Definitions for Distilling Non-Locality

QAlgorithms.Papers.Independent.Intakethird100.P48.Behavior

A behavior is any function $P:\{\mathrm{false},\mathrm{true}\}^4\to\mathbb R$, assigning a real number $P(x,y,a,b)$ to each quadruple of Boolean values $x,y,a,b$. This abbreviation imposes no nonnegativity, normalization, or other conditions.

QAlgorithms.Papers.Independent.Intakethird100.P48.HilbertSystem

A Hilbert system consists of a carrier type $H$, a normed additive commutative group structure on $H$, a complex inner product space structure on $H$ compatible with that normed group structure, and the assumption that $H$ is complete for the resulting metric. No finite dimensionality, separability, or existence of a nonzero vector is required; the zero space is allowed.

QAlgorithms.Papers.Independent.Intakethird100.P48.IsBinaryNS

For any function $P:\{\mathrm{false},\mathrm{true}\}^4\to\mathbb R$, the predicate $\operatorname{IsBinaryNS}(P)$ means that all four conditions hold: $P(x,y,a,b)\geq0$ for every Boolean $x,y,a,b$; $\sum_{a,b}P(x,y,a,b)=1$ for every Boolean $x,y$; $\sum_bP(x,y,a,b)=\sum_bP(x,z,a,b)$ for every Boolean $x,y,z,a$; and $\sum_aP(x,y,a,b)=\sum_aP(z,y,a,b)$ for every Boolean $x,z,y,b$. Every sum runs over both Boolean values.

QAlgorithms.Papers.Independent.Intakethird100.P48.IsBinaryPOVM

Given a complete complex inner-product space $A$ and a pair $M_0,M_1:A\to A$ of continuous complex-linear operators, indexed by the two Boolean values $0=\mathrm{false}$ and $1=\mathrm{true}$, the predicate says that both operators are positive and that

$$M_0+M_1=I_A.$$

Neither operator is required to be a projection or to be nonzero. The definition also applies to the zero space, where the identity operator is zero.

QAlgorithms.Papers.Independent.Intakethird100.P48.IsQuantum

For any real-valued array $P(x,y,a,b)$ with Boolean indices $x,y,a,b$, the predicate says that there exist complete complex inner-product spaces $A,B$, a vector $\psi$ in the uniform-space completion $\widehat{A\otimes_{\mathbb C}B}$, and continuous complex-linear operators $M_{x,a}:A\to A$ and $N_{y,b}:B\to B$, for every Boolean choice of their indices, such that $\|\psi\|=1$, each $M_{x,a}$ and $N_{y,b}$ is positive, $M_{x,0}+M_{x,1}=I_A$ for every $x$, $N_{y,0}+N_{y,1}=I_B$ for every $y$, and

$$P(x,y,a,b)=\operatorname{Re}\left\langle\psi,\widehat{M_{x,a}\otimes N_{y,b}}\,\psi\right\rangle \quad\text{for all }x,y,a,b.$$

Here $0,1$ denote the two Boolean values, and the operator with a hat is the map on the completion obtained from the tensor-product map $u\otimes v\mapsto M_{x,a}u\otimes N_{y,b}v$. No finite-dimensionality restriction is imposed; the required unit vector excludes witnesses whose joint space is zero.

QAlgorithms.Papers.Independent.Intakethird100.P48.JointSpace

For any two systems $A$ and $B$, each consisting of a complex inner-product space with its compatible norm and an assumption of completeness, their joint space is the uniform-space completion of the complex algebraic tensor product of their underlying spaces:

$$\operatorname{JointSpace}(A,B)=\widehat{A\otimes_{\mathbb C}B}.$$

No finite-dimensionality assumption is imposed, and the individual spaces are allowed to be zero spaces.

