Definitions for Steering, Entanglement, Nonlocality, and the EPR Paradox

QAlgorithms.Papers.Independent.Intakethird100.P62.BellNonlocal

For any natural numbers $a,b$ and any complex matrix $W$ on $\mathbb C^a\otimes\mathbb C^b$, being Bell nonlocal means that there do not exist a measurable space $\Omega$, a probability measure $\mu$, and real responses $r(A,x,w),s(B,y,w)$ for all projective resolutions $A=(P_x^A)_{x<a}$, $B=(Q_y^B)_{y<b}$, such that each response is measurable in $w$, all responses are pointwise nonnegative, their outcome sums are pointwise $1$, and, for every $A,B,x,y$, the product $r(A,x,\cdot)s(B,y,\cdot)$ is integrable and

$$\operatorname{Re}\operatorname{tr}[(P_x^A\otimes Q_y^B)W] =\int_\Omega r(A,x,w)s(B,y,w)\,d\mu(w).$$

A projective resolution in dimension $d$ consists of $d$ Hermitian idempotent $d\times d$ matrices with pairwise zero products for distinct indices and sum $I_d$; individual projectors may be zero. No density condition on $W$ is required. The definition includes zero dimensions, where such responses cannot be normalized on a probability space, so the negated existence condition holds.

QAlgorithms.Papers.Independent.Intakethird100.P62.HasBellLocal

For any natural numbers $a,b$ and any complex matrix $W$ on $\mathbb C^a\otimes\mathbb C^b$, this condition means that there exist a type $\Omega$ with a measurable-space structure, a probability measure $\mu$, and real response functions $r(A,x,w)$ and $s(B,y,w)$ defined for every projective resolution $A=(P_x^A)_{x<a}$ and $B=(Q_y^B)_{y<b}$, respectively. For every fixed resolution and outcome, the response as a function of $w$ must be measurable; all responses must be pointwise nonnegative, and $\sum_{x<a}r(A,x,w)=1$ and $\sum_{y<b}s(B,y,w)=1$ for every resolution and every $w$. For all $A,B,x,y$, the product response must be integrable and satisfy

$$\operatorname{Re}\operatorname{tr}[(P_x^A\otimes Q_y^B)W] =\int_\Omega r(A,x,w)s(B,y,w)\,d\mu(w).$$

Each projective resolution in dimension $d$ consists of $d$ Hermitian idempotent $d\times d$ matrices with distinct members multiplying to zero and sum $I_d$; zero projectors are allowed. No hidden matrix ensemble, measurability in the resolution, or density assumption on $W$ is required. If either $a=0$ or $b=0$, the condition is impossible: the empty projective resolution exists, but its response normalization cannot hold on a probability space.

QAlgorithms.Papers.Independent.Intakethird100.P62.HasHybrid

For any natural numbers $a,b$, complex matrix $W$ on $\mathbb C^a\otimes\mathbb C^b$, and set $\mathcal M$ of projective resolutions on $\mathbb C^a$, this condition means that there exist a measurable space $\Omega$, a probability measure $\mu$, a measurable Bochner-integrable function $\rho:\Omega\to\mathbb C^{b\times b}$, and real functions $r(A,x,w)$, such that every $\rho(w)$ is positive semidefinite with trace $1$, every $r(A,x,\cdot)$ is measurable and pointwise nonnegative, and $\sum_{x<a}r(A,x,w)=1$ for every $A\in\mathcal M,w$. For every $A\in\mathcal M,x<a$, every projective resolution $Q=(Q_y)_{y<b}$ on $\mathbb C^b$, and every $y<b$, the scalar function $w\mapsto r(A,x,w)\operatorname{Re}\operatorname{tr}[Q_y\rho(w)]$ must be integrable, and

$$\operatorname{Re}\operatorname{tr}[(P_x^A\otimes Q_y)W] =\int_\Omega r(A,x,w)\operatorname{Re}\operatorname{tr}[Q_y\rho(w)]\,d\mu(w).$$

A projective resolution in dimension $d$ consists of $d$ Hermitian idempotent $d\times d$ matrices with pairwise zero products for distinct indices and sum $I_d$; zero projectors are allowed. No density assumption on $W$ is imposed. If $\mathcal M$ is empty, only the hidden-ensemble requirements remain; $b=0$ is impossible, and $a=0$ with $\mathcal M\ne\varnothing$ is impossible because response normalization would require an empty sum to equal $1$.

