Definitions for I-concurrence and tangle for isotropic states

QAlgorithms.Papers.Independent.Intakethird100.P21.IsConvexEnvelope

For any set $D\subseteq\mathbb R$ and functions $f,g:\mathbb R\to\mathbb R$, this predicate requires that $D$ be convex, that $g$ be convex on $D$, that $g(x)\le f(x)$ for every $x\in D$, and that every function $h:\mathbb R\to\mathbb R$ which is convex on $D$ and satisfies $h(x)\le f(x)$ throughout $D$ also satisfy $h(x)\le g(x)$ throughout $D$. It imposes no restrictions on function values outside $D$. For $D=\varnothing$, the predicate holds for every $f$ and $g$.

QAlgorithms.Papers.Independent.Intakethird100.P21.IsMaximallyEntangled

For any natural numbers $a,b$ and any vector $\psi\in\mathbb C^{\{0,\ldots,a-1\}\times\{0,\ldots,b-1\}}$, put $m=\min(a,b)$. This predicate means that there exist real numbers $p_i\ge0$ for $0\le i<m$, summing to one, and orthonormal families $u_i\in\mathbb C^a$, $v_i\in\mathbb C^b$, such that

$$\psi_{(x,y)}=\sum_{i=0}^{m-1}\sqrt{p_i}\,(u_i)_x(v_i)_y, \qquad p_i=\frac1m\quad(0\le i<m).$$

If $m=0$, the empty probability sum cannot equal one, so the predicate is false.

QAlgorithms.Papers.Independent.Intakethird100.P21.IsOptimalPureEnsemble

For any finite type $I$, any real-valued function $c$ on $\mathbb C^I$, any complex matrix $\rho$ indexed by $I$, and any supplied finite ensemble $e=(K,w,\psi)$ whose weights are nonnegative and sum to one and whose vectors all have norm one, this predicate means

$$\rho=\sum_{k=0}^{K-1}w_k\psi_k\psi_k^\dagger, \qquad \sum_{k=0}^{K-1}w_k c(\psi_k) = \inf_{\substack{(L,p,\phi):\ p_\ell\ge0,\ \sum_\ell p_\ell=1\\ \|\phi_\ell\|=1,\ \rho=\sum_\ell p_\ell\phi_\ell\phi_\ell^\dagger}} \sum_{\ell=0}^{L-1}p_\ell c(\phi_\ell),$$

where each matrix $\phi_\ell\phi_\ell^\dagger$ has entries $(\phi_\ell)_i\overline{(\phi_\ell)_j}$. Thus the supplied ensemble must reproduce $\rho$ and have cost equal to this real infimum. No boundedness assumption is imposed on $c$; if the set of competing costs is unbounded below, the real infimum convention used here gives $0$. The supplied ensemble necessarily has $K>0$, and no such ensemble exists when $I$ is empty.

QAlgorithms.Papers.Independent.Intakethird100.P21.IsPureProductState

For any finite types $I,J$ and any vector $\psi\in\mathbb C^{I\times J}$, this predicate means that there exist vectors $a\in\mathbb C^I$ and $b\in\mathbb C^J$, each of norm one, such that

$$\psi_{(i,j)}=a_i b_j \quad\text{for every }i\in I,\ j\in J.$$

If either type is empty, the predicate cannot hold because its corresponding vector space has no norm-one vector.

QAlgorithms.Papers.Independent.Intakethird100.P21.ScalarMixture

For any real endpoints $\mathrm{lo}$ and $\mathrm{hi}$, an object of this structure consists of a natural number $q$, real weights $w_i$, and real points $x_i$, indexed by $i=0,\ldots,q-1$, together with the requirements $w_i\ge0$, $\sum_iw_i=1$, and $\mathrm{lo}\le x_i\le\mathrm{hi}$ for every $i$. Zero weights and repeated points are permitted, and the point restriction applies even at zero weights. The normalization excludes $q=0$; if $\mathrm{lo}>\mathrm{hi}$, no such object exists.