QAlgorithms.Papers.Independent.Intakethird100.P48.ParallelProtocol

For every natural number $n$, including $0$, a parallel protocol consists of four functions: Alice’s input function $A_{\mathrm{in}}:\{\mathrm{false},\mathrm{true}\}\times I_n\to\{\mathrm{false},\mathrm{true}\}$, Bob’s input function $B_{\mathrm{in}}$ with the same domain and codomain, Alice’s output function $A_{\mathrm{out}}:\{\mathrm{false},\mathrm{true}\}\times\{\mathrm{false},\mathrm{true}\}^{I_n}\to\{\mathrm{false},\mathrm{true}\}$, and Bob’s output function $B_{\mathrm{out}}$ with that same domain and codomain, where $I_n=\{0,\ldots,n-1\}$ and $\{\mathrm{false},\mathrm{true}\}^{I_n}$ is the set of Boolean strings indexed by $I_n$. There are no further conditions on these functions. When $n=0$, $I_n$ is empty, there is exactly one such string, the input functions have empty domains, and each output function can still depend on its Boolean first argument.

QAlgorithms.Papers.Independent.Intakethird100.P48.StrictlyDistills

For any natural number $n$, any protocol $W$ consisting of Boolean input functions $A_{\mathrm{in}}(x,i),B_{\mathrm{in}}(y,i)$ and Boolean output functions $A_{\mathrm{out}}(x,u),B_{\mathrm{out}}(y,v)$ on length-$n$ Boolean strings, and any function $P:\{\mathrm{false},\mathrm{true}\}^4\to\mathbb R$, define $Q(x,y,a,b)$ to be the sum, over all length-$n$ Boolean strings $u,v$ satisfying $A_{\mathrm{out}}(x,u)=a$ and $B_{\mathrm{out}}(y,v)=b$, of $\prod_{i=0}^{n-1}P(A_{\mathrm{in}}(x,i),B_{\mathrm{in}}(y,i),u_i,v_i)$. For either function $R=P$ or $R=Q$, put $E_R(s,t)=R(s,t,\mathrm{false},\mathrm{false})+R(s,t,\mathrm{true},\mathrm{true})-R(s,t,\mathrm{false},\mathrm{true})-R(s,t,\mathrm{true},\mathrm{false})$ and

$$S(R)=\max_{s,t\in\{\mathrm{false},\mathrm{true}\}} \left|E_R(s,t)+E_R(s,\neg t)+E_R(\neg s,t)-E_R(\neg s,\neg t)\right|.$$

Then $W$ strictly distills $P$ means that, for each $R\in\{P,Q\}$, all its entries are nonnegative, $\sum_{a,b}R(x,y,a,b)=1$ for every $x,y$, $\sum_bR(x,y,a,b)=\sum_bR(x,z,a,b)$ for every $x,y,z,a$, and $\sum_aR(x,y,a,b)=\sum_aR(z,y,a,b)$ for every $x,z,y,b$, and additionally $2<S(P)$ and $S(P)<S(Q)$. All these variables and sums are Boolean. The definition permits $n=0$ syntactically; in that case the empty product makes $Q$ the deterministic law of the two output functions on the empty strings, for which $S(Q)=2$, so the strict inequalities cannot hold.

QAlgorithms.Papers.Independent.Intakethird100.P48.binaryCorrelator

For any function $P:\{\mathrm{false},\mathrm{true}\}^4\to\mathbb R$ and Boolean values $x,y$, its binary correlator at $(x,y)$ is the real number

$$P(x,y,\mathrm{false},\mathrm{false}) +P(x,y,\mathrm{true},\mathrm{true}) -P(x,y,\mathrm{false},\mathrm{true}) -P(x,y,\mathrm{true},\mathrm{false}).$$

No conditions on $P$ are required.

QAlgorithms.Papers.Independent.Intakethird100.P48.chshAt

For any function $P:\{\mathrm{false},\mathrm{true}\}^4\to\mathbb R$ and Boolean values $x,y$, define $E_P(s,t)=P(s,t,\mathrm{false},\mathrm{false})+P(s,t,\mathrm{true},\mathrm{true})-P(s,t,\mathrm{false},\mathrm{true})-P(s,t,\mathrm{true},\mathrm{false})$ for Boolean $s,t$. Then the real number $\operatorname{chshAt}(P,x,y)$ is

$$\left|E_P(x,y)+E_P(x,\neg y)+E_P(\neg x,y)-E_P(\neg x,\neg y)\right|,$$

where $\neg$ exchanges the two Boolean values. No conditions on $P$ are required.

