Definitions for The resource theory of steering

QAlgorithms.Papers.Independent.Intakethird100.P36.AssemblageBlocks

For natural numbers $d,r,s$, this denotes an arbitrary family $\rho=(\rho_{a,x})$ of complex $d\times d$ matrices, indexed by $0\le a<r$ and $0\le x<s$. No positivity, normalization, or compatibility conditions are imposed. All three parameters may be zero; if $r=0$ or $s=0$, there are no blocks, and if $d=0$, each block is an empty matrix.

QAlgorithms.Papers.Independent.Intakethird100.P36.EntropyMaximum

For arbitrary natural numbers $d,r,s$ and arbitrary complex $d\times d$ matrix families $\rho,\sigma$ indexed by $a<r,x<s$, this predicate says that there exists a strategy whose score equals the supremum of all strategy scores. A strategy consists of a natural count $h$, matrices $K_g$ satisfying $\sum_gK_g^\dagger K_g=I_d$, and real weights $u_{g,x}\ge0$ satisfying $\sum_xu_{g,x}=1$. For each such strategy, put $B_\tau=s^{-1}\sum_{x,a}\tau_{a,x}$, $p_g=\operatorname{Re}\operatorname{Tr}(K_gB_\rho K_g^\dagger)$, $q_g=\operatorname{Re}\operatorname{Tr}(K_gB_\sigma K_g^\dagger)$, and $F^\tau_{g,x}=\bigoplus_aK_g\tau_{a,x}K_g^\dagger$; its score is

$$\mathcal D(\operatorname{diag}p,\operatorname{diag}q) +\sum_{g,x}u_{g,x}\,C(p_g,q_g;F^\rho_{g,x},F^\sigma_{g,x}).$$

The divergence $\mathcal D(R,S)$ is infinite unless $R,S$ are positive semidefinite with trace $1$ and $\ker S\subseteq\ker R$; when these conditions hold it is $\max\{0,\operatorname{Re}\operatorname{Tr}(f(R)-R\log S)\}$, with $f(t)=t\log t$ and continuous functional calculus. The conditional term $C$ is $0$ for $p\le0$, $+\infty$ for $p>0,q\le0$, and $p\mathcal D(A/p,B/q)$ for $p,q>0$. No assemblage assumptions are imposed, and the attained supremum may be infinite. If there is no strategy, the supremum is $0$, but this existence predicate is false. Zero dimensions and empty sums are included with total reciprocal and extended arithmetic conventions.

QAlgorithms.Papers.Independent.Intakethird100.P36.EntropyStrategy

For arbitrary natural numbers $d,s$, an object of this structure consists of a natural number $h$ of outcomes, complex $d\times d$ matrices $K_g$ for $0\le g<h$, and real numbers $u_{g,x}$ for $0\le g<h$ and $0\le x<s$, together with the requirements

$$\sum_{g=0}^{h-1}K_g^\dagger K_g=I_d, \qquad u_{g,x}\ge0, \qquad \sum_{x=0}^{s-1}u_{g,x}=1\quad\text{for every }g.$$

Here $\dagger$ denotes conjugate transpose. None of $d,s,h$ is explicitly required to be positive. If $h=0$, completeness is possible only when $d=0$; if $s=0$, the input laws are impossible for any existing outcome, so $h$ must be $0$. In particular, $d=s=h=0$ is permitted.

QAlgorithms.Papers.Independent.Intakethird100.P36.HasFiniteLHSModel

For natural numbers $d,r,s$ and a family $\rho_{a,x}$ of complex $d\times d$ matrices, this predicate first requires $d,r,s>0$, positive semidefinite blocks, setting-independent sums $\sum_a\rho_{a,x}$, and total mass $m_\rho=\operatorname{Re}\operatorname{tr}(s^{-1}\sum_x\sum_a\rho_{a,x})$ in $[0,1]$; it additionally requires the existence of a natural number $n$, real numbers $p_\ell$, complex $d\times d$ matrices $\xi_\ell$, and real numbers $R_{\ell,x,a}$, indexed by $0\le\ell<n$, $0\le x<s$, $0\le a<r$, such that

$$p_\ell\ge0,\quad \sum_\ell p_\ell=1,\qquad \xi_\ell\succeq0,\quad\operatorname{tr}\xi_\ell=1,\qquad R_{\ell,x,a}\ge0,\quad\sum_aR_{\ell,x,a}=1, \qquad \rho_{a,x}=\sum_{\ell=0}^{n-1}m_\rho p_\ell R_{\ell,x,a}\,\xi_\ell.$$

Although $n=0$ is quantified over, it cannot satisfy $\sum_\ell p_\ell=1$. Mass zero is allowed and makes every block in the displayed representation zero; mass one is not required.

QAlgorithms.Papers.Independent.Intakethird100.P36.IsAssemblage

For natural numbers $d,r,s$ and a family $\rho_{a,x}$ of complex $d\times d$ matrices indexed by $0\le a<r$ and $0\le x<s$, this predicate requires $d,r,s>0$, every $\rho_{a,x}$ to be positive semidefinite, and

$$\sum_{a=0}^{r-1}\rho_{a,x}=\sum_{a=0}^{r-1}\rho_{a,y} \quad\text{for every }0\le x,y<s, \qquad 0\le \operatorname{Re}\operatorname{tr}\left( \frac1s\sum_{x=0}^{s-1}\sum_{a=0}^{r-1}\rho_{a,x}\right)\le1.$$

Thus zero total mass is allowed, while a zero value of any dimension makes the predicate false.

QAlgorithms.Papers.Independent.Intakethird100.P36.IsConvexSteeringMonotone

For an arbitrary function $S$ assigning a real number to every array $\rho=(\rho_{ax})_{a<r,x<s}$ of complex $d\times d$ matrices for every $d,r,s\in\mathbb N$, this predicate is the conjunction of four requirements. In these requirements, an array is normalized when $d,r,s>0$, every $\rho_{ax}$ is positive semidefinite, $\sum_a\rho_{ax}$ is independent of $x$, and its mass $M_\rho=\operatorname{Re}\operatorname{tr}(s^{-1}\sum_x\sum_a\rho_{ax})$ lies in $[0,1]$ and equals $1$. First, $S(d,r,s,\rho)\ge0$ for every normalized array. Second, $S(d,r,s,\rho)=0$ for every normalized array that admits some finite representation

$$\rho_{ax}=\sum_{\ell<n}M_\rho\,p_\ell\,h_{\ell,x,a}\,\xi_\ell,$$

where $n\in\mathbb N$, $p_\ell\ge0$, $\sum_\ell p_\ell=1$, every $\xi_\ell$ is a positive semidefinite complex $d\times d$ matrix with trace $1$, and $h_{\ell,x,a}\ge0$, $\sum_{a<r}h_{\ell,x,a}=1$ for every $\ell,x$; the finite-model condition also requires positive dimensions, positive semidefinite blocks, setting-independent outcome sums, and mass in $[0,1]$. Third, for every normalized $\rho$, every positive output dimensions $d_f,r_f,s_f$, every natural $m$, matrices $K_w\in\mathbb C^{d_f\times d}$ with $\sum_{w<m}K_w^\dagger K_w=I_d$ and $I_d-\sum_wK_w^\dagger K_w\succeq0$, and nonnegative arrays $u_{w,x_f,x},v_{w,a_f,a,x,x_f}$ summing to $1$ over $x<s$ and $a_f<r_f$, respectively, put