QAlgorithms.Papers.Independent.Intakethird100.P21.SchmidtProbabilities

For every natural number $d$, this structure consists of a function $p:\{0,\ldots,d-1\}\to\mathbb R$, together with the requirements

$$p_i\ge0\quad\text{for every }i, \qquad \sum_{i=0}^{d-1}p_i=1.$$

There are no such structures for $d=0$, since the empty sum is zero.

QAlgorithms.Papers.Independent.Intakethird100.P21.canonicalVector

For every natural number $d$ and every family $\mu=(\mu_i)_{i=0}^{d-1}$ of nonnegative real numbers with $\sum_i\mu_i=1$, this defines the complex Euclidean vector indexed by pairs $i,j\in\{0,\ldots,d-1\}$ whose $(i,j)$-coordinate is $\sqrt{\mu_i}$ when $i=j$, and $0$ otherwise. For $d=0$, no such family $\mu$ exists, since its empty sum would have to equal $1$.

QAlgorithms.Papers.Independent.Intakethird100.P21.concurrenceFormula

For every natural number $d$ and every real $F$, this defines

$$\begin{cases} 0,&F\le 1/d,\\[2pt] \displaystyle\sqrt{\frac{2d}{d-1}}\left(F-\frac1d\right),&F>1/d. \end{cases}$$

No positivity assumption on $d$ and no interval restriction on $F$ is imposed. Subtraction is in $\mathbb R$, division by zero is zero, and real square roots use their total-function convention. Consequently, for $d=0$ or $d=1$, the value is zero for every $F$.

QAlgorithms.Papers.Independent.Intakethird100.P21.constrainedSchmidtTangle

For every natural number $d$ and real number $F$, this defines the real infimum of all numbers $2(1-\sum_{i=0}^{d-1}\mu_i^2)$ obtained from nonnegative real families $(\mu_i)_{i=0}^{d-1}$ satisfying $\sum_i\mu_i=1$ and $\sum_i\sqrt{\mu_i}=\sqrt{dF}$. No range assumption on $F$ is required. If there is no such family—including when $d=0$, or when $d>0$ and $F\notin[1/d,1]$—the real infimum used here is $0$.

QAlgorithms.Papers.Independent.Intakethird100.P21.explicitQ

For every natural number $d$ and real number $F$, this defines

$$2\left[ 1-\left(\sqrt{\frac Fd}+\sqrt{\frac{(d-1)(1-F)}d}\right)^4 -(d-1)\left(\sqrt{\frac Fd}-\sqrt{\frac{1-F}{d(d-1)}}\right)^4 \right].$$

All occurrences of $d$ in this expression are real-valued, division by zero is $0$, and square roots of negative numbers are $0$. No range or sign condition is imposed on the result; in particular, its value at $d=0$ is $2$.

QAlgorithms.Papers.Independent.Intakethird100.P21.fidelityProfile

For every natural number $d$, every supplied probability vector $p_i\ge0$ with $\sum_{i=0}^{d-1}p_i=1$, and every complex $d\times d$ matrix $V$, this defines

$$\frac1d\left|\sum_{k=0}^{d-1}\sqrt{p_k}\,V_{kk}\right|^2.$$

The matrix $V$ is not required to be unitary. Although $d=0$ is included in the natural-number parameter, no probability vector satisfying the required conditions exists in that case.

QAlgorithms.Papers.Independent.Intakethird100.P21.hullValues

For every natural number $d$, function $f:\mathbb R\to\mathbb R$, and real number $F$, this defines the set of real numbers $v$ for which there exist a natural number $q$, nonnegative real weights $w_i$ summing to $1$, and points $x_i\in[1/d,1]$, indexed by $i=0,\ldots,q-1$, such that $\sum_iw_ix_i=F$ and $v=\sum_iw_if(x_i)$. No regularity assumption is imposed on $f$. The weight normalization excludes $q=0$. Since $1/0=0$, the permitted interval is $[0,1]$ when $d=0$; when $d=1$, it is the singleton $\{1\}$. The set is empty if $F$ lies outside the permitted interval.

