Definitions for Pure-state transformations and catalysis under operations that completely preserve positivity of partial transpose

QAlgorithms.Papers.Independent.Intakethird100.P95.AntiIndex

For each natural number $d$, this abbreviates the type of ordered pairs $(i,j)$ of integers with $0\le i,j<d$ and $j<i$. Diagonal pairs are excluded, and the type is empty when $d=0$ or $d=1$.

QAlgorithms.Papers.Independent.Intakethird100.P95.CatalyticPreparation

For any finite type $I$, any natural number $K$, and any function $\lambda:I\to\mathbb R$, this predicate means that there exist a natural number $C\ge2$ and a complex-linear map $\mathcal L$ from matrices on $\{0,\ldots,C-1\}\times\{0,\ldots,K-1\}$ to matrices on $\{0,\ldots,C-1\}\times I$ such that $\mathcal L$ is completely positive, preserves the trace of every matrix, its conjugation by partial transpose on the second subsystem is completely positive, and

$$\mathcal L(\Omega_C\otimes\Omega_K)=\Omega_C\otimes D_\lambda.$$

The products here group the two first subsystems together and the two second subsystems together. Explicitly, $(\Omega_m)_{(i,j),(k,l)}=1/m$ if $i=j$ and $k=l$, and is zero otherwise; $(D_\lambda)_{(i,j),(k,l)}=\sqrt{\lambda_i\lambda_k}$ if $i=j$ and $k=l$, and is zero otherwise. Partial transpose sends $X_{(a,b),(c,d)}$ to $X_{(a,d),(c,b)}$; complete positivity requires every finite ancillary block amplification, including size zero, to preserve positive semidefiniteness. No nonnegativity or normalization of $\lambda$ is assumed. Empty $I$ and $K=0$ are included, and negative real square-root arguments are evaluated as zero.

QAlgorithms.Papers.Independent.Intakethird100.P95.CostRates

For a finite type $I$, an arbitrary function $\lambda:I\to\mathbb R$, and a Boolean flag $\mathrm{zeroError}$, this defines a set of extended real numbers containing precisely those $r$ for which there exist a finite real number $R$, sequences $n,K:\mathbb N\to\mathbb N$, and maps $\Phi_j$ from matrices on $\operatorname{Fin}(K_j)\times\operatorname{Fin}(K_j)$ to matrices on $I^{n_j}\times I^{n_j}$, such that $r=R$, every $n_j$ and $K_j$ is positive, $n_j\to+\infty$, and $\log(K_j)/(n_j\log2)\to R$. Each $\Phi_j$ must be complex-linear and trace preserving on every matrix, and both $\Phi_j$ and its composite with partial transposition on the second input and second output factors must be completely positive, meaning positivity preserving under every finite identity amplification, including dimension $0$. Define

$$\Omega_K\bigl((a,b),(c,d)\bigr)= \begin{cases} 1/K,&a=b\ \text{and}\ c=d,\\ 0,&\text{otherwise}, \end{cases} \qquad w_n(x)=\prod_{i=0}^{n-1}\lambda_{x_i},$$

and

$$\rho_{\lambda,n}\bigl((x,y),(u,v)\bigr)= \begin{cases} \sqrt{w_n(x)w_n(u)},&x=y\ \text{and}\ u=v,\\ 0,&\text{otherwise}. \end{cases}$$

If the flag is true, the requirement is $\Phi_j(\Omega_{K_j})=\rho_{\lambda,n_j}$ for every $j$; if it is false, the requirement is $\|\Phi_j(\Omega_{K_j})-\rho_{\lambda,n_j}\|_1\to0$, where $\|M\|_1=\operatorname{Re}\operatorname{tr}\sqrt{M^\dagger M}$. No nonnegativity or normalization assumption is imposed on $\lambda$, no monotonicity of the sequences is required, and neither infinite extended-real values nor an explicit restriction $R\ge0$ appears among the witnesses. If $I$ is empty, the set is empty because positive $n_j$ would make the output matrix space have identically zero trace, preventing trace preservation from the nonempty input space.

QAlgorithms.Papers.Independent.Intakethird100.P95.DistillationRates

For any finite type $I$, any function $\lambda:I\to\mathbb R$, and any Boolean $z$, this defines a set of extended real numbers consisting precisely of the finite numbers $R\in\mathbb R$ for which there are sequences of positive integers $n_j,K_j$, with $n_j\to\infty$ and $\log_2(K_j)/n_j\to R$, and maps $\mathcal L_j$ from matrices on $(I^{n_j})\times(I^{n_j})$ to matrices on $\{0,\ldots,K_j-1\}^2$, satisfying the following requirements. Each $\mathcal L_j$ is complex linear, trace preserving on every matrix, and completely positive, and its conjugation by partial transpose on the second subsystem is also completely positive; complete positivity means preservation of positive semidefiniteness after adjoining an identity map of every finite dimension, including dimension zero. Write $\lambda_n(u)=\prod_{i=0}^{n-1}\lambda(u_i)$, and define $D_{\lambda,n}$ by

$$(D_{\lambda,n})_{(u,v),(u',v')}= \begin{cases} \sqrt{\lambda_n(u)\lambda_n(u')},&u=v\text{ and }u'=v',\\ 0,&\text{otherwise}. \end{cases}$$