$$p_w=\operatorname{Re}\operatorname{tr}\!\left(K_w\left(\frac1s\sum_x\sum_a\rho_{ax}\right)K_w^\dagger\right),\qquad \tau^{(w)}_{a_fx_f}=\sum_x\sum_a v_{w,a_f,a,x,x_f}u_{w,x_f,x}K_w\rho_{ax}K_w^\dagger,\qquad M_w=\operatorname{Re}\operatorname{tr}\!\left(\frac1{s_f}\sum_{x_f}\sum_{a_f}\tau^{(w)}_{a_fx_f}\right);$$

then the sum over $w<m$ of $p_wS(d_f,r_f,s_f,M_w^{-1}\tau^{(w)})$ when $p_w>0$, and $0$ otherwise, must be at most $S(d,r,s,\rho)$, with entrywise rescaling and $0^{-1}=0$. Fourth, for every pair of normalized arrays $\rho,\sigma$ of the same dimensions and every $t\in[0,1]$,

$$S(d,r,s,t\rho+(1-t)\sigma) \le tS(d,r,s,\rho)+(1-t)S(d,r,s,\sigma).$$

All natural dimensions are quantified over, but normalization and protocol hypotheses exclude zero dimensions from these obligations; there is no restriction on the values of $S$ on other arrays.

QAlgorithms.Papers.Independent.Intakethird100.P36.IsNormalized

For natural numbers $d,r,s$ and a family $\rho_{a,x}$ of complex $d\times d$ matrices indexed by $0\le a<r$ and $0\le x<s$, this predicate requires $d,r,s>0$, positive semidefiniteness of every block, equality of $\sum_a\rho_{a,x}$ for every pair of settings $x,y$, and, writing

$$m_\rho=\operatorname{Re}\operatorname{tr}\left( \frac1s\sum_{x=0}^{s-1}\sum_{a=0}^{r-1}\rho_{a,x}\right),$$

both $0\le m_\rho\le1$ and $m_\rho=1$. It cannot hold when any dimension is zero.

QAlgorithms.Papers.Independent.Intakethird100.P36.IsPureAssemblage

For natural numbers $d,r,s$ and a family $\rho=(\rho_{a,x})_{a<r,\ x<s}$ of complex $d\times d$ matrices, this predicate means that there exist real numbers $p_x(a)$ and complex vectors $v_{a,x}\in\mathbb C^d$ such that, for every setting $x<s$, $p_x(a)\geq0$ for every $a<r$ and $\sum_a p_x(a)=1$; for every $a,x$, $\sum_{i<d}|v_{a,x,i}|^2=1$; and, for every $a,x,i,j$,

$$(\rho_{a,x})_{ij}=p_x(a)v_{a,x,i}\overline{v_{a,x,j}}.$$

The vector normalization is required even when $p_x(a)=0$. No condition equating the outcome sums for different settings is included. If $s=0$, all the indexed requirements are vacuous; if $s>0$ and $r=0$, the probability requirement is impossible, and if $s,r>0$ and $d=0$, vector normalization is impossible.

QAlgorithms.Papers.Independent.Intakethird100.P36.IsPureOrthogonal

For natural numbers $d,r,s$ and a family $\rho=(\rho_{a,x})_{a<r,\ x<s}$ of complex $d\times d$ matrices, this predicate means that there exist real probabilities $p_x(a)\geq0$ with $\sum_{a<r}p_x(a)=1$ for every $x<s$, and complex vectors $v_{a,x}\in\mathbb C^d$, such that for every $a,b<r$ and $x<s$,

$$\sum_{i<d}\overline{v_{a,x,i}}v_{b,x,i} =\begin{cases}1,&a=b,\\0,&a\ne b,\end{cases} \qquad (\rho_{a,x})_{ij}=p_x(a)v_{a,x,i}\overline{v_{a,x,j}} \quad(i,j<d).$$

Thus all the outcome vectors at each setting are required to be orthonormal, including vectors assigned zero probability. No condition equating the outcome sums between settings is included. For $s=0$ the requirements are vacuous; for $s>0$, they are impossible if $r=0$, and otherwise require an orthonormal family of $r$ vectors in $\mathbb C^d$.

QAlgorithms.Papers.Independent.Intakethird100.P36.MeasureConvexity

For an arbitrary real-valued function $S$ on arrays of complex matrices of all natural-number dimensions, this predicate requires that, for every $d,r,s\in\mathbb N$, every two arrays $\rho,\sigma$ of complex $d\times d$ matrices indexed by $a<r,x<s$, and every real $t$, if both arrays are normalized and $0\le t\le1$, then

$$S\!\left(d,r,s,(t\rho_{ax}+(1-t)\sigma_{ax})_{a,x}\right) \le t\,S(d,r,s,\rho)+(1-t)\,S(d,r,s,\sigma).$$

For each array, normalized means $d,r,s>0$, all its matrices are positive semidefinite, its sums over $a$ are independent of $x$, and the real trace of $s^{-1}\sum_x\sum_a\rho_{ax}$, or of the corresponding expression for $\sigma$, lies in $[0,1]$ and equals $1$. The endpoints $t=0,1$ are included. No condition is imposed by this predicate when either array is unnormalized, when a dimension is zero, or when $t\notin[0,1]$.

QAlgorithms.Papers.Independent.Intakethird100.P36.RobustnessCandidate

For natural numbers $d,r,s$, an arbitrary family $\rho=(\rho_{a,x})_{a<r,\ x<s}$ of complex $d\times d$ matrices, and $\nu\in\mathbb R$, this predicate holds precisely when $\nu\geq0$ and there exists a normalized family $\eta$ such that the family

$$Y_{a,x}=\frac{\rho_{a,x}}{1+\nu}+\frac{\nu}{1+\nu}\eta_{a,x}$$

has a finite hidden-state model. Explicitly, normalization of $\eta$ requires $d,r,s>0$, positive semidefinite blocks, setting-independent sums $\sum_a\eta_{a,x}$, and mass $1$, where the mass of any family $F$ is $\mu(F)=\operatorname{Re}\operatorname{tr}(s^{-1}\sum_x\sum_aF_{a,x})$. The condition on $Y$ requires positive semidefinite blocks, setting-independent outcome sums, $0\leq\mu(Y)\leq1$, and the existence of a natural number $h$, probabilities $p_\ell$ summing to $1$, positive semidefinite matrices $\xi_\ell$ of trace $1$, and probability distributions $q_{\ell,x}$ on the $r$ outcomes such that $Y_{a,x}=\mu(Y)\sum_{\ell<h}p_\ell q_{\ell,x}(a)\xi_\ell$ for all $a,x$. No normalization or assemblage condition is separately imposed on $\rho$, and $\nu$ has no upper bound. The denominator $1+\nu$ is positive for a candidate; if any of $d,r,s$ is zero, the predicate cannot hold.