QAlgorithms.Papers.Independent.Intakethird100.P21.isotropicDensityAtFidelity

For every natural number $d$ and every real number $F$, this defines a complex matrix indexed by $\{0,\ldots,d-1\}^2$. For $d>0$, let $P_d$ have entry $1/d$ between $(i,j)$ and $(k,l)$ when $i=j$ and $k=l$, and entry zero otherwise. The defined matrix is

$$F P_d+\frac{1-F}{d^2-1}(I-P_d).$$

For $d=0$, it is the zero matrix. No restriction on $F$ is imposed. Division is totalized with division by zero equal to zero, so for $d=1$ the second term vanishes and the matrix is $F I$.

QAlgorithms.Papers.Independent.Intakethird100.P21.isotropicReferenceVector

For every natural number $d$, this defines a vector $\Omega_d\in\mathbb C^{\{0,\ldots,d-1\}^2}$. For $d>0$, its coordinates are

$$(\Omega_d)_{(i,j)}= \begin{cases} 1/\sqrt d,&i=j,\\ 0,&i\ne j. \end{cases}$$

For $d=0$, it is the zero vector in the space indexed by the empty set.

QAlgorithms.Papers.Independent.Intakethird100.P21.marginalTangle

For any finite type $I$ and any complex matrix $A$ indexed by $I\times I$, this defines the real number

$$2\bigl(1-\operatorname{Re}\operatorname{tr}(A^2)\bigr).$$

No positivity, Hermitian, or trace normalization assumption is imposed on $A$. If $I$ is empty, the trace is zero and the value is $2$.

QAlgorithms.Papers.Independent.Intakethird100.P21.maximallyEntangledOverlap

For every natural number $d$ and every complex matrix $\rho$ indexed by $\{0,\ldots,d-1\}^2$, this defines $\operatorname{Re}\operatorname{tr}(P_d\rho)$, where, for $d>0$, $P_d$ has entry $1/d$ between $(i,i)$ and $(j,j)$ and zero elsewhere. Equivalently, for $d>0$, the value is

$$\frac1d\operatorname{Re}\sum_{i,j=0}^{d-1}\rho_{(i,i),(j,j)}.$$

For $d=0$, the value is zero. No positivity or normalization assumption is imposed on $\rho$.

QAlgorithms.Papers.Independent.Intakethird100.P21.minimizingLargeAmplitude

For every natural number $d$ and real number $F$, this defines

$$\sqrt{\frac{F}{d}}+\sqrt{\frac{(d-1)(1-F)}{d}},$$

where $d$ is interpreted as a real number, so $d-1$ is real subtraction. There are no positivity or range assumptions on $d$ or $F$; division by zero is defined to be $0$, and the real square root of a negative number is $0$. Thus the value is $0$ when $d=0$, and $\sqrt F$ when $d=1$.

QAlgorithms.Papers.Independent.Intakethird100.P21.minimizingProbabilities

For every natural number $d$ and real number $F$, this defines a real-valued function on $i\in\{0,\ldots,d-1\}$ by

$$i\longmapsto \begin{cases} \left(\sqrt{\frac Fd}+\sqrt{\frac{(d-1)(1-F)}d}\right)^2,&i=0,\\ \left(\sqrt{\frac Fd}-\sqrt{\frac{1-F}{d(d-1)}}\right)^2,&i\ne0. \end{cases}$$

Here subtraction involving $d$ is real subtraction, division by zero is $0$, and square roots of negative numbers are $0$. No normalization condition is included in the definition. For $d=0$ this is the function on the empty index set; for $d=1$, its sole value is $(\sqrt F)^2$.

QAlgorithms.Papers.Independent.Intakethird100.P21.minimizingSmallAmplitude

For every natural number $d$ and real number $F$, this defines

$$\sqrt{\frac{F}{d}}-\sqrt{\frac{1-F}{d(d-1)}},$$

where $d$ is interpreted as a real number and $d-1$ is real subtraction. There are no positivity or range assumptions, division by zero is $0$, and real square roots of negative numbers are $0$. The value is therefore $0$ when $d=0$, and $\sqrt F$ when $d=1$; no nonnegativity of the resulting difference is required.