QAlgorithms.Papers.Independent.Intakethird100.P62.HasProjectiveLHS

For any natural numbers $a,b$, complex matrix $W$ on $\mathbb C^a\otimes\mathbb C^b$, and set $\mathcal M$ of projective resolutions on $\mathbb C^a$, this condition means that there exist a type $\Omega$, a measurable-space structure on it, a probability measure $\mu$, a measurable Bochner-integrable matrix function $\rho:\Omega\to\mathbb C^{b\times b}$, and real functions $r(A,x,w)$ for $A\in\mathcal M,x<a,w\in\Omega$, such that every $\rho(w)$ is positive semidefinite with trace $1$, every $r(A,x,\cdot)$ is measurable, $r(A,x,w)\ge0$, $\sum_{x<a}r(A,x,w)=1$, and, for every $A\in\mathcal M,x<a$, the matrix function $w\mapsto r(A,x,w)\rho(w)$ is integrable and

$$\operatorname{Tr}_A[W(P_x^A\otimes I_b)] =\int_\Omega r(A,x,w)\rho(w)\,d\mu(w).$$

Here each $A$ supplies $a$ Hermitian idempotent matrices $P_x^A$, with pairwise zero products for distinct indices and sum $I_a$, and $(\operatorname{Tr}_AZ)_{uv}=\sum_{i<a}Z_{(i,u),(i,v)}$. All response and density conditions are pointwise; no density condition on $W$, or measurability in $A$, is required. An empty $\mathcal M$ removes the response and assemblage requirements but retains the hidden-ensemble requirements. In particular, $b=0$ is impossible; if $a=0$ and $\mathcal M\ne\varnothing$, response normalization is impossible.

QAlgorithms.Papers.Independent.Intakethird100.P62.IsHiddenEnsemble

For any measurable space $\Omega$, natural number $b$, measure $\mu$ on $\Omega$, and function $\rho$ assigning a complex $b\times b$ matrix to each $w\in\Omega$, the hidden-ensemble condition means that $\mu$ is a probability measure, that $\rho$ is measurable and Bochner integrable with respect to $\mu$, and that every $\rho(w)$ is positive semidefinite with trace $1$. The density-matrix condition is required at every point, including points of measure zero. The condition cannot hold when $\Omega$ is empty, because no probability measure exists there, or when $b=0$, because a zero-dimensional matrix has trace $0$.

QAlgorithms.Papers.Independent.Intakethird100.P62.IsResponse

For any types $\Omega$ and $S$, any specified measurable-space structure on $\Omega$, any natural number $d$, and any function $r:S\times\{0,\ldots,d-1\}\times\Omega\to\mathbb R$, the response condition means that $w\mapsto r(s,x,w)$ is measurable for every $s,x$, that $r(s,x,w)\ge0$ for every $s,x,w$, and that $\sum_xr(s,x,w)=1$ for every $s,w$. No measurable-space structure or measurability in $s$ is required. These requirements hold pointwise, rather than merely almost everywhere. If $S$ or $\Omega$ is empty, all requirements are vacuous; if $d=0$ and both $S$ and $\Omega$ are nonempty, the normalization requirement is impossible.

QAlgorithms.Papers.Independent.Intakethird100.P62.ProjectiveResolution

For every natural number $d$, a projective resolution consists of $d$ complex $d\times d$ matrices $P_x$, indexed by $x\in\{0,\ldots,d-1\}$, satisfying $P_x^\dagger=P_x$, $P_x^2=P_x$, $P_xP_y=0$ whenever $x\ne y$, and $\sum_xP_x=I_d$. No rank or nonzero condition is imposed on any individual projector. The case $d=0$ is included: there are no projectors, and completeness holds because the empty sum and the identity matrix on the zero-dimensional space are the same empty matrix.

QAlgorithms.Papers.Independent.Intakethird100.P62.Steerable

For any natural numbers $a,b$ and any complex matrix $W$ on $\mathbb C^a\otimes\mathbb C^b$, being steerable means that there do not exist a measurable space $\Omega$, a probability measure $\mu$, a measurable Bochner-integrable matrix function $\rho:\Omega\to\mathbb C^{b\times b}$ whose values are all positive semidefinite with trace $1$, and responses $r(A,x,w)$ for every projective resolution $A$ on $\mathbb C^a$, such that each $r(A,x,\cdot)$ is measurable, all responses are pointwise nonnegative, $\sum_{x<a}r(A,x,w)=1$, every $w\mapsto r(A,x,w)\rho(w)$ is integrable, and

$$\operatorname{Tr}_A[W(P_x^A\otimes I_b)] =\int_\Omega r(A,x,w)\rho(w)\,d\mu(w)$$

for every $A,x$. Here a projective resolution is a family of $a$ Hermitian idempotent $a\times a$ matrices $P_x^A$ with pairwise zero products for distinct indices and sum $I_a$, and $(\operatorname{Tr}_AZ)_{uv}=\sum_{i<a}Z_{(i,u),(i,v)}$. No density assumption on $W$ is part of this definition. Zero dimensions are included: if $a=0$, response normalization is impossible, and if $b=0$, the hidden density matrices are impossible; hence the negated existence condition holds in either case.

