Definitions for Improved Semidefinite Programming Upper Bound on Distillable Entanglement

QAlgorithms.Papers.Independent.Intakethird100.P83.DualUVFeasible

For finite index types $A,B$ and complex matrices $\rho,U,V$ indexed by $A\times B$, this predicate means

$$U\succeq0,\qquad V\succeq0,\qquad (U-V)^\Gamma-\rho\succeq0,$$

where $\succeq0$ means positive semidefinite and $(X^\Gamma)_{(a,b),(a',b')}=X_{(a,b'),(a',b)}$. No trace or normalization conditions are imposed on any of the three matrices. In particular, the last condition entails that $\rho$ is Hermitian, since $U$ and $V$ are Hermitian. Empty index types are included, in which case all conditions hold.

QAlgorithms.Papers.Independent.Intakethird100.P83.FidelityFeasible

For finite index types $A,B$, a real number $k$, and a complex matrix $Q$ indexed by $A\times B$, this predicate means

$$k>0,\qquad Q\succeq0,\qquad I-Q\succeq0,\qquad Q^\Gamma+k^{-1}I\succeq0,\qquad k^{-1}I-Q^\Gamma\succeq0,$$

where $I$ is the identity matrix, $\succeq0$ means positive semidefinite, and the partial transpose is defined entrywise by

$$(Q^\Gamma)_{(a,b),(a',b')}=Q_{(a,b'),(a',b)}.$$

The parameter $k$ is any positive real number, without an integrality requirement. If $k\le0$, the predicate is false. Empty index types are included; in that case all matrix conditions hold automatically, leaving only $k>0$.

QAlgorithms.Papers.Independent.Intakethird100.P83.PPTAchievableRate

For finite index types $A,B$, any complex matrix $\rho$ indexed by $A\times B$, and any real number $r$, this predicate means that $F_n\to1$ as the natural number $n\to\infty$, where

$$F_n=\sup\left\{\operatorname{Re}\operatorname{tr}(\rho^{[n]}Q): 0\preceq Q\preceq I,\ -2^{-nr}I\preceq Q^\Gamma\preceq2^{-nr}I\right\}, \qquad (\rho^{[n]})_{(a,b),(a',b')} =\prod_{t=0}^{n-1}\rho_{(a_t,b_t),(a'_t,b'_t)}.$$

Here $a,a'$ are $n$-tuples in $A$, $b,b'$ are $n$-tuples in $B$, $X\preceq Y$ means $Y-X$ is positive semidefinite, and $(Q^\Gamma)_{(a,b),(a',b')}=Q_{(a,b'),(a',b)}$. No positivity or normalization assumption is imposed on $\rho$, and no nonnegativity assumption is imposed on $r$. The sequence includes $n=0$, when the product defining $\rho^{[0]}$ is $1$ and $2^{nr}=1$.

QAlgorithms.Papers.Independent.Intakethird100.P83.PPTInstrument

For finite index types $A,B,C,D$, an arbitrary outcome type $O$, and functions $N_i$ from complex matrices on $A\times B$ to complex matrices on $C\times D$, this predicate requires the following: for every $i\in O$, both $N_i$ and $X\mapsto(N_i(X^\Gamma))^\Gamma$ agree on every matrix with complex-linear completely positive maps, and $\operatorname{Re}\operatorname{tr}N_i(X)\le\operatorname{Re}\operatorname{tr}X$ for every positive semidefinite $X$. Complete positivity here means that, for every natural number $r$, applying the map to each matrix block while retaining a reference index in $\{0,\ldots,r-1\}$ preserves positive semidefiniteness. Partial transpose exchanges the second-party row and column indices. The predicate also requires a function $T$ and finitely many matrices $E_j$, for some natural number $m$, such that $\sum_{j=0}^{m-1}E_j^\dagger E_j=I$, $T(X)=\sum_jE_jXE_j^\dagger$ for every $X$, and $\sum_{i\in O}N_i(X)=T(X)$ for every $X$. This last sum must converge as the net of sums over finite subsets of $O$; no finiteness or countability assumption is imposed on $O$.