QAlgorithms.Papers.Independent.Intakethird100.P21.profileValues

For every natural number $d$ and real number $F$, this defines the set

$$\left\{\,2\left(1-\sum_{i=0}^{d-1}\mu_i^2\right)\;\middle|\; \mu_i\ge0,\quad \sum_{i=0}^{d-1}\mu_i=1,\quad \sum_{i=0}^{d-1}\sqrt{\mu_i}=\sqrt{dF}\,\right\}.$$

The parameter $F$ has no imposed range, and the real square root is $0$ on negative arguments. The set is empty when $d=0$; for $d>0$, it is empty unless $1/d\le F\le1$.

QAlgorithms.Papers.Independent.Intakethird100.P21.pureTangle

For any finite types $I,J$ and any vector $\psi\in\mathbb C^{I\times J}$, define the matrix $R$ indexed by $I\times I$ by

$$R_{ij}=\sum_{h\in J}\psi_{(i,h)}\overline{\psi_{(j,h)}}.$$

This definition assigns to $\psi$ the real number $2\bigl(1-\operatorname{Re}\operatorname{tr}(R^2)\bigr)$. The vector $\psi$ need not have norm one. If either index type is empty, $R$ is zero and the value is $2$.

QAlgorithms.Papers.Independent.Intakethird100.P21.rawC

For every natural number $d$ and real number $F$, this defines the real square root of the real infimum of

$$\left\{\,2\left(1-\sum_{i=0}^{d-1}\mu_i^2\right)\;\middle|\; \mu_i\ge0,\quad \sum_i\mu_i=1,\quad \sum_i\sqrt{\mu_i}=\sqrt{dF}\,\right\}.$$

The real infimum of an empty set is taken to be $0$, and the real square root of a negative number is also $0$. In particular, this value is $0$ when $d=0$ or when $d>0$ and $F\notin[1/d,1]$.

QAlgorithms.Papers.Independent.Intakethird100.P21.relaxedRoof

For every natural number $d$, every real-valued cost function $c$ on $\mathbb C^{d}\otimes\mathbb C^{d}$, and every real $F$, this defines the real infimum of all averages $\sum_kw_kc(\psi_k)$ over finite families of unit vectors $\psi_k$ with nonnegative weights summing to $1$ and satisfying $\sum_kw_k\operatorname{Re}\operatorname{tr}(|\Phi_d\rangle\langle\Phi_d|\,|\psi_k\rangle\langle\psi_k|)=F$, where $\Phi_d=d^{-1/2}\sum_{i<d}|i,i\rangle$ for $d>0$, and $\Phi_0=0$. There is no constraint on the ensemble’s average density matrix, and no regularity or boundedness assumption on $c$. This is a real-valued infimum: the convention gives $0$ if the set is empty or unbounded below. In particular, it is $0$ for $d=0$, for $F\notin[0,1]$, and for $d=1$ with $F\ne1$.

QAlgorithms.Papers.Independent.Intakethird100.P21.relaxedValues

For every natural number $d$, every function $c:\mathbb C^{d}\otimes\mathbb C^{d}\to\mathbb R$ on the corresponding Euclidean vector space, and every real number $F$, this defines the set of real numbers $v$ for which there exist a natural number $q$, weights $w_0,\ldots,w_{q-1}\ge0$ summing to $1$, and unit vectors $\psi_0,\ldots,\psi_{q-1}\in\mathbb C^{d}\otimes\mathbb C^{d}$, such that

$$\sum_{k=0}^{q-1}w_k\langle\Phi_d,|\psi_k\rangle\langle\psi_k|\Phi_d\rangle=F, \qquad v=\sum_{k=0}^{q-1}w_kc(\psi_k),$$

where, for $d>0$, $\Phi_d=d^{-1/2}\sum_{i=0}^{d-1}|i,i\rangle$, and for $d=0$ the reference vector is zero. Here $|\psi\rangle\langle\psi|$ has entries $\psi_i\overline{\psi_j}$, and the overlap means the real part of the trace of its product with $|\Phi_d\rangle\langle\Phi_d|$. No particular average density matrix is prescribed. Zero weights are allowed, but $q=0$ cannot satisfy the weight sum. For $d=0$ the set is empty because there are no unit vectors; for $d>0$, it is empty when $F\notin[0,1]$, and for $d=1$ it is empty unless $F=1$.