Define $B_K$ to have entry $1/K$ at $((a,b),(c,d))$ when $a=b$ and $c=d$, and zero otherwise. If $z$ is true, $\mathcal L_j(D_{\lambda,n_j})=B_{K_j}$ for every $j$; if $z$ is false, $\operatorname{tr}\sqrt{H_j^\dagger H_j}\to0$, where $H_j=B_{K_j}-\mathcal L_j(D_{\lambda,n_j})$, with the square root interpreted through functional calculus. No nonnegativity or normalization is assumed for $\lambda$, and real square roots of negative numbers are zero. The type $I$ may be empty, although all $n_j$ and $K_j$ must be positive; neither $+\infty$ nor $-\infty$ belongs to this set.

QAlgorithms.Papers.Independent.Intakethird100.P95.FaceStationary

For natural numbers $d,K$, any real vector $\lambda=(\lambda_i)_{i=0}^{d-1}$, and any subset $C\subseteq\{0,\ldots,d-1\}$, this predicate means that there exists a real vector $x$ such that $|x_i|\le\sqrt{\lambda_i}$ for every $i$, $x_i=\sqrt{\lambda_i}$ for every $i\in C$, and

$$\sum_{i=0}^{d-1}x_i-Kx_j=0 \qquad\text{for every }j\notin C.$$

There are no positivity or normalization assumptions on $\lambda$ and no restriction on $K$. Real square roots of negative inputs are defined to be zero. If $C$ is the entire index set, the final equations impose no conditions; if $d=0$, the predicate holds using the empty vector.

QAlgorithms.Papers.Independent.Intakethird100.P95.IsUniformRank

For any finite type $I$, equipped with a finite enumeration, any function $\lambda:I\to\mathbb R$, and any natural number $K$, this predicate means

$$\#\{i\in I:\lambda_i\ne0\}=K \quad\text{and}\quad \forall i\in I,\ \lambda_i=0\ \text{or}\ \lambda_i=K^{-1}.$$

For $K=0$, the inverse is zero and the predicate holds exactly when $\lambda$ is identically zero. For $K>0$, exactly $K$ entries equal $1/K$ and every other entry is zero. Empty $I$ is allowed and then the predicate holds exactly for $K=0$.

QAlgorithms.Papers.Independent.Intakethird100.P95.OnFace

For every natural number $d$, arbitrary real vectors $\lambda,x:\{0,\ldots,d-1\}\to\mathbb R$, and any subset $C\subseteq\{0,\ldots,d-1\}$, this predicate means

$$|x_i|\le\sqrt{\lambda_i}\quad\text{for every }i, \qquad x_i=\sqrt{\lambda_i}\quad\text{for every }i\in C.$$

No nonnegativity or normalization of $\lambda$ is assumed. The real square root of a negative number is zero, so $\lambda_i\le0$ forces $x_i=0$. If $C$ is empty, the equality requirements are vacuous; if $d=0$, the entire predicate holds vacuously.

QAlgorithms.Papers.Independent.Intakethird100.P95.PPTChannel

For finite types $A,B,C,D$, this structure consists of a complex-linear map $\Phi$ from matrices on $A\times B$ to matrices on $C\times D$, together with proofs that $\Phi$ is completely positive, preserves the complex trace of every matrix, and that $\Gamma_D\circ\Phi\circ\Gamma_B$ is completely positive. Here $(\Gamma_BX)_{(a,b),(a',b')}=X_{(a,b'),(a',b)}$, and $\Gamma_D$ is defined analogously on the output. Complete positivity means that, for every natural number $h$, including $0$, the amplification acting on each matrix block by the given map sends every positive semidefinite matrix on $\operatorname{Fin}(h)\times(A\times B)$ to a positive semidefinite matrix on $\operatorname{Fin}(h)\times(C\times D)$; the same condition applies to the transposed composite. The four finite types may be empty.