QAlgorithms.Papers.Independent.Intakethird100.P36.SteeringConverts

For natural numbers $d,r,s,d_f,r_f,s_f$, and arrays $\rho=(\rho_{ax})_{a<r,x<s}$ of complex $d\times d$ matrices and $\sigma=(\sigma_{a_fx_f})_{a_f<r_f,x_f<s_f}$ of complex $d_f\times d_f$ matrices, this predicate holds precisely when both arrays are normalized and the following protocol exists. Here normalized means that the three dimensions of the array are positive, every matrix entry of the array is positive semidefinite, its sums over outcomes are independent of the setting, and its mass—defined as the real trace of the average of those sums—is in $[0,1]$ and equals $1$. There must exist a natural number $m$, matrices $K_w\in\mathbb C^{d_f\times d}$ for $w<m$ satisfying $I_d-\sum_wK_w^\dagger K_w\succeq0$, nonnegative real numbers $u_{w,x_f,x}$ summing to $1$ over $x<s$ for every $w,x_f$, nonnegative real numbers $v_{w,a_f,a,x,x_f}$ summing to $1$ over $a_f<r_f$ for every $w,a,x,x_f$, and a real number $p$ with $0<p\le1$, such that

$$\forall a_f<r_f,\ \forall x_f<s_f,\qquad \sum_{w<m}\sum_{x<s}\sum_{a<r} v_{w,a_f,a,x,x_f}\,u_{w,x_f,x}\, K_w\rho_{ax}K_w^\dagger =p\,\sigma_{a_fx_f}.$$

The protocol is not required to satisfy $\sum_wK_w^\dagger K_w=I_d$. Although $m=0$ is included in the existential quantifier, its zero action cannot satisfy the displayed equality for positive $p$ and normalized $\sigma$.

QAlgorithms.Papers.Independent.Intakethird100.P36.SteeringMeasure

This is an abbreviation for the type of all functions $S$ that take three natural numbers $d,r,s$ and an arbitrary array $(\rho_{ax})_{a<r,x<s}$ of complex $d\times d$ matrices and return a real number $S(d,r,s,\rho)$. All natural-number dimensions, including zero, and all such arrays are allowed. The type imposes no positivity, normalization, invariance, convexity, or monotonicity requirement on $S$.

QAlgorithms.Papers.Independent.Intakethird100.P36.SteeringProtocol.Complete

For natural numbers $d,r,s,d_f,r_f,s_f,m$, let $P$ consist of positive dimensions $d,r,s,d_f,r_f,s_f$, matrices $K_w\in\mathbb C^{d_f\times d}$ satisfying $I_d-\sum_wK_w^\dagger K_w\succeq0$, nonnegative input weights $I_{w,x_f,x}$ summing to one over $0\le x<s$ for every $w,x_f$, and nonnegative output weights $O_{w,a_f,a,x,x_f}$ summing to one over $0\le a_f<r_f$ for every $w,a,x,x_f$. This predicate requires the additional equality

$$\sum_{w=0}^{m-1}K_w^\dagger K_w=I_d.$$

It cannot hold for $m=0$, since $d>0$ and the empty sum is zero.

QAlgorithms.Papers.Independent.Intakethird100.P36.SteeringProtocol

For natural numbers $d,r,s,d_f,r_f,s_f,m$, this structure consists of proofs that $d,r,s,d_f,r_f,s_f>0$, matrices $K_w\in\mathbb C^{d_f\times d}$ for $0\le w<m$, and real arrays $I_{w,x_f,x}$ and $O_{w,a_f,a,x,x_f}$, where $0\le x_f<s_f$, $0\le x<s$, $0\le a_f<r_f$, and $0\le a<r$, satisfying

$$I_d-\sum_{w=0}^{m-1}K_w^\dagger K_w\succeq0,\qquad I_{w,x_f,x}\ge0,\quad\sum_{x=0}^{s-1}I_{w,x_f,x}=1,\qquad O_{w,a_f,a,x,x_f}\ge0,\quad \sum_{a_f=0}^{r_f-1}O_{w,a_f,a,x,x_f}=1.$$

The input normalization holds for every $w,x_f$, and the output normalization for every $w,a,x,x_f$. The parameter $m$ may be zero: then these arrays have no entries, their normalization requirements are vacuous, and the matrix inequality reduces to $I_d\succeq0$.

QAlgorithms.Papers.Independent.Intakethird100.P36.StrongMonotonicity

For an arbitrary real-valued function $S$ on arrays of complex matrices of all natural-number dimensions, this predicate requires the following inequality for every choice of natural numbers $d,r,s,d_f,r_f,s_f,m$, every protocol $P$, and every normalized array $\rho=(\rho_{ax})_{a<r,x<s}$. Normalization means $d,r,s>0$, $\rho_{ax}\succeq0$, $\sum_a\rho_{ax}$ is independent of $x$, and $M_\rho=\operatorname{Re}\operatorname{tr}(s^{-1}\sum_x\sum_a\rho_{ax})$ lies in $[0,1]$ and equals $1$. The protocol has all six dimensions positive, matrices $K_w\in\mathbb C^{d_f\times d}$, nonnegative coefficients $u_{w,x_f,x}$ summing to $1$ over $x$, and nonnegative coefficients $v_{w,a_f,a,x,x_f}$ summing to $1$ over $a_f$; it satisfies $I_d-\sum_wK_w^\dagger K_w\succeq0$, and the hypothesis additionally requires completeness, $\sum_wK_w^\dagger K_w=I_d$. Define

$$p_w=\operatorname{Re}\operatorname{tr}\!\left(K_w\left(\frac1s\sum_x\sum_a\rho_{ax}\right)K_w^\dagger\right),\qquad \tau^{(w)}_{a_fx_f}=\sum_x\sum_a v_{w,a_f,a,x,x_f}u_{w,x_f,x}K_w\rho_{ax}K_w^\dagger,\qquad M_w=\operatorname{Re}\operatorname{tr}\!\left(\frac1{s_f}\sum_{x_f}\sum_{a_f}\tau^{(w)}_{a_fx_f}\right).$$

The required inequality is

$$\sum_{w<m} \begin{cases} p_w\,S(d_f,r_f,s_f,M_w^{-1}\tau^{(w)}),&p_w>0,\\ 0,&p_w\le0 \end{cases} \ \le\ S(d,r,s,\rho),$$

where rescaling is entrywise and $0^{-1}=0$. There is no requirement here for unnormalized input arrays or incomplete protocols; zero dimensions cannot satisfy the stated hypotheses, and $m=0$ cannot satisfy completeness when $d>0$.

QAlgorithms.Papers.Independent.Intakethird100.P36.WeightCandidate

For arbitrary natural numbers $d,r,s$, an arbitrary family $\rho=(\rho_{a,x})_{a<r,x<s}$ of complex $d\times d$ matrices, and a real number $\nu$, this predicate requires $\nu\ge0$ and the existence of two families $\eta,\sigma$ of the same shape such that

$$\rho_{a,x}=\nu\,\eta_{a,x}+(1-\nu)\sigma_{a,x} \quad\text{for every }a<r,\ x<s.$$

Both $\eta$ and $\sigma$ must be normalized in the following precise sense: $d,r,s>0$, every block is positive semidefinite, the sums over $a$ are identical for all inputs $x$, and the mass $\operatorname{Re}\operatorname{Tr}(s^{-1}\sum_{x,a}\tau_{a,x})$ equals $1$ for each family $\tau$. In addition, $\sigma$ must admit some finite representation

$$\sigma_{a,x}=\sum_{\ell<n}p_\ell t_{\ell,x,a}\xi_\ell,$$

where $n$ is a natural number, $p_\ell\ge0$ and $\sum_\ell p_\ell=1$, every $\xi_\ell$ is positive semidefinite with trace $1$, and $t_{\ell,x,a}\ge0$ with $\sum_a t_{\ell,x,a}=1$ for every $\ell,x$; the mass factor in the model definition is $1$ because $\sigma$ is normalized. There is no upper bound $\nu\le1$, and no separate assemblage assumption is imposed on $\rho$. The predicate is impossible if any of $d,r,s$ is zero; the probability law on the hidden indices also rules out $n=0$.