QAlgorithms.Papers.Independent.Intakethird100.P21.scalarConvexRoofAtFidelity

For every natural number $d$, function $f:\mathbb R\to\mathbb R$, and real number $F$, this defines the real infimum of all finite weighted sums $\sum_iw_if(x_i)$ with $w_i\ge0$, $\sum_iw_i=1$, $x_i\in[1/d,1]$, and $\sum_iw_ix_i=F$. No continuity, convexity, or boundedness assumption is imposed on $f$. Division by zero gives $1/0=0$, so the permitted interval for $d=0$ is $[0,1]$. If the set of weighted sums is empty or unbounded below, the real infimum used here is $0$.

QAlgorithms.Papers.Independent.Intakethird100.P21.schmidtCost

For every natural number $d$ and every function $\mu:\{0,\ldots,d-1\}\to\mathbb R$, this defines

$$2\left(1-\sum_{i=0}^{d-1}\mu_i^2\right).$$

No nonnegativity or sum-to-one condition is imposed on $\mu$. For $d=0$, the value is $2$.

QAlgorithms.Papers.Independent.Intakethird100.P21.spectralTrace

For every natural number $d$, every function $f:\mathbb R\to\mathbb R$, and every complex $d\times d$ matrix $\rho$, this defines $\operatorname{Re}\operatorname{tr}(f(\rho))$, where $f(\rho)$ is obtained using the real continuous functional calculus. For Hermitian $\rho$, this is $\sum_{i=0}^{d-1}f(\lambda_i)$, with eigenvalues counted with multiplicity. The definition assumes neither that $\rho$ is Hermitian, positive semidefinite, or trace one, nor that $f$ is globally continuous; the functional calculus’s total-function conventions apply outside its applicable hypotheses. The value is zero when $d=0$.

QAlgorithms.Papers.Independent.Intakethird100.P21.tangentPoint

For every natural number $d$, this defines the real number

$$\frac{4(d-1)}{d^2},$$

with all arithmetic performed in $\mathbb R$, so $d-1$ is real subtraction. The definition includes $d=0$ and $d=1$; its value is $0$ in both cases, using the convention that division by zero is zero.

QAlgorithms.Papers.Independent.Intakethird100.P21.tangle

For any finite types $I,J$ and any complex matrix $\rho$ indexed by $I\times J$, this defines the infimum of

$$\sum_{k=0}^{K-1}w_k\,2\bigl(1-\operatorname{Re}\operatorname{tr}(R_k^2)\bigr), \qquad (R_k)_{ij}=\sum_{h\in J}(\psi_k)_{(i,h)}\overline{(\psi_k)_{(j,h)}},$$

over all natural numbers $K$, real weights $w_k\ge0$ summing to one, and vectors $\psi_k\in\mathbb C^{I\times J}$ with $\|\psi_k\|=1$, subject to $\rho=\sum_k w_k\psi_k\psi_k^\dagger$, where $(\psi_k\psi_k^\dagger)_{xy}=(\psi_k)_x\overline{(\psi_k)_y}$. Every listed vector must be normalized, including those with zero weight; $K=0$ cannot satisfy the weight condition. No density matrix assumption is imposed on $\rho$. If there is no such ensemble, the real infimum of the empty set used here is $0$; in particular, this occurs if either index type is empty.

QAlgorithms.Papers.Independent.Intakethird100.P21.tangleFormula

For every natural number $d$ and every real $F$, define

$$u=\sqrt{\frac Fd}+\sqrt{\frac{(d-1)(1-F)}d}, \qquad v=\sqrt{\frac Fd}-\sqrt{\frac{1-F}{d(d-1)}}.$$

The declaration defines the real-valued function

$$\begin{cases} 0,&F\le 1/d,\\[2pt] 2\bigl(1-u^4-(d-1)v^4\bigr), &F>1/d\ \text{and}\ F\le 4(d-1)/d^2,\\[2pt] \dfrac{2d(F-1)}{d-1}+\dfrac{2(d-1)}d, &F>1/d\ \text{and}\ F>4(d-1)/d^2. \end{cases}$$

There is no restriction that $F\in[0,1]$ or that $d$ be positive. All subtractions here are real, division by zero is zero, and the real square root of a negative argument is zero. In particular, for $d=0$ or $d=1$, the resulting function is zero for every $F$.