is the least nonnegative integer greater than or equal to $r$, and is $0$ when $r\le0$. The logarithm is total, with $\log_2 0=0$ and $\log_2 t=\log|t|/\log2$ for $t\ne0$. In particular, the size is $0$ when $p(x)=0$ or $|p(x)|\ge1$; $p$ is not assumed to be a probability distribution.

QAlgorithms.Papers.Independent.Intakethird100.P80.shannonWord

For any type $Z$ equipped with a finite enumeration, any function $p:Z\to\mathbb R$, and any $x\in Z$, let $e:Z\to\{0,\ldots,|Z|-1\}$ be the supplied enumeration and define

$$\ell=\left\lceil-\log_2(p(x))\right\rceil_+,\qquad c=\sum_{\substack{y\in Z\\ p(x)<p(y)\ \text{or}\ (p(y)=p(x)\ \text{and}\ e(y)<e(x))}}p(y).$$

Then $\operatorname{shannonWord}(p,x)$ is the Boolean word of length $\ell$ whose entry at each index $j=0,\ldots,\ell-1$ is true exactly when

$$\left\lfloor 2^{j+1}c\right\rfloor_+\bmod2=1.$$

Here $\lfloor r\rfloor_+$ is the natural-number floor, equal to $0$ for negative $r$, and $\lceil r\rceil_+$ is the natural-number ceiling, equal to $0$ for $r\le0$. The total logarithm satisfies $\log_2 0=0$ and uses $\log|t|/\log2$ for nonzero $t$. No probability assumptions are imposed on $p$; when $\ell=0$, the word is empty.

QAlgorithms.Papers.Independent.Intakethird100.P80.simplexVolume

For every Bell experiment $S$ with a positive number $k$ of parties, positive setting counts $m_i$, and positive outcome counts $o_{ia}$, let

$$V=\prod_{i=0}^{k-1}\prod_{a=0}^{m_i-1}\{0,\ldots,o_{ia}-1\}$$

be the finite, nonempty space of assignments of an outcome to every party’s every setting, and let $\Delta_V=\{p:V\to\mathbb R:\ p(v)\ge0\text{ for all }v,\ \sum_vp(v)=1\}$. This definition assigns the measure on $\mathbb R^V$ obtained by restricting its $(|V|-1)$-dimensional Hausdorff measure to $\Delta_V$. The dimension is computed by natural-number subtraction before conversion to a real number; here $|V|\ge1$, so it equals the ordinary integer $|V|-1$, including dimension $0$ when $V$ is a singleton. No normalization of this restricted measure is included.

QAlgorithms.Papers.Independent.Intakethird100.P80.statisticalPayoff

For a Bell experiment $S$ with $k>0$ parties, positive setting counts $m_i$, and positive outcome counts $o_{i,a}$, put $A=\prod_i\{0,\ldots,m_i-1\}$, $X_a=\prod_i\{0,\ldots,o_{i,a_i}-1\}$, and $V=\prod_i\prod_{b=0}^{m_i-1}\{0,\ldots,o_{i,b}-1\}$. Given arbitrary real functions $Q(a,x)$, $s(a)$, and $p(v)$, define $L_p(a,x)=\sum_{v\in V:\,\forall i,\ v_{i,a_i}=x_i}p(v)$, $q_{a,x}=s(a)Q(a,x)$, and $r_{a,x}=s(a)L_p(a,x)$. The payoff is $+\infty$ if some trial satisfies $q_{a,x}>0$ and $r_{a,x}=0$; otherwise it is

$$\max\left\{0,\ \sum_{a\in A}\sum_{x\in X_a} \begin{cases} 0,&q_{a,x}=0,\\ q_{a,x}\log_2(q_{a,x}/r_{a,x}),&q_{a,x}\ne0 \end{cases}\right\},$$

regarded as a value in $[0,\infty]$. Here division by zero is zero, $\log_2(0)=0$, and $\log_2(t)=\log|t|/\log2$ for nonzero real $t$. None of $Q,s,p$ is assumed nonnegative or normalized.

