Definitions for Irreversibility of Asymptotic Entanglement Manipulation Under Quantum Operations Completely Preserving Positivity of Partial Transpose

QAlgorithms.Papers.Independent.Intakethird100.P90.alphaKet

For every real number $\alpha$ and every $j\in\{0,1,2\}$, this defines a vector in $\mathbb C^{\{0,1,2\}\times\{0,1,2\}}$ by

$$v_{\alpha,j}=\sqrt{\alpha}\,|0,j\rangle-\sqrt{1-\alpha}\,|j,0\rangle,$$

where $|a,b\rangle$ has coordinate $1$ at $(a,b)$ and coordinate $0$ everywhere else. There is no restriction that $\alpha\in[0,1]$; the real square root used here equals $0$ on negative inputs. The index $j=0$ is included, in which case the two basis vectors coincide and $v_{\alpha,0}=(\sqrt{\alpha}-\sqrt{1-\alpha})|0,0\rangle$. No unit-norm condition is part of this definition.

QAlgorithms.Papers.Independent.Intakethird100.P90.alphaState

For every real number $\alpha$, this defines the complex matrix

$$\rho_\alpha=\frac12\left(v_{\alpha,1}v_{\alpha,1}^{\dagger}+v_{\alpha,2}v_{\alpha,2}^{\dagger}\right), \qquad v_{\alpha,j}=\sqrt{\alpha}\,|0,j\rangle-\sqrt{1-\alpha}\,|j,0\rangle,$$

indexed by pairs from $\{0,1,2\}$. Here $|a,b\rangle$ is the coordinate basis vector, and $vv^\dagger$ means the matrix with entry $v_u\overline{v_v}$ at row $u$, column $v$. Equivalently,

$$\rho_\alpha=\frac12\sum_{j=1}^{2}\left[ (\sqrt{\alpha})^2|0,j\rangle\langle0,j| +(\sqrt{1-\alpha})^2|j,0\rangle\langle j,0| -\sqrt{\alpha}\sqrt{1-\alpha}\bigl(|0,j\rangle\langle j,0|+|j,0\rangle\langle0,j|\bigr) \right].$$

The parameter $\alpha$ is unrestricted, including values outside $[0,1]$, and real square roots of negative numbers are $0$. The definition itself imposes no trace-one hypothesis.

QAlgorithms.Papers.Independent.Intakethird100.P90.antisymmetricKet

For every $j\in\{0,1,2\}$, this definition gives the vector in $\mathbb C^{\{0,1,2\}^2}$

$$v_j=\frac{e_{(0,j)}-e_{(j,0)}}{\sqrt2},$$

where $e_{(a,b)}$ has coordinate one at $(a,b)$ and zero elsewhere. The allowed value $j=0$ gives the zero vector; the definition imposes no condition excluding it.

QAlgorithms.Papers.Independent.Intakethird100.P90.antisymmetricSupported

For an arbitrary complex matrix $\rho$ indexed by $\{0,1,2\}\times\{0,1,2\}$, this predicate requires

$$\rho_{(b,a),j}=-\rho_{(a,b),j}$$

for every $a,b\in\{0,1,2\}$ and every column index $j\in\{0,1,2\}^2$. The condition swaps only the two components of the row index. In particular, every row indexed by $(a,a)$ must be zero. No positivity, Hermiticity, trace normalization, or separate column-swap condition is assumed.

QAlgorithms.Papers.Independent.Intakethird100.P90.asymptoticEntanglementCost

For finite types $A,B$ with decidable equality, a Boolean selector $b$, and any complex matrix $\rho$ on $A\times B$, this defines the extended real infimum of all real $r\ge0$ with the following property: for every $\epsilon,\delta>0$, all sufficiently large positive integers $n$ admit $m\in\mathbb N$ and a map $\Phi$ such that

$$m\le n(r+\delta), \qquad \left\|\Phi(T_{2^m})-\widehat{\rho^{[n]}}\right\|_1\le\epsilon.$$