QAlgorithms.Papers.Independent.Intakethird100.P48.epsilonBox

For every real number $\epsilon$, the epsilon box is the function $P_\epsilon:\{\mathrm{false},\mathrm{true}\}^4\to\mathbb R$ given by

$$P_\epsilon(x,y,a,b)= \begin{cases} (1-\epsilon)/2,&x=y=\mathrm{true}\ \text{and }a=b,\\ \epsilon/2,&x=y=\mathrm{true}\ \text{and }a\ne b,\\ 1/2,&\text{at least one of }x,y\text{ is false, and }a=b,\\ 0,&\text{at least one of }x,y\text{ is false, and }a\ne b. \end{cases}$$

All four arguments are Boolean, and no bounds on $\epsilon$ are imposed; values outside $[0,1]$ are included even though they produce negative entries.

QAlgorithms.Papers.Independent.Intakethird100.P48.nonlocality

For any function $P:\{\mathrm{false},\mathrm{true}\}^4\to\mathbb R$, put $E_P(s,t)=P(s,t,\mathrm{false},\mathrm{false})+P(s,t,\mathrm{true},\mathrm{true})-P(s,t,\mathrm{false},\mathrm{true})-P(s,t,\mathrm{true},\mathrm{false})$. Its nonlocality is the real number

$$\max_{x,y\in\{\mathrm{false},\mathrm{true}\}} \left|E_P(x,y)+E_P(x,\neg y)+E_P(\neg x,y)-E_P(\neg x,\neg y)\right|,$$

where $\neg$ exchanges the Boolean values. The maximum includes all four pairs $(x,y)$, and the definition imposes no nonnegativity or normalization conditions on $P$.

QAlgorithms.Papers.Independent.Intakethird100.P48.optimalDelta

The real constant named $\operatorname{optimalDelta}$ is defined by

$$\operatorname{optimalDelta}=\frac{1-\cos(\pi/8)}{2}.$$

QAlgorithms.Papers.Independent.Intakethird100.P48.optimalEpsilon

The real constant named $\operatorname{optimalEpsilon}$ is defined by

$$\operatorname{optimalEpsilon}=\frac{1-\cos(3\pi/8)}{2}.$$

QAlgorithms.Papers.Independent.Intakethird100.P48.parallelOutputLaw

For any natural number $n$, any protocol $W$ consisting of Boolean input functions $A_{\mathrm{in}}(x,i),B_{\mathrm{in}}(y,i)$ for $i\in I_n=\{0,\ldots,n-1\}$ and Boolean output functions $A_{\mathrm{out}}(x,u),B_{\mathrm{out}}(y,v)$ for Boolean strings $u,v$ on $I_n$, and any function $P:\{\mathrm{false},\mathrm{true}\}^4\to\mathbb R$, the parallel output law is the function $Q:\{\mathrm{false},\mathrm{true}\}^4\to\mathbb R$ defined, for all Boolean $x,y,a,b$, by

$$Q(x,y,a,b)= \sum_{u,v\in\{\mathrm{false},\mathrm{true}\}^{I_n}} \begin{cases} \displaystyle\prod_{i\in I_n} P\bigl(A_{\mathrm{in}}(x,i),B_{\mathrm{in}}(y,i),u_i,v_i\bigr), & A_{\mathrm{out}}(x,u)=a\ \text{and}\ B_{\mathrm{out}}(y,v)=b,\\ 0,&\text{otherwise}. \end{cases}$$

No conditions on $P$ or on the protocol functions are assumed. When $n=0$, each string sum has a single empty string and the product is $1$, so $Q(x,y,a,b)$ is $1$ exactly when the two output functions, evaluated on the empty strings, return $a$ and $b$, and is $0$ otherwise.