QAlgorithms.Papers.Independent.Intakethird100.P80.trialJointLaw

For a Bell experiment $S$ with $k>0$ parties, positive setting counts $m_i$, and positive outcome counts $o_{i,a}$, let $A=\prod_i\{0,\ldots,m_i-1\}$ and $X_a=\prod_i\{0,\ldots,o_{i,a_i}-1\}$. Given any real table $Q(a,x)$ for $a\in A,x\in X_a$ and any function $s:A\to\mathbb R$, this definition assigns the trial $(a,x)$ the real value $s(a)Q(a,x)$. No nonnegativity or normalization condition is required of either $Q$ or $s$.

QAlgorithms.Papers.Independent.Intakethird100.P80.trialMeasurableSpace

For every Bell experiment $S$, consisting of a positive number $k$ of parties, a positive number $m_i$ of settings for each party $i$, and a positive number $o_{ia}$ of outcomes for every party $i$ and its setting $a$, let $A=\prod_{i=0}^{k-1}\{0,\ldots,m_i-1\}$ and $X_a=\prod_{i=0}^{k-1}\{0,\ldots,o_{i,a_i}-1\}$. A trial is a pair $(a,x)$ with $a\in A$ and $x\in X_a$. This instance equips the finite trial space $\coprod_{a\in A}X_a$ with the measurable structure in which every subset is measurable.

QAlgorithms.Papers.Independent.Intakethird100.P80.trialMeasure

For every Bell experiment $S$ with a positive number $k$ of parties, positive setting counts $m_i$, and positive outcome counts $o_{ia}$, write $A=\prod_i\{0,\ldots,m_i-1\}$, $X_a=\prod_i\{0,\ldots,o_{i,a_i}-1\}$, and $Z=\coprod_{a\in A}X_a$ for its trial space, equipped with all subsets measurable. For every real-valued table $Q(a,x)$ on $Z$ and every real-valued function $s$ on $A$, this definition assigns the measure

$$\mu_{Q,s}=\sum_{(a,x)\in Z}\max\{s(a)Q(a,x),0\}\,\delta_{(a,x)},$$

where each coefficient is regarded as an extended nonnegative real number and $\delta_{(a,x)}$ is the Dirac measure at that trial. There are no nonnegativity or normalization hypotheses on $Q$ or $s$; negative products receive weight $0$, and the resulting measure need not have total mass $1$.

QAlgorithms.Papers.Independent.Intakethird100.P80.uncorrelatedStrength

For a Bell experiment $S$ with $k>0$ parties, positive setting counts $m_i$, and positive outcome counts $o_{i,a}$, put $A=\prod_i\{0,\ldots,m_i-1\}$, $X_a=\prod_i\{0,\ldots,o_{i,a_i}-1\}$, and $V=\prod_i\prod_{b=0}^{m_i-1}\{0,\ldots,o_{i,b}-1\}$. For any real table $Q(a,x)$, this definition is the supremum in $[0,\infty]$, over all nonnegative $s:A\to\mathbb R$ with $\sum_as(a)=1$ that admit a representation $s(a)=\prod_i p_i(a_i)$ by nonnegative functions $p_i$ satisfying $\sum_{b=0}^{m_i-1}p_i(b)=1$, of the infimum over all nonnegative $p:V\to\mathbb R$ with $\sum_vp(v)=1$ of the following quantity: set $q_{a,x}=s(a)Q(a,x)$ and $r_{a,x}=s(a)\sum_{v:\,\forall i,\ v_{i,a_i}=x_i}p(v)$; the quantity is $+\infty$ if some $q_{a,x}>0$ has $r_{a,x}=0$, and otherwise is $\max\{0,\sum_{a,x:\,q_{a,x}\ne0}q_{a,x}\log_2(q_{a,x}/r_{a,x})\}$. Division by zero is zero, $\log_2(0)=0$, and $\log_2(t)=\log|t|/\log2$ for $t\ne0$. Zero setting probabilities are allowed and make all corresponding trial summands zero. No nonnegativity or normalization assumption on $Q$, or attainment of either extremum, is included.