The input matrix $T_k$ has entry $1/k$ when both its row and column pairs have equal components, and zero otherwise; $\rho^{[n]}$ has entries $\prod_{t=0}^{n-1}\rho_{(a_t,b_t),(a'_t,b'_t)}$, and the hat reindexes $A^n,B^n$ by their chosen finite ordinal enumerations. The norm is $\operatorname{Re}\operatorname{Tr}\sqrt{X^\dagger X}$. If $b$ is true, $\Phi$ must be built recursively from identities and complete finite local Kraus instruments on either party, with an outcome-dependent recursive continuation and summed branches. If $b$ is false, $\Phi$ and its conjugation by the second-register partial transpose $\Gamma(X)_{(a,b),(a',b')}=X_{(a,b'),(a',b)}$ must both have finite Kraus representations $X\mapsto\sum_jK_jXK_j^\dagger$ with $\sum_jK_j^\dagger K_j=I$. The quantifiers allow $m=0$, but require $n>0$; no density-matrix condition is imposed on $\rho$, and $A,B$ may be empty. If there are no such rates, the infimum is $+\infty$; attainment is not asserted.

QAlgorithms.Papers.Independent.Intakethird100.P90.etaDual

For finite types $A,B$ equipped with decidable equality and an arbitrary complex matrix $\rho$ indexed by $A\times B$, this definition is the real supremum

$$\sup\left\{ \operatorname{Re}\operatorname{Tr}\!\left(P_\rho\,\Gamma(V-F)\right): V,F,W,X\succeq0,\ \Gamma(W-X)-(V+F)\succeq0,\ \operatorname{Re}\operatorname{Tr}(W+X)\le1 \right\},$$

where all four optimization variables are complex matrices on $A\times B$, $(\Gamma Z)_{(a,b),(a',b')}=Z_{(a,b'),(a',b)}$, and $P_\rho=\operatorname{cfc}(f,\rho)$ for $f(0)=0$ and $f(t)=1$ when $t\ne0$, using real continuous functional calculus. No state or Hermiticity assumption is imposed on $\rho$. The all-zero choice is feasible and gives objective zero. Empty index types are included and give value zero.

QAlgorithms.Papers.Independent.Intakethird100.P90.etaDualFeasible

For finite types $A,B$, possibly empty, and complex matrices $V,F,W,X$ indexed by $A\times B$, this predicate means that all four matrices are positive semidefinite and

$$\Gamma(W-X)-(V+F)\succeq0, \qquad \operatorname{Re}\operatorname{Tr}(W+X)\le1,$$

where the operation $\Gamma$ is defined entrywise by $(\Gamma Z)_{(a,b),(a',b')}=Z_{(a,b'),(a',b)}$. Here $\succeq0$ denotes positive semidefiniteness. There are no trace-one requirements on these matrices; in particular, taking all four to be zero satisfies the predicate.

QAlgorithms.Papers.Independent.Intakethird100.P90.etaDualObjective

For finite types $A,B$ equipped with decidable equality, and arbitrary complex matrices $\rho,V,F$ indexed by $A\times B$, this definition returns

$$\operatorname{Re}\operatorname{Tr}\!\left(P_\rho\,\Gamma(V-F)\right),$$

where $P_\rho=\operatorname{cfc}(f,\rho)$, with $f(0)=0$ and $f(t)=1$ for every nonzero real $t$, using the real continuous functional calculus, and $(\Gamma Z)_{(a,b),(a',b')}=Z_{(a,b'),(a',b)}$. No density, positivity, Hermiticity, or feasibility hypothesis is imposed on $\rho,V,F$. Empty index types are permitted, in which case the trace is zero.