QAlgorithms.Papers.Independent.Intakethird100.P36.angleAssemblage

For every real $\theta$, this definition gives a family of complex $2\times2$ matrices indexed by outcome $a\in\{0,1\}$ and setting $x\in\{0,1\}$:

$$\rho_{0,0}=\frac12\begin{pmatrix}1&0\\0&0\end{pmatrix}, \quad \rho_{1,0}=\frac12\begin{pmatrix}0&0\\0&1\end{pmatrix}, \quad \rho_{0,1}=\frac12\begin{pmatrix}\cos^2\theta&\cos\theta\sin\theta\\ \cos\theta\sin\theta&\sin^2\theta\end{pmatrix}, \quad \rho_{1,1}=\frac12\begin{pmatrix}\sin^2\theta&-\sin\theta\cos\theta\\ -\sin\theta\cos\theta&\cos^2\theta\end{pmatrix}.$$

Equivalently, each block is one half of $vv^\dagger$, using respectively $v=(1,0),(0,1),(\cos\theta,\sin\theta),(-\sin\theta,\cos\theta)$. There is no restriction on $\theta$.

QAlgorithms.Papers.Independent.Intakethird100.P36.angleKet

For every real $\theta$ and indices $a,x\in\{0,1\}$, this definition gives a vector in $\mathbb C^2$, with coordinates ordered as $0,1$, by

$$v_{\theta,0,0}=(1,0),\qquad v_{\theta,1,0}=(0,1),\qquad v_{\theta,0,1}=(\cos\theta,\sin\theta),\qquad v_{\theta,1,1}=(-\sin\theta,\cos\theta).$$

The real coordinates are regarded as complex numbers, and there is no restriction on $\theta$.

QAlgorithms.Papers.Independent.Intakethird100.P36.assemblageCQMatrix

For any natural numbers $d,r,s$, any array $\rho=(\rho_{ax})_{a<r,x<s}$ of complex $d\times d$ matrices, and any $x<s$, the defined matrix is indexed by pairs $(a,i)$ with $a<r$ and $i<d$, and has entries

$$C_{(a,i),(b,j)} = \begin{cases} (\rho_{ax})_{ij},&a=b,\\ 0,&a\ne b. \end{cases}$$

Thus its diagonal blocks are the matrices $\rho_{ax}$ at the chosen setting $x$. No positivity, normalization, or setting-independence assumption is imposed. If $r=0$ or $d=0$, the matrix has an empty index set; if $s=0$, there is no possible argument $x$.

QAlgorithms.Papers.Independent.Intakethird100.P36.assemblageEntropy

For arbitrary natural numbers $d,r,s$ and arbitrary matrix families $\rho,\sigma=(\rho_{a,x}),(\sigma_{a,x})$, with $a<r,x<s$ and complex $d\times d$ blocks, this defines the supremum in $[0,+\infty]$ of the following scores over all natural outcome counts $h$, complex $d\times d$ matrices $K_g$, and real weights $u_{g,x}$ satisfying $\sum_gK_g^\dagger K_g=I_d$, $u_{g,x}\ge0$, and $\sum_xu_{g,x}=1$. Put $B_\tau=s^{-1}\sum_{x,a}\tau_{a,x}$, $p_g=\operatorname{Re}\operatorname{Tr}(K_gB_\rho K_g^\dagger)$, $q_g=\operatorname{Re}\operatorname{Tr}(K_gB_\sigma K_g^\dagger)$, and $F^\tau_{g,x}=\bigoplus_{a<r}K_g\tau_{a,x}K_g^\dagger$. The score being maximized is

$$\mathcal D(\operatorname{diag}p,\operatorname{diag}q) +\sum_{g,x}u_{g,x}\,C(p_g,q_g;F^\rho_{g,x},F^\sigma_{g,x}).$$

Here $\mathcal D(R,S)$ is $+\infty$ unless $R,S$ are positive semidefinite of trace $1$ and $\ker S\subseteq\ker R$; otherwise it is $\max\{0,\operatorname{Re}\operatorname{Tr}(f(R)-R\log S)\}$, where $f(t)=t\log t$ and matrix functions use continuous functional calculus. Also, $C(p,q;A,B)$ is zero for $p\le0$, infinite for $p>0,q\le0$, and $p\mathcal D(A/p,B/q)$ for $p,q>0$. Neither family is required to satisfy positivity, normalization, or input consistency. The supremum need not be attained; an empty set of scores has supremum $0$. Zero dimensions and outcome counts are allowed subject to the strategy equations, and all reciprocal and extended arithmetic conventions apply.

QAlgorithms.Papers.Independent.Intakethird100.P36.assemblageFamilyMixture

For any natural numbers $n,d,r,s$, any real coefficients $p_j$ indexed by $j<n$, and any family $\rho^{(j)}=(\rho^{(j)}_{ax})_{a<r,x<s}$ of arrays of complex $d\times d$ matrices, the defined array has entries

$$\sum_{j<n}p_j\,\rho^{(j)}_{ax},$$

where each coefficient is regarded as a complex scalar. No nonnegativity or sum-to-one condition is imposed on the coefficients, and no positivity or normalization condition is imposed on the arrays. In particular, $n=0$ gives the zero array; the other dimensions may also be zero.

QAlgorithms.Papers.Independent.Intakethird100.P36.assemblageMass

For arbitrary natural numbers $d,r,s$ and an arbitrary family of complex $d\times d$ matrices $\rho_{a,x}$, indexed by $0\le a<r$ and $0\le x<s$, this defines the real number

$$m_\rho=\operatorname{Re}\operatorname{tr}\left((s:\mathbb C)^{-1} \sum_{x=0}^{s-1}\sum_{a=0}^{r-1}\rho_{a,x}\right).$$

No positivity or normalization is required, so this number is not required to be nonnegative. Empty sums and empty traces are zero, and $0^{-1}=0$; consequently $m_\rho=0$ if any of $d,r,s$ is zero.

QAlgorithms.Papers.Independent.Intakethird100.P36.assemblageMixture

For any natural numbers $d,r,s$, any two arrays $\rho=(\rho_{ax})_{a<r,x<s}$ and $\sigma=(\sigma_{ax})_{a<r,x<s}$ of complex $d\times d$ matrices, and any real number $t$, the defined array has entries

$$t\,\rho_{ax}+(1-t)\,\sigma_{ax}.$$

Both real coefficients are regarded as complex scalars. There is no assumption that $0\le t\le1$, or that either input array is positive, normalized, or has setting-independent outcome sums. Zero dimensions and empty arrays are included.