QAlgorithms.Papers.Independent.Intakethird100.P95.PreparationFeasible

For finite types $A,B$, possibly empty, a natural number $K$, and arbitrary complex matrices $X,P$ indexed by $A\times B$, this predicate requires $P$ to be Hermitian and positive semidefinite and requires

$$X^\Gamma+(K-1)P^\Gamma\succeq0, \qquad (K+1)P^\Gamma-X^\Gamma\succeq0,$$

where $M^\Gamma_{(a,b),(c,d)}=M_{(a,d),(c,b)}$, and $\succeq0$ means positive semidefinite. The subtraction $K-1$ is performed in the real numbers: $K=0$ gives the two conditions $X^\Gamma-P^\Gamma\succeq0$ and $P^\Gamma-X^\Gamma\succeq0$, while $K=1$ gives $X^\Gamma\succeq0$ and $2P^\Gamma-X^\Gamma\succeq0$. No trace normalization or separate hypothesis on $X$ is included.

QAlgorithms.Papers.Independent.Intakethird100.P95.RateLimit

For sequences $n,K:\mathbb N\to\mathbb N$ and a real number $R$, this predicate means

$$\lim_{j\to\infty}\frac{\log(K_j)/\log 2}{n_j}=R.$$

This predicate itself imposes no positivity, divergence, or monotonicity conditions on either sequence. The total-function conventions assign $\log 0=0$ and division by zero the value $0$, so terms with $K_j=0$ or $n_j=0$ are included and have value $0$.

QAlgorithms.Papers.Independent.Intakethird100.P95.Reach

For finite types $A,B,C,D$ and arbitrary complex matrices $X$ on $A\times B$ and $Y$ on $C\times D$, this predicate means that there exists a complex-linear map $\Phi$ with $\Phi(X)=Y$ which preserves the trace of every matrix and for which both $\Phi$ and $\Gamma_D\circ\Phi\circ\Gamma_B$ are completely positive. Here $\Gamma_B$ exchanges the two $B$ indices of a matrix entry, and $\Gamma_D$ exchanges the two $D$ indices. Complete positivity requires positivity preservation after amplification by the identity on $\operatorname{Fin}(h)$ for every $h\ge0$. Neither $X$ nor $Y$ is assumed to be positive semidefinite or normalized, and empty indexing types are allowed.

QAlgorithms.Papers.Independent.Intakethird100.P95.SchmidtDualFeasible

For every natural number $d$, and real arrays $\mu_{ij}$, $0\le j\le i<d$, and $t_{ij}$, $0\le j<i<d$, this predicate means that every $\mu_{ij}\le1$, every $t_{ij}\le1$, and the following matrix on the standard basis $|ij\rangle$ of $\mathbb C^{d\times d}$ is positive semidefinite:

$$C(\mu,t)= \sum_i\mu_{ii}|ii\rangle\langle ii| +\frac12\sum_{j<i}\left[ (\mu_{ij}+t_{ij})(|ij\rangle\langle ij|+|ji\rangle\langle ji|) +(\mu_{ij}-t_{ij})(|ii\rangle\langle jj|+|jj\rangle\langle ii|) \right].$$

The coordinate bounds contain no explicit lower bounds. For $d=0$, the arrays and matrix are empty and the predicate holds.

QAlgorithms.Papers.Independent.Intakethird100.P95.SchmidtPrimalFeasible

For any natural numbers $d,K$, arbitrary real function $\lambda:\{0,\ldots,d-1\}\to\mathbb R$, real coefficients $s_{ij}$ for $0\le j\le i<d$, and real coefficients $a_{ij}$ for $0\le j<i<d$, this predicate requires

$$s_{ij}\ge\frac{\sqrt{\lambda_i\lambda_j}}{K+1}\quad(j\le i), \qquad a_{ij}\ge\frac{\sqrt{\lambda_i\lambda_j}}{K-1}\quad(j<i),$$

and requires the following matrix to be positive semidefinite:

$$\sum_i s_{ii}e_{ii}e_{ii}^\dagger+ \frac12\sum_{j<i} \left[ (s_{ij}+a_{ij})(e_{ij}e_{ij}^\dagger+e_{ji}e_{ji}^\dagger) +(s_{ij}-a_{ij})(e_{ii}e_{jj}^\dagger+e_{jj}e_{ii}^\dagger) \right].$$

Here $e_{ij}$ is the coordinate vector at $(i,j)$, and $uv^\dagger$ has entries $u_p\overline{v_q}$. No nonnegativity or normalization assumption is imposed on $\lambda$, and a negative real square-root argument gives zero. The denominators use real subtraction and total division: at $K=1$, every antisymmetric lower bound is zero; at $K=0$, it is $-\sqrt{\lambda_i\lambda_j}$. When $d=0$, the coefficient conditions are vacuous and the empty matrix is positive semidefinite.

QAlgorithms.Papers.Independent.Intakethird100.P95.SymIndex

For each natural number $d$, this abbreviates the type of ordered pairs $(i,j)$ of integers with $0\le i,j<d$ and $j\le i$. Diagonal pairs are included, and the type is empty when $d=0$.