QAlgorithms.Papers.Independent.Intakethird100.P90.etaEntanglement

For finite types $A,B$ with decidable equality and any complex matrix $\rho$ on $A\times B$, this defines the real number

$$-\log_2\!\left( \inf_{\substack{Y=Y^\dagger\\ Y-\Gamma(P_\rho)\succeq0\\ Y+\Gamma(P_\rho)\succeq0}} \|\Gamma(Y)\|_{\mathrm{op}} \right).$$

Here $\Gamma(X)_{(a,b),(a',b')}=X_{(a,b'),(a',b)}$, the norm is the Euclidean operator norm, and $P_\rho=f(\rho)$ uses real continuous functional calculus with $f(0)=0$ and $f(x)=1$ for $x\ne0$; it selects nonzero eigenspaces for Hermitian $\rho$ and gives zero for non-Hermitian $\rho$. No positivity or normalization is assumed for $\rho$. The logarithm is the totalized real base-two logarithm, so its value at zero is zero: in particular, $\rho=0$ and empty underlying types give value zero, rather than an infinite value.

QAlgorithms.Papers.Independent.Intakethird100.P90.etaFeasible

For finite types $A,B$ with decidable equality and any complex matrices $\rho,Y$ on $A\times B$, this predicate means

$$Y=Y^\dagger,\qquad Y-\Gamma(P_\rho)\succeq0,\qquad Y+\Gamma(P_\rho)\succeq0,$$

where $\Gamma(X)_{(a,b),(a',b')}=X_{(a,b'),(a',b)}$ and $P_\rho=f(\rho)$ is defined by real continuous functional calculus with $f(0)=0$ and $f(x)=1$ for $x\ne0$. For Hermitian $\rho$, $P_\rho$ projects onto its nonzero eigenspaces; for non-Hermitian $\rho$, this real functional calculus gives zero. No positivity or normalization hypothesis is imposed on $\rho$. Empty $A$ or $B$ are permitted, in which case all three conditions hold for the unique matrices.

QAlgorithms.Papers.Independent.Intakethird100.P90.etaPrimal

For finite types $A,B$ with decidable equality and any complex matrix $\rho$ on $A\times B$, this defines the real infimum

$$\inf\left\{t\in\mathbb R: \exists Y,\ Y=Y^\dagger,\ Y\pm\Gamma(P_\rho)\succeq0,\ tI\pm\Gamma(Y)\succeq0 \right\},$$

where each $\pm$ requires both signs, $I$ is the identity, $\Gamma(X)_{(a,b),(a',b')}=X_{(a,b'),(a',b)}$, and $P_\rho=f(\rho)$ uses real continuous functional calculus with $f(0)=0$ and $f(x)=1$ for $x\ne0$. For Hermitian $\rho$, $P_\rho$ projects onto its nonzero eigenspaces; for non-Hermitian $\rho$, it is zero. No density-matrix assumption is imposed on $\rho$, and the defining set does not separately restrict $t$ to be nonnegative. If $\rho=0$, the value is zero. If either underlying type is empty, every real $t$ is feasible, and the real infimum operation assigns zero to this unbounded-below set.

QAlgorithms.Papers.Independent.Intakethird100.P90.etaPrimalFeasible

For finite types $A,B$ with decidable equality, arbitrary complex matrices $\rho,Y$ on $A\times B$, and any real $t$, this predicate means

$$Y=Y^\dagger,\qquad Y-\Gamma(P_\rho)\succeq0,\qquad Y+\Gamma(P_\rho)\succeq0,\qquad tI-\Gamma(Y)\succeq0,\qquad \Gamma(Y)+tI\succeq0.$$

Here $I$ is the identity matrix, $\Gamma(X)_{(a,b),(a',b')}=X_{(a,b'),(a',b)}$, and $P_\rho=f(\rho)$ is real continuous functional calculus with $f(0)=0$ and $f(x)=1$ for $x\ne0$. It projects onto nonzero eigenspaces when $\rho$ is Hermitian and is zero otherwise. There is no explicit assumption $t\ge0$ and no positivity or normalization assumption on $\rho$. If either underlying type is empty, the conditions hold for every real $t$.