QAlgorithms.Papers.Independent.Intakethird100.P36.bobStep

For natural numbers $d,r,s,d_f,r_f,s_f,m$, take a protocol consisting of positive dimensions $d,r,s,d_f,r_f,s_f$, matrices $K_w\in\mathbb C^{d_f\times d}$ with $\sum_wK_w^\dagger K_w\preceq I_d$, nonnegative input weights $I_{w,x_f,x}$ normalized over $x<s$, and nonnegative output weights $O_{w,a_f,a,x,x_f}$ normalized over $a_f<r_f$, for every choice of their other indices. Given any family $\rho_{a,x}\in\mathbb C^{d\times d}$, $a<r,x<s$, and a branch index $w<m$, this defines a family of complex $d_f\times d_f$ matrices by

$$(\rho^{(w)})_{a,x}=K_w\rho_{a,x}K_w^\dagger.$$

No positivity, compatibility, or normalization is assumed for $\rho$. If $m=0$, no branch index $w$ exists.

QAlgorithms.Papers.Independent.Intakethird100.P36.bobTotal

For arbitrary natural numbers $d,r,s$ and an arbitrary family of complex $d\times d$ matrices $\rho_{a,x}$, indexed by $0\le a<r$ and $0\le x<s$, this defines the complex $d\times d$ matrix

$$B_\rho=(s:\mathbb C)^{-1}\sum_{x=0}^{s-1}\sum_{a=0}^{r-1}\rho_{a,x}.$$

There are no positivity or normalization assumptions. Empty sums are zero, and the inverse of zero is defined to be zero, so $B_\rho=0$ whenever $s=0$ or $r=0$.

QAlgorithms.Papers.Independent.Intakethird100.P36.conditionalD

For any finite type $I$ equipped with decidable equality, arbitrary real numbers $p,q$, and arbitrary complex matrices $A,B$ indexed by $I\times I$, this defines

$$C(p,q;A,B)= \begin{cases} p\,\mathcal D(A/p,B/q),&p>0,\ q>0,\\ +\infty,&p>0,\ q\le0,\\ 0,&p\le0. \end{cases}$$

The values lie in $[0,+\infty]$, and $\mathcal D(R,S)$ means $+\infty$ unless $R,S$ are positive semidefinite, both have trace $1$, and $\ker S\subseteq\ker R$; when all these conditions hold, it means

$$\mathcal D(R,S)= \max\!\left\{0,\operatorname{Re}\operatorname{Tr}\bigl(f(R)-R\log S\bigr)\right\}, \qquad f(t)=t\log t,$$

using continuous functional calculus and the total scalar convention $\log0=0$. The displayed divisions mean scalar multiplication by the complex reciprocals. No sign, trace, or positivity conditions are hypotheses. In particular, $p\le0$ gives zero regardless of the other arguments, while for an empty $I$ and $p,q>0$, the density trace conditions fail and the value is $+\infty$.

QAlgorithms.Papers.Independent.Intakethird100.P36.conditionalKetProjector

For every natural number $d$ and every complex vector $v=(v_i)_{i<d}$, this definition gives the complex $d\times d$ matrix whose $(i,j)$ entry is $v_i\overline{v_j}$ for all $i,j<d$. No normalization or nonzero condition is imposed on $v$; in particular, $v=0$ gives the zero matrix, and $d=0$ gives the empty matrix.

QAlgorithms.Papers.Independent.Intakethird100.P36.densityD

For any finite index type $I$ with decidable equality, including the empty type, and any complex matrices $\rho,\sigma$ indexed by $I\times I$, this definition returns a nonnegative extended real number as follows. If either matrix fails to be positive semidefinite with trace exactly $1$, the value is $+\infty$. If both satisfy those conditions but $\ker\sigma\not\subseteq\ker\rho$, where the kernels are those of their matrix actions on complex coordinate vectors, the value is also $+\infty$. Otherwise the value is

$$\max\!\left\{0,\, \operatorname{Re}\operatorname{tr}\!\left(F(\rho)-\rho\,L(\sigma)\right) \right\},$$

regarded as a nonnegative extended real number, where $F$ and $L$ are applied by continuous functional calculus and are given by $F(t)=t\log t$ and $L(t)=\log t$, with the natural logarithm and the total-function convention $\log0=0$. The maximum with zero is the conversion from a real number to a nonnegative extended real number. For empty $I$, every matrix has trace zero, so the value is $+\infty$.

QAlgorithms.Papers.Independent.Intakethird100.P36.entropyBranch

Let $d,r,s,d_f,r_f,s_f,m$ be natural numbers, and let $P$ consist of positive dimensions $d,r,s,d_f,r_f,s_f$, complex $d_f\times d$ matrices $K_w$ for $w<m$ satisfying $I_d-\sum_wK_w^\dagger K_w\ge0$, and real coefficients $u_{w,y,x}\ge0$, $v_{w,b,a,x,y}\ge0$ satisfying $\sum_{x<s}u_{w,y,x}=1$ and $\sum_{b<r_f}v_{w,b,a,x,y}=1$. For arbitrary input matrix families $\rho,\sigma$ indexed by $a<r,x<s$, and an outcome $w<m$, put

$$B_\tau=\frac1s\sum_{x,a}\tau_{a,x},\quad p=\operatorname{Re}\operatorname{Tr}(K_wB_\rho K_w^\dagger),\quad q=\operatorname{Re}\operatorname{Tr}(K_wB_\sigma K_w^\dagger),\quad \Gamma^\tau_{b,y}=\sum_{x,a}v_{w,b,a,x,y}u_{w,y,x}K_w\tau_{a,x}K_w^\dagger,$$

and $M_\tau=\operatorname{Re}\operatorname{Tr}(s_f^{-1}\sum_{y,b}\Gamma^\tau_{b,y})$. This defines the value to be $0$ when $p\le0$, $+\infty$ when $p>0,q\le0$, and $p\,H(\Gamma^\rho/M_\rho,\Gamma^\sigma/M_\sigma)$ when $p,q>0$, with division applied to every block. For families $\alpha,\beta$ of $d_f\times d_f$ blocks indexed by $b<r_f,y<s_f$, $H(\alpha,\beta)$ is the supremum over finite matrices $L_g$ satisfying $\sum_gL_g^\dagger L_g=I_{d_f}$ and weights $z_{g,y}\ge0$, $\sum_yz_{g,y}=1$, of $\mathcal D(\operatorname{diag}a,\operatorname{diag}b)+\sum_{g,y}z_{g,y}C(a_g,b_g;F^\alpha_{g,y},F^\beta_{g,y})$, where $a_g=\operatorname{Re}\operatorname{Tr}(L_g(s_f^{-1}\sum_{y,b}\alpha_{b,y})L_g^\dagger)$, $b_g$ is the same expression for $\beta$, and $F^\tau_{g,y}=\bigoplus_bL_g\tau_{b,y}L_g^\dagger$. Here $\mathcal D(R,S)$ is infinite unless both matrices are positive semidefinite with trace $1$ and $\ker S\subseteq\ker R$; otherwise it is $\max\{0,\operatorname{Re}\operatorname{Tr}(f(R)-R\log S)\}$, with $f(t)=t\log t$ and continuous functional calculus. The term $C(a,b;A,B)$ is zero for $a\le0$, infinite for $a>0,b\le0$, and $a\mathcal D(A/a,B/b)$ for $a,b>0$. Neither $\rho$ nor $\sigma$ is assumed normalized or positive, the protocol need not satisfy $\sum_wK_w^\dagger K_w=I_d$, and normalization uses each branch’s own mass $M_\tau$, with $0^{-1}=0$. The index $w$ requires $m>0$; extended arithmetic includes $0\cdot(+\infty)=0$.