QAlgorithms.Papers.Independent.Intakethird100.P95.ValidLengths

For two sequences $n,K:\mathbb N\to\mathbb N$, this predicate means that $n_j>0$ and $K_j>0$ for every $j$, and that $n_j\to+\infty$ as $j\to+\infty$: for every natural bound, all sufficiently late $n_j$ meet or exceed that bound. Neither sequence is required to be monotone, and no growth condition is imposed on $K$.

QAlgorithms.Papers.Independent.Intakethird100.P95.antisymmetricSchmidtProjector

For any natural number $d$ and indices $0\le i,j<d$, this defines

$$A_{ij}=\frac12(e_{ij}-e_{ji})(e_{ij}-e_{ji})^\dagger,$$

where $e_{ij}$ is the coordinate vector equal to one at $(i,j)$ and zero elsewhere, and $vv^\dagger$ has entries $v_p\overline{v_q}$. There is no assumption that $i\ne j$, so $A_{ii}=0$; when $d=0$, there are no possible indices.

QAlgorithms.Papers.Independent.Intakethird100.P95.bobTransposeLinear

For arbitrary types $A$ and $B$, this defines a complex-linear endomorphism $\Gamma_B$ of the complex matrices indexed by $A\times B$, given by

$$(\Gamma_BX)\bigl((a,b),(a',b')\bigr) =X\bigl((a,b'),(a',b)\bigr).$$

Thus it exchanges the two $B$ indices while retaining the $A$ indices. No finiteness assumptions are required.

QAlgorithms.Papers.Independent.Intakethird100.P95.coordinateMatrix

For any natural number $d$, arbitrary real coefficients $s_{ij}$ indexed by $0\le j\le i<d$, and arbitrary real coefficients $a_{ij}$ indexed by $0\le j<i<d$, this defines the complex matrix

$$C(s,a)= \sum_{i=0}^{d-1}s_{ii}e_{ii}e_{ii}^\dagger+ \frac12\sum_{0\le j<i<d} \left[ (s_{ij}+a_{ij})(e_{ij}e_{ij}^\dagger+e_{ji}e_{ji}^\dagger) +(s_{ij}-a_{ij})(e_{ii}e_{jj}^\dagger+e_{jj}e_{ii}^\dagger) \right],$$

where $e_{ij}$ is the coordinate vector at $(i,j)$, and $uv^\dagger$ has entries $u_p\overline{v_q}$. Equivalently, the definition takes partial transpose on the second subsystem of the weighted symmetric and antisymmetric vector outer products. No sign conditions are imposed on the coefficients; for $d=0$, both sums are empty and give the zero matrix.

QAlgorithms.Papers.Independent.Intakethird100.P95.exampleSpectrum

This defines the real vector indexed by $\{0,\ldots,5\}$ whose entries, in index order, are

$$\left(\frac1{20},\frac1{20},\frac1{20}, \frac4{20},\frac4{20},\frac9{20}\right).$$

QAlgorithms.Papers.Independent.Intakethird100.P95.faceQuadratic

For arbitrary natural numbers $d,K$ and an arbitrary real vector $x=(x_0,\ldots,x_{d-1})$, this function is

$$\sum_{0\le j<i<d}\frac{x_i x_j}{K-1} -\sum_{0\le j\le i<d}\frac{x_i x_j}{K+1}.$$

There are no sign assumptions on $x$ and no lower bound on $K$. Division by zero gives zero, so when $K=1$ the entire first sum is zero. When $d=0$, both sums are empty and the value is zero.

QAlgorithms.Papers.Independent.Intakethird100.P95.faceValue

For natural numbers $d,K$, any real vector $\lambda=(\lambda_i)_{i=0}^{d-1}$, and any subset $C\subseteq\{0,\ldots,d-1\}$, put $D=K+|C|-d$, with this subtraction performed in $\mathbb R$. The defined real number is

$$\frac{K}{(K^2-1)D} \left[ \left(\sum_{i\in C}\sqrt{\lambda_i}\right)^2 -D\sum_{i\in C}\lambda_i \right].$$

No positivity, normalization, or stationarity condition is required. Real square roots of negative numbers are zero, and division by zero is defined to be zero; consequently the value is zero when $K=1$ or $D=0$, and also when $K=0$ or $C$ is empty. The definition includes $d=0$.

QAlgorithms.Papers.Independent.Intakethird100.P95.largestProbability

For any finite type $I$ and any function $\lambda:I\to\mathbb R$, this defines the real supremum of $\{\lambda_i:i\in I\}$. When $I$ is nonempty, this is its maximum; when $I$ is empty, the real supremum convention gives $0$. The entries are not assumed to be nonnegative or to sum to $1$.