QAlgorithms.Papers.Independent.Intakethird100.P90.exteriorKet

For arbitrary vectors $a,b\in\mathbb C^{\{0,1,2\}}$, this definition gives the vector $v\in\mathbb C^{\{0,1,2\}^2}$ with coordinates

$$v_{(i,j)}=\frac{a_i b_j-b_i a_j}{\sqrt2} \qquad(i,j\in\{0,1,2\}).$$

There are no normalization, orthogonality, or nonzero assumptions on $a,b$; in particular, linearly dependent inputs give the zero vector.

QAlgorithms.Papers.Independent.Intakethird100.P90.loccDistillable

For finite types $A,B$ with decidable equality and any complex matrix $\rho$ on $A\times B$, this defines the extended real supremum of all real $r\ge0$ such that, for every $\epsilon,\delta>0$, every sufficiently large positive integer $n$ admits $m\in\mathbb N$ and a finite recursive local-operation map $\Phi$ with

$$n(r-\delta)\le m, \qquad \left\|\Phi(\widehat{\rho^{[n]}})-T_{2^m}\right\|_1\le\epsilon.$$

Here $\rho^{[n]}_{(a,b),(a',b')}=\prod_{t=0}^{n-1}\rho_{(a_t,b_t),(a'_t,b'_t)}$; the hat reindexes $A^n,B^n$ through their chosen finite ordinal enumerations; $T_k$ has entry $1/k$ when both index pairs have equal components and zero otherwise; and $\|X\|_1=\operatorname{Re}\operatorname{Tr}\sqrt{X^\dagger X}$. A permitted map is an identity or is obtained by applying a finite complete local Kraus family on either party, following each outcome by another permitted map, and summing branches; completeness means $\sum_jK_j^\dagger K_j=I$. No density-matrix assumption is imposed on $\rho$, empty underlying types are included, and $m=0$ is allowed. An empty set of achievable rates has supremum $-\infty$, while an unbounded set has supremum $+\infty$; attainment is not asserted.

QAlgorithms.Papers.Independent.Intakethird100.P90.loccDistillationRate

For finite types $A,B$ with decidable equality, any complex matrix $\rho$ on $A\times B$, and $r\in\mathbb R$, this predicate means that $r\ge0$ and, for every $\epsilon,\delta>0$, there exists $N\in\mathbb N$ such that every integer $n\ge N$ with $n>0$ admits $m\in\mathbb N$ and a map $\Phi$ satisfying

$$n(r-\delta)\le m, \qquad \left\|\Phi(\widehat{\rho^{[n]}})-T_{2^m}\right\|_1\le\epsilon.$$

The input is indexed by finite ordinal enumerations of $A^n$ and $B^n$, and the output by two registers of size $2^m$. Here $\rho^{[n]}_{(a,b),(a',b')}=\prod_{t=0}^{n-1}\rho_{(a_t,b_t),(a'_t,b'_t)}$, the hat denotes reindexing by the chosen bijections, and $(T_k)_{(i,j),(i',j')}=1/k$ when $i=j$ and $i'=j'$, and zero otherwise. The norm is $\|X\|_1=\operatorname{Re}\operatorname{Tr}\sqrt{X^\dagger X}$. The map $\Phi$ must be built by the following finite recursion: identities are allowed; either party may apply a finite local Kraus family with $\sum_jK_j^\dagger K_j=I$, follow each outcome by another recursively allowed map, and sum the resulting branches. No positivity or normalization assumption is made on $\rho$, and empty $A,B$ are permitted. The case $m=0$ is allowed; when $r-\delta<0$, the displayed rate inequality alone imposes no positive lower bound on $m$.

QAlgorithms.Papers.Independent.Intakethird100.P90.pptRelativeEntropy

For finite types $A,B$ with decidable equality, and any complex matrix $\rho$ indexed by $A\times B$, this defines the extended real number

$$\inf_{\substack{\sigma\succeq0,\ \operatorname{Tr}\sigma=1\\ \Gamma(\sigma)\succeq0}}D(\rho\Vert\sigma), \qquad \Gamma(X)_{(a,b),(a',b')}=X_{(a,b'),(a',b)}.$$