QAlgorithms.Papers.Independent.Intakethird100.P62.assemblage

For any natural numbers $a,b$, any complex matrix $W$ indexed by $\{0,\ldots,a-1\}\times\{0,\ldots,b-1\}$, any family $P_x$ of $a$ Hermitian idempotent $a\times a$ matrices with $P_xP_{x'}=0$ for $x\ne x'$ and $\sum_xP_x=I_a$, and any outcome $x<a$, the assemblage is the $b\times b$ matrix $\operatorname{Tr}_A[W(P_x\otimes I_b)]$, where $(\operatorname{Tr}_AZ)_{uv}=\sum_{i<a}Z_{(i,u),(i,v)}$. Explicitly, its $(u,v)$-entry is $\sum_{i,j<a}W_{(i,u),(j,v)}(P_x)_{ji}$. No positivity, Hermiticity, or trace normalization of $W$ is assumed. When $a=0$, there is no possible outcome $x$; $b=0$ is allowed and gives an empty output matrix.

QAlgorithms.Papers.Independent.Intakethird100.P62.jointProbability

For any natural numbers $a,b$, any complex matrix $W$ on the index set $\{0,\ldots,a-1\}\times\{0,\ldots,b-1\}$, projective resolutions $P=(P_x)_{x<a}$ and $Q=(Q_y)_{y<b}$, and outcomes $x<a,y<b$, the joint-probability expression is $\operatorname{Re}\operatorname{tr}[(P_x\otimes Q_y)W]$. Here each resolution is a family of Hermitian idempotent matrices, with distinct members multiplying to zero and with sum equal to the identity; zero projectors are permitted. No density-matrix assumption is imposed on $W$, so this definition alone imposes no probability bounds on the resulting real number. If either dimension is zero, the corresponding outcome cannot be supplied.

QAlgorithms.Papers.Independent.Intakethird100.P62.lhs_correlation_equivalence

For every pair of natural numbers $a,b$ with $a>0$ and $b>0$, every positive semidefinite matrix $W$ of trace $1$ on $\mathbb C^a\otimes\mathbb C^b$, and every set $M$ of projective measurements on $\mathbb C^a$, the following two existence statements are equivalent. A projective measurement on $\mathbb C^d$ here consists of exactly $d$ Hermitian idempotents $(P_x)_{x=0}^{d-1}$, with $P_xP_y=0$ for $x\ne y$ and $\sum_xP_x=I_d$; zero projectors are allowed. The first existence statement asks for a measurable space $\Omega$, a probability measure $\mu$, a measurable and $\mu$-integrable family $\sigma_\omega$ of density matrices on $\mathbb C^b$, and measurable real functions $r_{A,x}(\omega)$ for $A\in M$, with $r_{A,x}(\omega)\ge0$ and $\sum_xr_{A,x}(\omega)=1$ for every $A,\omega$, such that every matrix-valued function $r_{A,x}\sigma$ is integrable and

$$\operatorname{Tr}_1\!\bigl(W(A_x\otimes I_b)\bigr) =\int_\Omega r_{A,x}(\omega)\sigma_\omega\,d\mu(\omega) \quad(A\in M,\ 0\le x<a),$$

where $(\operatorname{Tr}_1 Z)_{u,v}=\sum_{i=0}^{a-1}Z_{(i,u),(i,v)}$. The second existence statement asks for a measurable space, probability measure, density-matrix family, and response functions with those same measurability, integrability of the density-matrix family, nonnegativity, and normalization requirements, such that, for every $A\in M$, every $x<a$, every projective measurement $B$ on $\mathbb C^b$, and every $y<b$, the scalar function $r_{A,x}(\omega)\operatorname{Re}\operatorname{tr}(B_y\sigma_\omega)$ is integrable and

$$\operatorname{Re}\operatorname{tr}\!\bigl((A_x\otimes B_y)W\bigr) =\int_\Omega r_{A,x}(\omega)\operatorname{Re}\operatorname{tr}(B_y\sigma_\omega)\,d\mu(\omega).$$

The witnesses in the two existence statements need not be identical. The set $M$ may be empty, in which case all requirements indexed by $A\in M$ are vacuous.