QAlgorithms.Papers.Independent.Intakethird100.P95.pptDistillableEntanglement

For any finite type $I$, arbitrary $\lambda:I\to\mathbb R$, and Boolean $z$, this defines the extended real supremum of all finite real numbers $R$ for which positive integer sequences $n_j,K_j$ satisfy $n_j\to\infty$ and $\log_2(K_j)/n_j\to R$, and there exist complex linear, trace-preserving maps $\mathcal L_j$ from matrices on $I^{n_j}\times I^{n_j}$ to matrices on $\{0,\ldots,K_j-1\}^2$, with both $\mathcal L_j$ and its conjugation by partial transpose on the second subsystem completely positive. Complete positivity requires preservation of positive semidefiniteness after adjoining an identity map of every finite dimension. Put $\lambda_n(u)=\prod_{i=0}^{n-1}\lambda(u_i)$, define $D_{\lambda,n}$ to have entry $\sqrt{\lambda_n(u)\lambda_n(u')}$ when its indices are $((u,u),(u',u'))$ and zero elsewhere, and define $B_K$ to have entry $1/K$ when both its row pair and its column pair have equal coordinates and zero elsewhere. If $z$ is true, require $\mathcal L_j(D_{\lambda,n_j})=B_{K_j}$ for every $j$; otherwise require $\operatorname{Re}\operatorname{tr}\sqrt{H_j^\dagger H_j}\to0$, where $H_j=B_{K_j}-\mathcal L_j(D_{\lambda,n_j})$ and the square root uses functional calculus. There is no nonnegativity or normalization assumption on $\lambda$; negative real square-root arguments give zero, and $I$ may be empty. An empty set of rates has supremum $-\infty$, while an unbounded set has supremum $+\infty$.

QAlgorithms.Papers.Independent.Intakethird100.P95.pptEntanglementCost

For any finite type $I$, arbitrary $\lambda:I\to\mathbb R$, and Boolean $z$, this defines an extended real number as the infimum of all finite real rates $R$ with the following property: there are positive integer sequences $n_j,K_j$, with $n_j\to\infty$ and $\log_2(K_j)/n_j\to R$, and complex linear maps $\mathcal L_j$ from matrices on $\{0,\ldots,K_j-1\}^2$ to matrices on $I^{n_j}\times I^{n_j}$, such that each map preserves trace on every matrix and both the map and its conjugation by partial transpose on the second subsystem are completely positive. Here complete positivity requires positive semidefiniteness to be preserved after adjoining an identity map of every finite dimension. Let $B_K$ have entries $1/K$ when both its row pair and its column pair have equal coordinates, and zero otherwise; let $\lambda_n(u)=\prod_{i=0}^{n-1}\lambda(u_i)$, and let $D_{\lambda,n}$ have entry $\sqrt{\lambda_n(u)\lambda_n(u')}$ at $((u,v),(u',v'))$ when $u=v$ and $u'=v'$, and zero otherwise. For $z$ true, $\mathcal L_j(B_{K_j})=D_{\lambda,n_j}$ for every $j$; for $z$ false, the trace norm of $\mathcal L_j(B_{K_j})-D_{\lambda,n_j}$ tends to zero, where the trace norm of $H$ is $\operatorname{Re}\operatorname{tr}\sqrt{H^\dagger H}$, using functional calculus. No normalization or sign condition is imposed on $\lambda$, $I$ may be empty, and negative real square-root arguments give zero. The infimum is $+\infty$ if there are no such rates.

QAlgorithms.Papers.Independent.Intakethird100.P95.preparationValue

For finite types $A,B$, possibly empty, a natural number $K$, and an arbitrary complex matrix $X$ indexed by $A\times B$, this defines a real number as the infimum of

$$\left\{\operatorname{Re}\operatorname{tr}P: P=P^\dagger,\ P\succeq0,\ X^\Gamma+(K-1)P^\Gamma\succeq0,\ (K+1)P^\Gamma-X^\Gamma\succeq0\right\},$$

where $M^\Gamma_{(a,b),(c,d)}=M_{(a,d),(c,b)}$, and $K-1$ is real subtraction. There are no normalization assumptions on $X$ or $P$, and $K=0$ and $K=1$ are included. This is the real-valued infimum operation, which returns zero when the displayed set is empty; the definition does not assert attainment.

QAlgorithms.Papers.Independent.Intakethird100.P95.preparationValues

For finite types $A,B$, possibly empty, a natural number $K$, and an arbitrary complex matrix $X$ indexed by $A\times B$, this defines the set of real numbers $v$ for which there exists a Hermitian positive semidefinite matrix $P$ on the same index set satisfying

$$X^\Gamma+(K-1)P^\Gamma\succeq0, \qquad (K+1)P^\Gamma-X^\Gamma\succeq0, \qquad \operatorname{Re}\operatorname{tr}P=v.$$

Here $M^\Gamma_{(a,b),(c,d)}=M_{(a,d),(c,b)}$, the inequalities mean positive semidefiniteness, and $K-1$ is real subtraction, including the value $-1$ at $K=0$. Neither $X$ nor $P$ is required to have trace one; the set can be empty.