Here

$$D(\rho\Vert\sigma)= \begin{cases} \dfrac{\operatorname{Re}\operatorname{Tr}\!\left(f(\rho)-\rho\,\ell(\sigma)\right)}{\log 2}, &\ker\sigma\subseteq\ker\rho,\\[2mm] +\infty,&\text{otherwise}, \end{cases}$$

where $f(x)=x\log x$ and $\ell(x)=\log x$ are applied through real continuous functional calculus. The real logarithm is totalized, with $\log0=0$ and $\log x=\log|x|$ for nonzero real $x$; this real functional calculus gives zero on non-Hermitian matrices. No positivity or normalization is required of $\rho$, and finite values of $D$ are not truncated to be nonnegative. Empty $A$ or $B$ are permitted; then no trace-one $\sigma$ exists, so the infimum is $+\infty$.

QAlgorithms.Papers.Independent.Intakethird100.P90.preparationRate

For finite types $A,B$ with decidable equality, a Boolean selector $b$, any complex matrix $\rho$ on $A\times B$, and $r\in\mathbb R$, this predicate means that $r\ge0$ and, for every $\epsilon>0$ and $\delta>0$, there is $N\in\mathbb N$ such that every integer $n\ge N$ with $n>0$ admits an integer $m\ge0$ and a map $\Phi$ from matrices on $\{0,\ldots,2^m-1\}^2$ to matrices on $\{0,\ldots,|A^n|-1\}\times\{0,\ldots,|B^n|-1\}$ satisfying

$$m\le n(r+\delta), \qquad \left\|\Phi(T_{2^m})-\widehat{\rho^{[n]}}\right\|_1\le\epsilon.$$

Here $(T_k)_{(i,j),(i',j')}=1/k$ when $i=j$ and $i'=j'$, and is zero otherwise; $\rho^{[n]}_{(a,b),(a',b')}=\prod_{t=0}^{n-1}\rho_{(a_t,b_t),(a'_t,b'_t)}$; and the hat reindexes the two tuple registers using the chosen bijections with their finite ordinal enumerations. The norm is $\|X\|_1=\operatorname{Re}\operatorname{Tr}\sqrt{X^\dagger X}$. When $b$ is true, $\Phi$ must be obtainable by finitely many recursive local operations: either an identity map, or a complete finite local Kraus family on either party followed, for each outcome, by another such map, with the branches summed. When $b$ is false, both $\Phi$ and $X\mapsto\Gamma(\Phi(\Gamma(X)))$ must be channels, where $\Gamma(X)_{(a,b),(a',b')}=X_{(a,b'),(a',b)}$, and a channel means a map $X\mapsto\sum_jK_jXK_j^\dagger$ with finitely many $K_j$ satisfying $\sum_jK_j^\dagger K_j=I$. No positivity or normalization hypothesis is imposed on $\rho$; empty underlying types are included. The case $m=0$, whose input is the one-dimensional matrix $T_1=(1)$, is allowed.

QAlgorithms.Papers.Independent.Intakethird100.P90.regularizedPptRelativeEntropy

For finite types $A,B$ with decidable equality, and any complex matrix $\rho$ on $A\times B$, this defines

$$\inf_{k\ge1}\frac1k \inf_{\substack{\sigma\succeq0,\ \operatorname{Tr}\sigma=1\\\Gamma(\sigma)\succeq0}} D(\rho^{[k]}\Vert\sigma)$$

in the extended reals. The matrix $\rho^{[k]}$, indexed by $A^k\times B^k$, has entries

$$\rho^{[k]}_{(a,b),(a',b')}=\prod_{i=0}^{k-1}\rho_{(a_i,b_i),(a'_i,b'_i)},$$

and $\Gamma(X)_{(a,b),(a',b')}=X_{(a,b'),(a',b)}$, with the same formula for tuple indices. The quantity $D(X\Vert Y)$ is $+\infty$ unless $\ker Y\subseteq\ker X$; otherwise it is $\operatorname{Re}\operatorname{Tr}(f(X)-X\ell(Y))/\log2$, using real continuous functional calculus with $f(x)=x\log x$, $\ell(x)=\log x$, the totalized real logarithm, and zero functional-calculus values on non-Hermitian matrices. No density-matrix assumption is made on $\rho$. Only positive copy numbers occur, so there is no division by zero and no zero-copy term. If either underlying type is empty, the value is $+\infty$. The definition takes an infimum and does not assert convergence of the normalized quantities.