QAlgorithms.Papers.Independent.Intakethird100.P62.oneProjective

This defines the projective resolution in dimension $1$ whose sole projector is the $1\times1$ identity matrix $[1]$. It is Hermitian and idempotent, and its sum is the identity. The requirement that projectors with distinct indices multiply to zero is vacuous because the outcome index set has only one member.

QAlgorithms.Papers.Independent.Intakethird100.P62.strict_hierarchy

For positive subsystem dimensions $s,t$, call a matrix a density matrix when it is positive semidefinite and has trace $1$. A projective measurement on $\mathbb C^s$ means a family $(A_x)_{x=0}^{s-1}$ of Hermitian idempotent matrices satisfying $A_xA_{x'}=0$ for $x\ne x'$ and $\sum_xA_x=I_s$; zero projectors are allowed. Say that $W$ has a projective hidden-state model if there exist a measurable space $\Omega$, a probability measure $\mu$, a measurable, $\mu$-integrable family of density matrices $\sigma_\omega$ on $\mathbb C^t$, and response functions $r_{A,x}(\omega)$, indexed by every such projective measurement $A$, which are measurable, nonnegative, and satisfy $\sum_xr_{A,x}(\omega)=1$ for every $A,\omega$, such that each $r_{A,x}\sigma$ is integrable and

$$\operatorname{Tr}_1\!\bigl(W(A_x\otimes I_t)\bigr) =\int_\Omega r_{A,x}(\omega)\sigma_\omega\,d\mu(\omega).$$

Here $(\operatorname{Tr}_1 Z)_{u,v}=\sum_iZ_{(i,u),(i,v)}$. “Steerable” means that no such model exists. Say that $W$ is Bell local if there exist a measurable probability space and measurable, nonnegative response functions $r_{A,x},s_{B,y}$, normalized pointwise over their respective outcomes, for all projective measurements $A,B$ on the two subsystems, such that every product $r_{A,x}s_{B,y}$ is integrable and

$$\operatorname{Re}\operatorname{tr}\!\bigl((A_x\otimes B_y)W\bigr) =\int_\Omega r_{A,x}(\omega)s_{B,y}(\omega)\,d\mu(\omega);$$

“Bell nonlocal” means the absence of this model. Say that $W$ is separable positive if $W=\sum_{j=0}^{q-1}w_j\rho_j\otimes\sigma_j$ for some natural number $q$, nonnegative real weights $w_j$, and density matrices $\rho_j,\sigma_j$; this definition imposes no separate normalization on the weights and allows an empty sum. The declaration asserts the conjunction of three statements: for every $a,b\in\mathbb N$ and every density matrix $W$ on $\mathbb C^{a+1}\otimes\mathbb C^{b+1}$, Bell nonlocality implies steerability, and steerability implies that $W$ is not separable positive; there exists a real $\eta$ with $1/3<\eta\le1/2$ for which $W_\eta=\frac{1+\eta}{4}I_4-\frac{\eta}{2}F$, where $F(u\otimes v)=v\otimes u$, is a density matrix that is not separable positive and is not steerable; and there exists a real $\eta$ with $1/2<\eta\le1$ for which this same formula gives a density matrix that is steerable and Bell local. The universal statement includes $a=0$ or $b=0$, corresponding to a one-dimensional subsystem.

QAlgorithms.Papers.Independent.Intakethird100.P62.wernerEta

For every natural number $d$ and every real number $\eta$, set $p=[d-1+(d+1)\eta]/(2d)$, with the arithmetic in this expression taking place in $\mathbb R$. Let $F$ be the complex matrix indexed by pairs from $\{0,\ldots,d-1\}$, with $F_{(i,j),(k,l)}=1$ when $i=l$ and $j=k$, and $0$ otherwise, and set $P_0=(I+F)/2$, $P_1=I-P_0$. The defined matrix is

$$\frac{2(1-p)}{d(d+1)}P_0+\frac{2p}{d(d-1)}P_1.$$

The real coefficients are regarded as complex scalars, and $d-1$ in the displayed denominator is real subtraction. There is no restriction on $\eta$ and no density-matrix requirement. All divisions use the total convention $x/0=0$; thus $d=0$ gives the empty matrix, and $d=1$ gives the $1\times1$ matrix with entry $1-\eta$.