QAlgorithms.Papers.Independent.Intakethird100.P36.entropyScore

For arbitrary natural numbers $d,r,s$, arbitrary families $\rho,\sigma$ of complex $d\times d$ matrices indexed by $a<r,x<s$, and a strategy $T$ consisting of $h$ outcomes, matrices $K_g$ with $\sum_gK_g^\dagger K_g=I_d$, and weights $u_{g,x}\ge0$ with $\sum_xu_{g,x}=1$, define $B_\tau=s^{-1}\sum_{x,a}\tau_{a,x}$, $p_g=\operatorname{Re}\operatorname{Tr}(K_gB_\rho K_g^\dagger)$, and $q_g=\operatorname{Re}\operatorname{Tr}(K_gB_\sigma K_g^\dagger)$. Let $F^\tau_{g,x}$ be the block diagonal matrix with blocks $K_g\tau_{a,x}K_g^\dagger$ indexed by $a<r$. The score is

$$\mathcal D(\operatorname{diag}p,\operatorname{diag}q) +\sum_{g<h}\sum_{x<s}\max\{u_{g,x},0\}\, C(p_g,q_g;F^\rho_{g,x},F^\sigma_{g,x}).$$

Here $\mathcal D(R,S)=+\infty$ unless $R,S$ are positive semidefinite with trace $1$ and $\ker S\subseteq\ker R$; otherwise it is $\max\{0,\operatorname{Re}\operatorname{Tr}(f(R)-R\log S)\}$, with $f(t)=t\log t$ and continuous functional calculus. The quantity $C(p,q;A,B)$ is $0$ if $p\le0$, $+\infty$ if $p>0,q\le0$, and $p\mathcal D(A/p,B/q)$ if $p,q>0$. All arithmetic in the score is extended nonnegative arithmetic, including $0\cdot(+\infty)=0$. No assemblage or normalization assumptions are imposed on $\rho,\sigma$; zero dimensions, empty sums, and the total reciprocal $0^{-1}=0$ are included.

QAlgorithms.Papers.Independent.Intakethird100.P36.filteredCQ

For natural numbers $d,r,s$, let $T$ have $h$ outcomes, complex $d\times d$ matrices $K_g$ with $\sum_gK_g^\dagger K_g=I_d$, and real input weights $u_{g,x}\ge0$ with $\sum_{x<s}u_{g,x}=1$ for each outcome. Given an arbitrary family $\rho=(\rho_{a,y})_{a<r,y<s}$ of complex $d\times d$ matrices, an outcome $g<h$, and an input $x<s$, this defines a complex matrix indexed by pairs $(a,i)\in\{0,\ldots,r-1\}\times\{0,\ldots,d-1\}$, whose entries are

$$F_{(a,i),(b,j)} = \begin{cases} (K_g\rho_{a,x}K_g^\dagger)_{i,j},&a=b,\\ 0,&a\ne b. \end{cases}$$

The input weights do not enter this formula. No positivity, normalization, or consistency condition is imposed on $\rho$. The binders $g$ and $x$ require existing outcome and input indices; $r=0$ or $d=0$ still permits an empty resulting matrix.

QAlgorithms.Papers.Independent.Intakethird100.P36.measureBranch

Let $S$ be any real-valued function on arrays of complex matrices of every natural-number dimension. For natural numbers $d,r,s,d_f,r_f,s_f,m$, let $P$ be a protocol with all six dimensions positive, matrices $K_w\in\mathbb C^{d_f\times d}$ satisfying $I_d-\sum_{w<m}K_w^\dagger K_w\succeq0$, and real arrays $u_{w,x_f,x},v_{w,a_f,a,x,x_f}$ that are nonnegative and sum to $1$ over $x<s$ and $a_f<r_f$, respectively, for every choice of their other indices. Given any array $\rho=(\rho_{ax})_{a<r,x<s}$ of complex $d\times d$ matrices and any $w<m$, put

$$b_\rho=\frac1s\sum_x\sum_a\rho_{ax},\qquad p_w=\operatorname{Re}\operatorname{tr}(K_wb_\rho K_w^\dagger),\qquad \tau^{(w)}_{a_fx_f}=\sum_x\sum_a v_{w,a_f,a,x,x_f}\,u_{w,x_f,x}\,K_w\rho_{ax}K_w^\dagger, \qquad M_w=\operatorname{Re}\operatorname{tr}\!\left(\frac1{s_f}\sum_{x_f}\sum_{a_f}\tau^{(w)}_{a_fx_f}\right).$$

The defined value is $p_w\,S(d_f,r_f,s_f,(M_w^{-1}\tau^{(w)}_{a_fx_f})_{a_f,x_f})$ if $p_w>0$, and is $0$ otherwise. No validity or normalization condition is imposed on $\rho$, and $S$ is unrestricted. In particular, zero and negative $p_w$ both select the zero value, and any zero mass in the rescaling uses $0^{-1}=0$.

QAlgorithms.Papers.Independent.Intakethird100.P36.normalizeAssemblage

For any natural numbers $d,r,s$ and any array $\rho=(\rho_{ax})_{a<r,x<s}$ of complex $d\times d$ matrices, define

$$b_\rho=\frac1s\sum_{x<s}\sum_{a<r}\rho_{ax}, \qquad M_\rho=\operatorname{Re}\operatorname{tr}(b_\rho).$$

The resulting array has entries $(M_\rho^{-1})\rho_{ax}$, with the real scalar regarded as complex. No positivity, normalization, or nonzero-mass assumption is imposed. In particular, inversion at zero is defined to be zero, so $M_\rho=0$ produces the zero array. Zero dimensions and empty index sets are included; when $s=0$, the scalar $1/s$ is also zero.

QAlgorithms.Papers.Independent.Intakethird100.P36.oneWayFormula

For natural numbers $d,r,s,d_f,r_f,s_f,m$, take a protocol with positive dimensions $d,r,s,d_f,r_f,s_f$, matrices $K_w\in\mathbb C^{d_f\times d}$ satisfying $\sum_wK_w^\dagger K_w\preceq I_d$, nonnegative input weights $I_{w,x_f,x}$ summing to one over $x<s$, and nonnegative output weights $O_{w,a_f,a,x,x_f}$ summing to one over $a_f<r_f$, for every choice of their other indices. For any family $\rho_{a,x}\in\mathbb C^{d\times d}$, $a<r,x<s$, this defines a family of complex $d_f\times d_f$ matrices, indexed by $a_f<r_f,x_f<s_f$, through

$$F(\rho)_{a_f,x_f} =\sum_{w=0}^{m-1}\sum_{a=0}^{r-1}\sum_{x=0}^{s-1} \left(I_{w,x_f,x}O_{w,a_f,a,x,x_f}\right) K_w\rho_{a,x}K_w^\dagger,$$

where the real product in parentheses is interpreted as a complex scalar. The input family is arbitrary, and protocol completeness is not assumed. If $m=0$, the sum gives the zero family.