QAlgorithms.Papers.Independent.Intakethird100.P90.regularizedRains

For finite types $A,B$ with decidable equality, and any complex matrix $\rho$ on $A\times B$, this defines the extended real number

$$\inf_{k\ge1}\frac1k \inf_{\substack{\sigma\succeq0\\\|\Gamma(\sigma)\|_1\le1}} D(\rho^{[k]}\Vert\sigma).$$

Here $\rho^{[k]}_{(a,b),(a',b')}=\prod_{i=0}^{k-1}\rho_{(a_i,b_i),(a'_i,b'_i)}$ on $A^k\times B^k$, $\Gamma(X)_{(a,b),(a',b')}=X_{(a,b'),(a',b)}$, and $\|X\|_1=\operatorname{Re}\operatorname{Tr}\sqrt{X^\dagger X}$. The quantity $D(X\Vert Y)$ equals $+\infty$ when $\ker Y\not\subseteq\ker X$, and otherwise equals $\operatorname{Re}\operatorname{Tr}(f(X)-X\ell(Y))/\log2$, where $f(x)=x\log x$ and $\ell(x)=\log x$ use real continuous functional calculus, the totalized real logarithm, and zero functional-calculus values on non-Hermitian matrices. Neither $\rho$ nor the feasible $\sigma$ is required to have trace one; $\rho$ is not required to be positive. Only positive copy numbers occur, and no limit is asserted. Empty $A$ or $B$ are allowed and give value zero.

QAlgorithms.Papers.Independent.Intakethird100.P90.rhoV

This definition gives the complex matrix on $\{0,1,2\}^2$

$$\rho_V= \frac14\left[ (e_{(0,1)}-e_{(1,0)})(e_{(0,1)}-e_{(1,0)})^\dagger+ (e_{(0,2)}-e_{(2,0)})(e_{(0,2)}-e_{(2,0)})^\dagger \right],$$

where $e_{(a,b)}$ is the coordinate vector at $(a,b)$, and a vector outer product $vv^\dagger$ has entry $v_i\overline{v_j}$. Equivalently, it is one half the sum of the outer products of the two vectors $(e_{(0,j)}-e_{(j,0)})/\sqrt2$ for $j=1,2$.

QAlgorithms.Papers.Independent.Intakethird100.P90.stateRank

For a finite type $I$ equipped with decidable equality and an arbitrary complex matrix $\rho$ indexed by $I$, this definition returns the complex dimension of the range of the linear map

$$v\longmapsto \rho v,\qquad (\rho v)_i=\sum_{j\in I}\rho_{ij}v_j.$$

It is therefore the matrix rank as a natural number, without any assumption that $\rho$ is a density matrix. The empty type $I$ is included and gives rank zero.

QAlgorithms.Papers.Independent.Intakethird100.P90.supportDual

For finite types $A,B$ equipped with decidable equality and an arbitrary complex matrix $\rho$ indexed by $A\times B$, this definition is the real infimum

$$\inf\left\{ t\in\mathbb R: \exists\,Y,\ Y=Y^\dagger,\ Y-\Gamma P_\rho\succeq0,\ tI-\Gamma Y\succeq0 \right\},$$

where $Y$ is a complex matrix on $A\times B$, $(\Gamma Z)_{(a,b),(a',b')}=Z_{(a,b'),(a',b)}$, and $P_\rho=\operatorname{cfc}(f,\rho)$, with $f(0)=0$ and $f(t)=1$ for nonzero real $t$, using real continuous functional calculus. No state assumption on $\rho$ or explicit restriction $t\ge0$ is imposed. If either index type is empty, every real $t$ is feasible; the real infimum of this unbounded-below set is defined to be zero.