QAlgorithms.Papers.Independent.Intakethird100.P95.primalValue

For arbitrary natural numbers $d,K$ and arbitrary real numbers $\lambda_i$, $0\le i<d$, this is the real infimum

$$\inf\left\{ \sum_{j\le i}s_{ij}+\sum_{j<i}a_{ij}: \frac{\sqrt{\lambda_i\lambda_j}}{K+1}\le s_{ij},\ \frac{\sqrt{\lambda_i\lambda_j}}{K-1}\le a_{ij},\ C(s,a)\succeq0 \right\},$$

where $s_{ij}$ ranges over real arrays with $0\le j\le i<d$, $a_{ij}$ ranges over real arrays with $0\le j<i<d$, and

$$C(s,a)= \sum_i s_{ii}|ii\rangle\langle ii| +\frac12\sum_{j<i}\left[ (s_{ij}+a_{ij})(|ij\rangle\langle ij|+|ji\rangle\langle ji|) +(s_{ij}-a_{ij})(|ii\rangle\langle jj|+|jj\rangle\langle ii|) \right]$$

on the standard basis $|ij\rangle$ of $\mathbb C^{d\times d}$. No restriction on $\lambda$ or $K$ is imposed; negative square-root arguments and zero denominators are evaluated as zero. This definition does not assert that a minimizing array exists. The real infimum operation returns zero for an empty set or a set unbounded below; in particular, $d=0$ gives value zero.

QAlgorithms.Papers.Independent.Intakethird100.P95.primalValues

For arbitrary natural numbers $d,K$ and an arbitrary function $\lambda:\{0,\ldots,d-1\}\to\mathbb R$, this is the set of real numbers $v$ for which there exist real arrays $s_{ij}$, $0\le j\le i<d$, and $a_{ij}$, $0\le j<i<d$, satisfying

$$\frac{\sqrt{\lambda_i\lambda_j}}{K+1}\le s_{ij},\qquad \frac{\sqrt{\lambda_i\lambda_j}}{K-1}\le a_{ij},\qquad C(s,a)\succeq0,\qquad v=\sum_{j\le i}s_{ij}+\sum_{j<i}a_{ij}.$$

Here, with $|ij\rangle$ the standard basis of $\mathbb C^{d\times d}$, the matrix appearing in the positivity condition is explicitly

$$C(s,a)= \sum_i s_{ii}|ii\rangle\langle ii| +\frac12\sum_{j<i}\left[ (s_{ij}+a_{ij})(|ij\rangle\langle ij|+|ji\rangle\langle ji|) +(s_{ij}-a_{ij})(|ii\rangle\langle jj|+|jj\rangle\langle ii|) \right].$$

No nonnegativity or normalization of $\lambda$, or lower bound on $K$, is assumed. Real square roots of negative arguments are zero, and division by zero is zero, including the denominator $K-1$ when $K=1$. When $d=0$, all constraints are vacuous or concern the empty matrix, and the set is $\{0\}$.

QAlgorithms.Papers.Independent.Intakethird100.P95.rankOneValue

For arbitrary natural numbers $d,K$ and an arbitrary real array $\lambda_i$, $0\le i<d$, this is the real supremum of

$$\frac{K\,2^{H_{1/2}(\lambda)}-1}{K^2-1} -\sum_{j\le i}\frac{\mu_{ij}\sqrt{\lambda_i\lambda_j}}{K+1} -\sum_{j<i}\frac{t_{ij}\sqrt{\lambda_i\lambda_j}}{K-1}, \qquad H_{1/2}(\lambda)= 2\frac{\log\!\left(\sum_i\operatorname{rpow}(\lambda_i,\tfrac12)\right)}{\log2},$$

over real arrays $\mu_{ij}$, $0\le j\le i<d$, and $t_{ij}$, $0\le j<i<d$, satisfying $\mu_{ij}\le1$, $t_{ij}\le1$, and requiring

$$C(\mu,t)= \sum_i\mu_{ii}|ii\rangle\langle ii| +\frac12\sum_{j<i}\left[ (\mu_{ij}+t_{ij})(|ij\rangle\langle ij|+|ji\rangle\langle ji|) +(\mu_{ij}-t_{ij})(|ii\rangle\langle jj|+|jj\rangle\langle ii|) \right]$$

to be positive semidefinite of complex rank exactly one, where $|ij\rangle$ is the standard basis of $\mathbb C^{d\times d}$. No restrictions on $\lambda$ or $K$ are imposed, and no maximizing arrays are asserted to exist. The total conventions give zero for zero-denominator divisions and negative square-root arguments, with $\log0=0$; the real supremum is zero for an empty set or a set unbounded above. In particular, the value is zero when $d=0$, because its feasible set is empty.