QAlgorithms.Papers.Independent.Intakethird100.P80.uniformSettings

For a Bell experiment $S$ with $k>0$ parties, positive setting counts $m_i$, and positive outcome counts at each party and setting, let $A=\prod_{i=0}^{k-1}\{0,\ldots,m_i-1\}$. This function assigns every joint setting $a\in A$ the real value $\lvert A\rvert^{-1}=(\prod_i m_i)^{-1}$. The positivity assumptions ensure that $A$ is nonempty and this denominator is positive.

QAlgorithms.Papers.Independent.Intakethird100.P80.uniformStrength

For a Bell experiment $S$ with $k>0$ parties, positive setting counts $m_i$, and positive outcome counts $o_{i,a}$, let $A=\prod_i\{0,\ldots,m_i-1\}$, $X_a=\prod_i\{0,\ldots,o_{i,a_i}-1\}$, $V=\prod_i\prod_{b=0}^{m_i-1}\{0,\ldots,o_{i,b}-1\}$, and $u=\lvert A\rvert^{-1}>0$. For any real table $Q(a,x)$, this definition is the infimum in $[0,\infty]$, over all nonnegative functions $p:V\to\mathbb R$ with $\sum_vp(v)=1$, of the following quantity: set $q_{a,x}=uQ(a,x)$ and $r_{a,x}=u\sum_{v:\,\forall i,\ v_{i,a_i}=x_i}p(v)$; the quantity is $+\infty$ if some $q_{a,x}>0$ has $r_{a,x}=0$, and otherwise is $\max\{0,\sum_{a,x:\,q_{a,x}\ne0}q_{a,x}\log_2(q_{a,x}/r_{a,x})\}$. Division by zero is zero, $\log_2(0)=0$, and $\log_2(t)=\log|t|/\log2$ for $t\ne0$. No nonnegativity or normalization assumption on $Q$, or attainment of the infimum, is included.

QAlgorithms.Papers.Independent.Intakethird100.P80.upperValue

Let $S$ consist of a positive finite number of parties, each with a positive finite number of settings and a positive finite number of outcomes at each setting. Let $\mathcal A$ be the joint settings, $\mathcal X_a$ the joint outcomes at $a$, and $\mathcal V$ the assignments of outcomes to every local setting, with $v(a)$ denoting the outcome selected at $a$. Given any real table $Q(a,x)$, any set $A\subseteq(\mathcal A\to\mathbb R)$, and any function $p:\mathcal V\to\mathbb R$, the upper value is

$$\operatorname{upperValue}(Q,A,p)=\sup_{s\in A}D_{Q,s}(p)\ \in[0,\infty].$$

To define the payoff, put $L_p(a,x)=\sum_{v:\,v(a)=x}p(v)$, $q(a,x)=s(a)Q(a,x)$, and $r(a,x)=s(a)L_p(a,x)$. Then $D_{Q,s}(p)=+\infty$ if some trial has $q(a,x)>0$ and $r(a,x)=0$; otherwise

$$D_{Q,s}(p)=\max\!\left\{0,\ \sum_{a,x} \begin{cases} 0,&q(a,x)=0,\\ q(a,x)\log_2\!\bigl(q(a,x)/r(a,x)\bigr),&q(a,x)\ne0. \end{cases}\right\}.$$

Division by zero is zero; the total real logarithm has value zero at zero and uses the absolute value at negative arguments. There are no probability, positivity, or normalization assumptions on $Q$, $p$, or the members of $A$. In particular, $A$ may be empty, in which case the upper value is zero.