QAlgorithms.Papers.Independent.Intakethird100.P48.parameterBox

For any real numbers $\epsilon,\delta$, the parameter box is the function $P_{\epsilon,\delta}:\{\mathrm{false},\mathrm{true}\}^4\to\mathbb R$ defined by taking $r_{x,y}=\epsilon$ when both $x$ and $y$ are true, and $r_{x,y}=\delta$ otherwise, and setting

$$P_{\epsilon,\delta}(x,y,a,b)= \begin{cases} (1-r_{x,y})/2,&a=b,\\ r_{x,y}/2,&a\ne b. \end{cases}$$

All four arguments are Boolean. There are no bounds on $\epsilon$ or $\delta$, so the definition also includes parameter choices that give negative entries.

QAlgorithms.Papers.Independent.Intakethird100.P48.parityProtocol

For every natural number $n$, the parity protocol is the quadruple of functions on Boolean inputs and length-$n$ Boolean strings given by $A_{\mathrm{in}}(x,i)=x$, $B_{\mathrm{in}}(y,i)=y$, $A_{\mathrm{out}}(x,u)=\bigoplus_{i=0}^{n-1}u_i$, and $B_{\mathrm{out}}(y,v)=\bigoplus_{i=0}^{n-1}v_i$. Here $\bigoplus$ is Boolean addition, so the output is true exactly when the string has an odd number of true entries; this expands the Boolean sum of each entry conjoined with true used in the definition. Both output functions ignore their first argument. The definition includes $n=0$, when both outputs are false because the Boolean sum is empty.

QAlgorithms.Papers.Independent.Intakethird100.P48.quantumDistilledValues

This is the set of real numbers $z$ for which there exist $\epsilon,\delta\in[0,1]$ and a natural number $n>1$ satisfying the following conditions. Identify the Boolean values with $0,1$, and define the real array $P_{\epsilon,\delta}$ by $P_{\epsilon,\delta}(x,y,a,b)=(1-r_{xy})/2$ when $a=b$, and $r_{xy}/2$ otherwise, where $r_{11}=\epsilon$ and $r_{xy}=\delta$ at the other three input pairs. For any real Boolean-indexed array $S$, define

$$(T_nS)(x,y,a,b)= \sum_{\substack{u,v\in\{0,1\}^n\\ \bigoplus_i u_i=a,\ \bigoplus_i v_i=b}} \prod_{i=0}^{n-1}S(x,y,u_i,v_i), \qquad C_S(x,y)=S(x,y,0,0)+S(x,y,1,1)-S(x,y,0,1)-S(x,y,1,0),$$

and

$$L(S)=\max_{x,y\in\{0,1\}} \left|C_S(x,y)+C_S(x,1-y)+C_S(1-x,y)-C_S(1-x,1-y)\right|.$$

The symbol $\bigoplus$ means addition modulo two. Both $P_{\epsilon,\delta}$ and $T_nP_{\epsilon,\delta}$ must be nonnegative at every entry, have $\sum_{a,b}S(x,y,a,b)=1$ for every $x,y$, have $\sum_bS(x,y,a,b)$ independent of $y$ for each $x,a$, and have $\sum_aS(x,y,a,b)$ independent of $x$ for each $y,b$. They must satisfy $2<L(P_{\epsilon,\delta})<L(T_nP_{\epsilon,\delta})$, and $z=L(T_nP_{\epsilon,\delta})$. In addition, $P_{\epsilon,\delta}$ must admit complete complex inner-product spaces $A,B$, a unit vector $\psi\in\widehat{A\otimes_{\mathbb C}B}$, and positive continuous complex-linear operators $M_{x,a}$ and $N_{y,b}$, with $M_{x,0}+M_{x,1}=I_A$ and $N_{y,0}+N_{y,1}=I_B$, such that $P_{\epsilon,\delta}(x,y,a,b)=\operatorname{Re}\langle\psi,\widehat{M_{x,a}\otimes N_{y,b}}\psi\rangle$ at every Boolean quadruple; the hat denotes the induced map on the completion. The parameter endpoints $0$ and $1$ are permitted, but $n=0,1$ are excluded.