QAlgorithms.Papers.Independent.Intakethird100.P95.rankOneValues

For arbitrary natural numbers $d,K$ and arbitrary real numbers $\lambda_i$, $0\le i<d$, this is the set of all real values

$$\frac{K\,2^{H_{1/2}(\lambda)}-1}{K^2-1} -\sum_{j\le i}\frac{\mu_{ij}\sqrt{\lambda_i\lambda_j}}{K+1} -\sum_{j<i}\frac{t_{ij}\sqrt{\lambda_i\lambda_j}}{K-1}, \qquad H_{1/2}(\lambda)= 2\frac{\log\!\left(\sum_i\operatorname{rpow}(\lambda_i,\tfrac12)\right)}{\log2},$$

for which there exist real arrays $\mu_{ij}$, $0\le j\le i<d$, and $t_{ij}$, $0\le j<i<d$, such that $\mu_{ij}\le1$, $t_{ij}\le1$, and the matrix

$$C(\mu,t)= \sum_i\mu_{ii}|ii\rangle\langle ii| +\frac12\sum_{j<i}\left[ (\mu_{ij}+t_{ij})(|ij\rangle\langle ij|+|ji\rangle\langle ji|) +(\mu_{ij}-t_{ij})(|ii\rangle\langle jj|+|jj\rangle\langle ii|) \right]$$

is positive semidefinite and has rank exactly one over $\mathbb C$. The vectors $|ij\rangle$ form the standard basis of $\mathbb C^{d\times d}$. There are no restrictions on $\lambda$ or $K$; real powers and logarithms use their total definitions, zero denominators give zero, and negative square-root arguments give zero. For $d=0$, no matrix has the required rank, so this set is empty.

QAlgorithms.Papers.Independent.Intakethird100.P95.renyiInfinity

For any finite type $I$ and any function $\lambda:I\to\mathbb R$, let $m=\sup\{\lambda_i:i\in I\}$, with $m=0$ when $I$ is empty. This defines $-\log(m)/\log 2$, using the total real logarithm. Thus its value is $0$ if $m=0$, and $-\log|m|/\log 2$ otherwise. No probability-distribution assumptions are imposed on $\lambda$.

QAlgorithms.Papers.Independent.Intakethird100.P95.rhoPower

For any type $I$, any function $\lambda:I\to\mathbb R$, and any natural number $n$, this defines a complex matrix on pairs of $n$-tuples. Writing $w(x)=\prod_{j=0}^{n-1}\lambda_{x_j}$, its entries are

$$\rho_{\lambda,n}\bigl((x,y),(u,v)\bigr)= \begin{cases} \sqrt{w(x)w(u)},&x=y\ \text{and}\ u=v,\\ 0,&\text{otherwise}. \end{cases}$$

The real square root is embedded into $\mathbb C$ and equals $0$ on negative arguments. No nonnegativity or normalization of $\lambda$ is assumed. For $n=0$, this is the one-by-one matrix with entry $1$, even if $I$ is empty.

QAlgorithms.Papers.Independent.Intakethird100.P95.schmidtDensity

For any type $I$ and any function $\lambda:I\to\mathbb R$, this defines the complex matrix indexed by $I\times I$ whose entries are

$$S_\lambda\bigl((i,j),(k,\ell)\bigr)= \begin{cases} \sqrt{\lambda_i\lambda_k},&i=j\ \text{and}\ k=\ell,\\ 0,&\text{otherwise}, \end{cases}$$

with the real square root embedded into $\mathbb C$. The real square root is $0$ on negative arguments, and the diagonal entry at $(i,i)$ is $|\lambda_i|$. No finiteness, nonnegativity, or normalization assumption is made.

QAlgorithms.Papers.Independent.Intakethird100.P95.schmidtDualObjective

For arbitrary natural numbers $d,K$, arbitrary real numbers $\lambda_i$, $0\le i<d$, and arbitrary real arrays $\mu_{ij}$, $0\le j\le i<d$, and $t_{ij}$, $0\le j<i<d$, define

$$H_{1/2}(\lambda)= 2\log_2\left(\sum_{i=0}^{d-1}\operatorname{rpow}(\lambda_i,\tfrac12)\right), \qquad \log_2 x=\frac{\log x}{\log2},$$

where $\operatorname{rpow}$ is the total real-power operation. The objective is

$$\frac{K\,2^{H_{1/2}(\lambda)}-1}{K^2-1} -\sum_{j\le i}\frac{\mu_{ij}\sqrt{\lambda_i\lambda_j}}{K+1} -\sum_{j<i}\frac{t_{ij}\sqrt{\lambda_i\lambda_j}}{K-1}.$$

No feasibility, sign, or normalization conditions are assumed. All operations are total: division by zero and square roots of negative real numbers give zero, and $\log0=0$. Thus $K=0$, $K=1$, and $d=0$ are included; when $d=0$, the objective is $(K-1)/(K^2-1)$, with the same zero-division convention.