QAlgorithms.Papers.Independent.Intakethird100.P36.protocolAction

For natural numbers $d,r,s,d_f,r_f,s_f,m$, take a protocol with positive dimensions $d,r,s,d_f,r_f,s_f$, matrices $K_w\in\mathbb C^{d_f\times d}$ satisfying $\sum_wK_w^\dagger K_w\preceq I_d$, nonnegative input weights $I_{w,x_f,x}$ normalized over $x<s$, and nonnegative output weights $O_{w,a_f,a,x,x_f}$ normalized over $a_f<r_f$, for every choice of their other indices. For any family $\rho_{a,x}\in\mathbb C^{d\times d}$, $a<r,x<s$, this defines the complex $d_f\times d_f$ blocks

$$\mathcal P(\rho)_{a_f,x_f} =\sum_{w=0}^{m-1}\left( \sum_{x=0}^{s-1}\sum_{a=0}^{r-1} O_{w,a_f,a,x,x_f} \left(I_{w,x_f,x}\left(K_w\rho_{a,x}K_w^\dagger\right)\right)\right), \quad a_f<r_f,\ x_f<s_f.$$

No assemblage or normalization condition is imposed on $\rho$, and no equality $\sum_wK_w^\dagger K_w=I_d$ is required. For $m=0$, every resulting block is zero.

QAlgorithms.Papers.Independent.Intakethird100.P36.protocolBranch

For natural numbers $d,r,s,d_f,r_f,s_f,m$, take a protocol with positive dimensions $d,r,s,d_f,r_f,s_f$, matrices $K_w\in\mathbb C^{d_f\times d}$ satisfying $\sum_wK_w^\dagger K_w\preceq I_d$, nonnegative weights $I_{w,x_f,x}$ summing to one over $x<s$, and nonnegative weights $O_{w,a_f,a,x,x_f}$ summing to one over $a_f<r_f$, for every choice of the remaining indices. For any family $\rho_{a,x}\in\mathbb C^{d\times d}$, $a<r,x<s$, and any $w<m$, this defines the family indexed by $a_f<r_f,x_f<s_f$ whose blocks are

$$B^{(w)}_{a_f,x_f} =\sum_{x=0}^{s-1}\sum_{a=0}^{r-1} O_{w,a_f,a,x,x_f}\left(I_{w,x_f,x} \left(K_w\rho_{a,x}K_w^\dagger\right)\right),$$

with real weights acting as complex scalars. The family $\rho$ is arbitrary and need not satisfy assemblage or normalization conditions; when $m=0$, there is no admissible $w$.

QAlgorithms.Papers.Independent.Intakethird100.P36.relativeEntropyMeasure

This defines a real-valued function of every triple of natural numbers $d,r,s$ and every family $\rho=(\rho_{a,x})_{a<r,x<s}$ of complex $d\times d$ matrices by taking the real conversion of the following extended nonnegative infimum. Minimize $H(\rho,\sigma)$ over families $\sigma$ with $d,r,s>0$, positive semidefinite blocks, input-independent sums $\sum_a\sigma_{a,x}$, mass $\operatorname{Re}\operatorname{Tr}(s^{-1}\sum_{x,a}\sigma_{a,x})=1$, and a finite representation $\sigma_{a,x}=\sum_{\ell<n}v_\ell t_{\ell,x,a}\xi_\ell$, where $v_\ell\ge0$, $\sum_\ell v_\ell=1$, each $\xi_\ell$ is positive semidefinite with trace $1$, and $t_{\ell,x,a}\ge0$, $\sum_a t_{\ell,x,a}=1$. Here $H(\rho,\sigma)$ is the supremum, over finite families $K_g$ with $\sum_gK_g^\dagger K_g=I_d$ and weights $u_{g,x}\ge0$, $\sum_xu_{g,x}=1$, of

$$\mathcal D(\operatorname{diag}p,\operatorname{diag}q) +\sum_{g,x}u_{g,x}\,C(p_g,q_g;F^\rho_{g,x},F^\sigma_{g,x}),$$

where $B_\tau=s^{-1}\sum_{x,a}\tau_{a,x}$, $p_g=\operatorname{Re}\operatorname{Tr}(K_gB_\rho K_g^\dagger)$, $q_g=\operatorname{Re}\operatorname{Tr}(K_gB_\sigma K_g^\dagger)$, and $F^\tau_{g,x}=\bigoplus_aK_g\tau_{a,x}K_g^\dagger$. The divergence $\mathcal D(R,S)$ is $+\infty$ unless both matrices are positive semidefinite of trace $1$ and $\ker S\subseteq\ker R$; otherwise it is $\max\{0,\operatorname{Re}\operatorname{Tr}(f(R)-R\log S)\}$, with $f(t)=t\log t$ and continuous functional calculus. The conditional term $C$ is zero when $p\le0$, infinite when $p>0,q\le0$, and $p\mathcal D(A/p,B/q)$ when $p,q>0$. Real conversion preserves finite values and sends $+\infty$ to $0$. Thus an empty minimizing set, including any case with $d=0$, $r=0$, or $s=0$, yields the real value $0$. No assemblage assumption on $\rho$, monotonicity property, or attainment condition is part of this definition.

QAlgorithms.Papers.Independent.Intakethird100.P36.steerableWeight

For arbitrary natural numbers $d,r,s$ and a family $\rho=(\rho_{a,x})_{a<r,\ x<s}$ of complex $d\times d$ matrices, the steerable weight is the real infimum of all $\nu\geq0$ for which there exist normalized families $\eta,\sigma$ of the same dimensions satisfying $\rho_{a,x}=\nu\eta_{a,x}+(1-\nu)\sigma_{a,x}$ for every $a,x$, with $\sigma$ admitting a finite hidden-state representation. Here a normalized family has $d,r,s>0$, positive semidefinite blocks, sums $\sum_a\rho_{a,x}$ independent of $x$, and $\operatorname{Re}\operatorname{tr}B(\rho)=1$, where $B(\rho)=s^{-1}\sum_x\sum_a\rho_{a,x}$. The required representation of the normalized $\sigma$ consists of a natural number $h$, probabilities $p_\ell\geq0$ summing to $1$, positive semidefinite matrices $\xi_\ell$ of trace $1$, and response probabilities $q_{\ell,x}(a)\geq0$ summing to $1$ over $a$, such that $\sigma_{a,x}=\sum_{\ell<h}p_\ell q_{\ell,x}(a)\xi_\ell$. There is no explicit restriction $\nu\leq1$ or assumption that $\rho$ is an assemblage. If the candidate set is empty, its real infimum is defined to be $0$; this occurs in particular if any of $d,r,s$ is zero.