QAlgorithms.Papers.Independent.Intakethird100.P90.supportDualFeasible

For finite types $A,B$ equipped with decidable equality, arbitrary complex matrices $\rho,Y$ indexed by $A\times B$, and a real number $t$, this predicate means

$$Y=Y^\dagger,\qquad Y-\Gamma P_\rho\succeq0,\qquad tI-\Gamma Y\succeq0,$$

where $I$ is the identity matrix, $(\Gamma Z)_{(a,b),(a',b')}=Z_{(a,b'),(a',b)}$, and $P_\rho=\operatorname{cfc}(f,\rho)$ for $f(0)=0$ and $f(t)=1$ for nonzero real $t$, using real continuous functional calculus. There is no explicit nonnegativity condition on $t$, nor any state assumption on $\rho$. Empty index types are allowed; then the predicate holds for every real $t$ and the unique matrix $Y$.

QAlgorithms.Papers.Independent.Intakethird100.P90.supportEta

For finite types $A,B$ with decidable equality and any complex matrix $\rho$ on $A\times B$, this defines the real infimum

$$\inf_{\substack{Y=Y^\dagger\\ Y-\Gamma(P_\rho)\succeq0\\ Y+\Gamma(P_\rho)\succeq0}} \|\Gamma(Y)\|_{\mathrm{op}},$$

where $\|\cdot\|_{\mathrm{op}}$ is the operator norm for Euclidean norms, $\Gamma(X)_{(a,b),(a',b')}=X_{(a,b'),(a',b)}$, and $P_\rho=f(\rho)$ uses real continuous functional calculus with $f(0)=0$ and $f(x)=1$ for $x\ne0$. For Hermitian $\rho$, this selects its nonzero eigenspaces; for non-Hermitian $\rho$, it gives zero. No density-matrix assumption is made on $\rho$, and no minimizing $Y$ is specified by the definition. If $\rho=0$, or either underlying type is empty, the value is zero.

QAlgorithms.Papers.Independent.Intakethird100.P90.supportOverlap

For finite types $A,B$ equipped with decidable equality and an arbitrary complex matrix $\rho$ indexed by $A\times B$, this definition is

$$\sup_{\substack{\sigma\succeq0,\ \operatorname{Tr}\sigma=1\\ \Gamma\sigma\succeq0}} \operatorname{Re}\operatorname{Tr}(P_\rho\sigma),$$

where $\sigma$ ranges over complex matrices on $A\times B$, $(\Gamma Z)_{(a,b),(a',b')}=Z_{(a,b'),(a',b)}$, and $P_\rho=\operatorname{cfc}(f,\rho)$, with $f(0)=0$ and $f(t)=1$ for nonzero real $t$, using real continuous functional calculus. No density or Hermiticity assumption is made on $\rho$. The supremum is taken in $\mathbb R$; if either index type is empty, no trace-one matrix $\sigma$ exists, and the real supremum of the empty set is zero.

QAlgorithms.Papers.Independent.Intakethird100.P90.wangDuanBound

For finite types $A,B$ equipped with decidable equality and an arbitrary complex matrix $\rho$ indexed by $A\times B$, this definition is

$$\inf_{\substack{X=X^\dagger\\X-\rho\succeq0}} \log_2\!\left( \operatorname{Re}\operatorname{Tr} \left[ \operatorname{cfc}\!\left(\sqrt{\cdot},(\Gamma X)^\dagger(\Gamma X)\right) \right]\right),$$

where $X$ ranges over complex matrices on $A\times B$, and $(\Gamma Z)_{(a,b),(a',b')}=Z_{(a,b'),(a',b)}$. Thus the logarithm is applied to the explicitly defined trace norm of $\Gamma X$. No density or Hermiticity hypothesis is imposed on $\rho$, and $X$ itself is required to be Hermitian but is not separately required to be positive semidefinite. The logarithm is total, with $\log_2 0=0$; the infimum is real-valued, with value zero for an empty or unbounded-below set of objective values. Empty index types are permitted and give value zero.