QAlgorithms.Papers.Independent.Intakethird100.P95.schmidtDualValues

For arbitrary natural numbers $d,K$ and an arbitrary real array $\lambda_i$, $0\le i<d$, this is the set of all values

$$\frac{K\,2^{H_{1/2}(\lambda)}-1}{K^2-1} -\sum_{j\le i}\frac{\mu_{ij}\sqrt{\lambda_i\lambda_j}}{K+1} -\sum_{j<i}\frac{t_{ij}\sqrt{\lambda_i\lambda_j}}{K-1}, \qquad H_{1/2}(\lambda)= 2\frac{\log\!\left(\sum_i\operatorname{rpow}(\lambda_i,\tfrac12)\right)}{\log2},$$

obtained from real arrays $\mu_{ij}$, $0\le j\le i<d$, and $t_{ij}$, $0\le j<i<d$, with $\mu_{ij}\le1$, $t_{ij}\le1$, and

$$\sum_i\mu_{ii}|ii\rangle\langle ii| +\frac12\sum_{j<i}\left[ (\mu_{ij}+t_{ij})(|ij\rangle\langle ij|+|ji\rangle\langle ji|) +(\mu_{ij}-t_{ij})(|ii\rangle\langle jj|+|jj\rangle\langle ii|) \right]\succeq0.$$

Here $|ij\rangle$ is the standard basis of $\mathbb C^{d\times d}$, and $\operatorname{rpow}$ is total real exponentiation. No assumptions on $\lambda$ or lower bound on $K$ are imposed; zero denominators and negative square-root arguments give zero, and $\log0=0$. For $d=0$, the set is the singleton containing $(K-1)/(K^2-1)$.

QAlgorithms.Papers.Independent.Intakethird100.P95.schmidtPrimalObjective

For every natural number $d$, including $d=0$, and arbitrary real arrays $s_{ij}$ indexed by $0\le j\le i<d$ and $a_{ij}$ indexed by $0\le j<i<d$, the objective is

$$\sum_{0\le j\le i<d}s_{ij}+\sum_{0\le j<i<d}a_{ij}.$$

There are no sign or feasibility assumptions on either array. Empty sums are zero, so the objective is zero when $d=0$.

QAlgorithms.Papers.Independent.Intakethird100.P95.schmidtRenyi

For any finite type $I$, any real number $t$, and any function $\lambda:I\to\mathbb R$, define $L(x)=\log(x)/\log 2$, using the total real logarithm: $L(0)=0$, and $L(x)=\log|x|/\log 2$ when $x\ne0$. The defined value is

$$\begin{cases} L\bigl(\#\{i\in I:\lambda_i\ne0\}\bigr),&t=0,\\ -\displaystyle\sum_{i\in I}\lambda_iL(\lambda_i),&t=1,\\ \displaystyle\frac{L\bigl(\sum_{i\in I}\lambda_i^{\,t}\bigr)}{1-t},&t\notin\{0,1\}, \end{cases}$$

where the powers in the last branch use the total real-power function. Neither nonnegativity nor normalization of $\lambda$ is assumed, and $t$ is unrestricted. If $I$ is empty, the value is $0$ in every branch.

QAlgorithms.Papers.Independent.Intakethird100.P95.symmetricSchmidtProjector

For any natural number $d$ and indices $0\le i,j<d$, let $e_{ij}\in\mathbb C^{\{0,\ldots,d-1\}^2}$ be the coordinate vector equal to one at $(i,j)$ and zero elsewhere. This defines the matrix

$$S_{ij}= \begin{cases} e_{ii}e_{ii}^\dagger,&i=j,\\[2pt] \frac12(e_{ij}+e_{ji})(e_{ij}+e_{ji})^\dagger,&i\ne j. \end{cases}$$

For a vector $v$, the matrix $vv^\dagger$ has entry $v_p\overline{v_q}$ at $(p,q)$. The indices are unrestricted apart from lying between zero and $d-1$; when $d=0$, there are no such indices.

QAlgorithms.Papers.Independent.Intakethird100.P95.tensorSpectrum

For any type $I$, any function $\lambda:I\to\mathbb R$, and any natural number $n$, this defines a real-valued function on $n$-tuples $x:\operatorname{Fin}(n)\to I$ by

$$\lambda^{(n)}(x)=\prod_{j=0}^{n-1}\lambda_{x_j}.$$

No assumptions are made on the entries or on the size of $I$. For $n=0$, the domain contains the unique empty tuple and its value is the empty product $1$.