QAlgorithms.Papers.Independent.Intakethird100.P36.steeringBranchProbability

For natural numbers $d,r,s,d_f,r_f,s_f,m$, let $\rho=(\rho_{ax})_{a<r,x<s}$ be any array of complex $d\times d$ matrices, and let $P$ be a protocol with all six dimensions $d,r,s,d_f,r_f,s_f$ positive. Its data are matrices $K_w\in\mathbb C^{d_f\times d}$, indexed by $w<m$, satisfying $I_d-\sum_wK_w^\dagger K_w\succeq0$, and real arrays $u_{w,x_f,x}$ and $v_{w,a_f,a,x,x_f}$, where for each $w,x_f$, the numbers $u_{w,x_f,x}$ are nonnegative and sum to $1$ over $x<s$, and for each $w,a,x,x_f$, the numbers $v_{w,a_f,a,x,x_f}$ are nonnegative and sum to $1$ over $a_f<r_f$. For any $w<m$, the defined real number is

$$\operatorname{Re}\operatorname{tr}\!\left( K_w\left(\frac1s\sum_{x<s}\sum_{a<r}\rho_{ax}\right)K_w^\dagger \right).$$

No positivity, normalization, or equality of the sums $\sum_a\rho_{ax}$ is required of $\rho$; consequently, the definition does not itself require this real number to lie in $[0,1]$. The protocol’s arrays $u,v$ do not enter this expression.

QAlgorithms.Papers.Independent.Intakethird100.P36.steeringClassicalDivergence

For every natural number $n$, including $0$, and arbitrary real functions $p,q:\{0,\ldots,n-1\}\to\mathbb R$, this defines an extended nonnegative real number as follows. Put $P=\operatorname{diag}(p_i)$ and $Q=\operatorname{diag}(q_i)$, regarded as complex matrices. If both matrices are positive semidefinite with trace $1$, and $\ker Q\subseteq\ker P$, the value is

$$\max\!\left\{0,\ \operatorname{Re}\operatorname{Tr}\bigl(f(P)-P\log Q\bigr)\right\}, \qquad f(t)=t\log t,$$

where functions of matrices use continuous functional calculus and the scalar logarithm has its total convention $\log 0=0$. Otherwise the value is $+\infty$. Thus the density conditions require $p_i,q_i\ge0$ and $\sum_i p_i=\sum_i q_i=1$, and the kernel condition requires $q_i=0\Rightarrow p_i=0$. No such conditions are assumed on the arguments beforehand. When $n=0$, the trace condition fails and the value is $+\infty$.

QAlgorithms.Papers.Independent.Intakethird100.P36.steeringEntropy

For arbitrary natural numbers $d,r,s$ and any family $\rho=(\rho_{a,x})_{a<r,x<s}$ of complex $d\times d$ matrices, this defines the infimum in $[0,+\infty]$, over families $\sigma$ described below, of a quantity $H(\rho,\sigma)$. An admissible $\sigma$ requires $d,r,s>0$, positive semidefinite blocks, equal sums $\sum_a\sigma_{a,x}$ for all inputs $x$, and mass $\operatorname{Re}\operatorname{Tr}(s^{-1}\sum_{x,a}\sigma_{a,x})=1$; it must also admit a finite model: for some natural $n$, numbers $v_\ell\ge0$ summing to $1$, positive semidefinite matrices $\xi_\ell$ of trace $1$, and response numbers $t_{\ell,x,a}\ge0$ summing to $1$ over $a$, one has $\sigma_{a,x}=\sum_{\ell<n}v_\ell t_{\ell,x,a}\xi_\ell$. To define $H(\rho,\sigma)$, take the supremum over natural $h$, matrices $K_g$ with $\sum_gK_g^\dagger K_g=I_d$, and weights $u_{g,x}\ge0$ with $\sum_xu_{g,x}=1$, of

$$\mathcal D(\operatorname{diag}p,\operatorname{diag}q) +\sum_{g,x}u_{g,x}\,C(p_g,q_g;F^\rho_{g,x},F^\sigma_{g,x}),$$

where $B_\tau=s^{-1}\sum_{x,a}\tau_{a,x}$, $p_g=\operatorname{Re}\operatorname{Tr}(K_gB_\rho K_g^\dagger)$, $q_g=\operatorname{Re}\operatorname{Tr}(K_gB_\sigma K_g^\dagger)$, and $F^\tau_{g,x}=\bigoplus_aK_g\tau_{a,x}K_g^\dagger$. Here $\mathcal D(R,S)$ is infinite unless $R,S$ are positive semidefinite of trace $1$ and $\ker S\subseteq\ker R$; otherwise it is $\max\{0,\operatorname{Re}\operatorname{Tr}(f(R)-R\log S)\}$, with $f(t)=t\log t$ and continuous functional calculus. The term $C(p,q;A,B)$ is zero for $p\le0$, infinite for $p>0,q\le0$, and $p\mathcal D(A/p,B/q)$ for $p,q>0$. There is no assumption on $\rho$, and no attainment is required for either extremum. An empty admissible set has infimum $+\infty$, in particular whenever any of $d,r,s$ is zero; an empty score set has supremum $0$, and extended arithmetic uses $0\cdot(+\infty)=0$.

QAlgorithms.Papers.Independent.Intakethird100.P36.steeringRobustness

For arbitrary natural numbers $d,r,s$ and a family $\rho=(\rho_{a,x})_{a<r,\ x<s}$ of complex $d\times d$ matrices, the steering robustness is the real infimum of the numbers $\nu\geq0$ for which there exists a normalized family $\eta$ such that $Y_{a,x}=(\rho_{a,x}+\nu\eta_{a,x})/(1+\nu)$ has a finite hidden-state model. For a family $F$, write $\mu(F)=\operatorname{Re}\operatorname{tr}(s^{-1}\sum_x\sum_aF_{a,x})$. Normalization of $\eta$ means that $d,r,s>0$, its blocks are positive semidefinite, its outcome sums are independent of the setting, and $\mu(\eta)=1$. The hidden-state condition on $Y$ means that its blocks are positive semidefinite, its outcome sums are setting-independent, $0\leq\mu(Y)\leq1$, and there are finitely many density matrices $\xi_\ell$, probabilities $p_\ell$ summing to $1$, and response probabilities $q_{\ell,x}(a)$ summing to $1$ over $a$, with $Y_{a,x}=\mu(Y)\sum_\ell p_\ell q_{\ell,x}(a)\xi_\ell$. Each density matrix is positive semidefinite with trace $1$. The definition assumes no such conditions on $\rho$ itself and places no upper bound on $\nu$. An empty candidate set has real infimum $0$, including when any dimension $d,r,s$ is zero.

QAlgorithms.Papers.Independent.Intakethird100.P36.strategyProbability

For natural numbers $d,r,s$, let $T$ consist of a natural outcome count $h$, complex $d\times d$ matrices $K_g$ satisfying $\sum_gK_g^\dagger K_g=I_d$, and real input weights $u_{g,x}\ge0$ satisfying $\sum_{x<s}u_{g,x}=1$ for each $g<h$. For any family $\rho=(\rho_{a,x})_{a<r,x<s}$ of complex $d\times d$ matrices and any outcome $g<h$, this defines the real number

$$\operatorname{Re}\operatorname{Tr}\!\left( K_g\left(\frac1s\sum_{x=0}^{s-1}\sum_{a=0}^{r-1}\rho_{a,x}\right)K_g^\dagger \right).$$

The input weights are part of $T$ but do not enter this formula. There is no assumption that $\rho$ is positive, normalized, or has equal outcome sums across inputs; consequently no probability bounds are asserted. Empty sums are zero, and the reciprocal of $0$ is interpreted as $0$.