QAlgorithms.Papers.Independent.Intakethird100.P83.WFeasible

For finite index types $A,B$ and any complex matrix $R$ indexed by $A\times B$, this predicate means that $R$ is Hermitian, its operator norm induced by the Euclidean norm on $\mathbb C^{A\times B}$ is at most $1$, and $R^\Gamma$ is positive semidefinite, where

$$(R^\Gamma)_{(a,b),(a',b')}=R_{(a,b'),(a',b)}.$$

There is no requirement that $R$ itself be positive semidefinite or have a specified trace. Empty index types are included, in which case all three conditions hold.

QAlgorithms.Papers.Independent.Intakethird100.P83.WZeroFeasible

For finite index types $A,B$ with decidable equality and complex matrices $\rho,R$ on $A\times B$, this predicate means, on putting $q=\operatorname{Re}\operatorname{tr}(\rho R)$, that all four matrices

$$R,\qquad qI-R,\qquad R^\Gamma+I,\qquad I-R^\Gamma$$

are positive semidefinite, where $R^\Gamma_{(a,b),(a',b')}=R_{(a,b'),(a',b)}$. Equivalently, $R\ge0$, $R\le qI$, and $-I\le R^\Gamma\le I$. No density-matrix condition on $\rho$ is required.

QAlgorithms.Papers.Independent.Intakethird100.P83.antisymmetricSupport

This defines the complex linear subspace of all functions $\{0,1,2\}\times\{0,1,2\}\to\mathbb C$ spanned by

$$e_{01}-e_{10},\qquad e_{02}-e_{20},\qquad e_{12}-e_{21},$$

where $e_{ab}(i,j)$ equals $1$ if $(i,j)=(a,b)$ and $0$ otherwise. Thus its elements are exactly the complex linear combinations of these three specified functions.

QAlgorithms.Papers.Independent.Intakethird100.P83.branchContribution

For finite index types $A,B,C,D$ with decidable equality, any function $N$ from complex matrices on $A\times B$ to complex matrices on $C\times D$, and any matrix $\rho$ on $A\times B$, put $p=\operatorname{Re}\operatorname{tr}N(\rho)$. This defines zero if $p\le0$, and otherwise defines

$$p\log_2\!\left( \sup_{\substack{R=R^\dagger,\ \|R\|_{\mathrm{op}}\le1\\R^\Gamma\ \text{positive semidefinite}}} \operatorname{Re}\operatorname{tr}\!\left[(p^{-1}N(\rho))^\Gamma R\right]\right).$$

Here $R$ is a complex matrix on $C\times D$, $\|\cdot\|_{\mathrm{op}}$ is the operator norm for the Euclidean norm, and $\Gamma$ exchanges the two $D$-indices. No channel condition on $N$ or density-matrix condition on $\rho$ is required; the real logarithm is total, including value zero at zero.

QAlgorithms.Papers.Independent.Intakethird100.P83.complexSpectralNorm

For any finite type $I$ with decidable equality and any complex matrix $X$ indexed by $I\times I$, this defines the real number $\|T_X\|_{\mathrm{op}}$, where $T_X:\mathbb C^I\to\mathbb C^I$ is the continuous complex-linear map

$$(T_Xv)_i=\sum_{j\in I}X_{ij}v_j,$$

and both vector spaces carry their Euclidean norms. Thus this is the operator norm induced by the Euclidean norm. The index type $I$ may be empty, in which case the value is $0$.

QAlgorithms.Papers.Independent.Intakethird100.P83.dualUVValue

For finite index types $A,B$ and any complex matrix $\rho$ indexed by $A\times B$, this defines

$$\inf\left\{ \operatorname{Re}\operatorname{tr}(U+V): U\succeq0,\ V\succeq0,\ (U-V)^\Gamma-\rho\succeq0 \right\},$$

where $\succeq0$ means positive semidefinite and $(X^\Gamma)_{(a,b),(a',b')}=X_{(a,b'),(a',b)}$. No density-matrix assumption is imposed on $\rho$. If $\rho$ is not Hermitian, no feasible pair exists, and the real-valued infimum of the empty set used here is $0$. If $A\times B$ is empty, the value is $0$.

QAlgorithms.Papers.Independent.Intakethird100.P83.dualXValue

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

$$\inf\left\{\left\|X^\Gamma\right\|_1:X-\rho\ \text{is positive semidefinite}\right\}.$$

Here $X$ ranges over all complex matrices on $A\times B$, $X^\Gamma_{(a,b),(a',b')}=X_{(a,b'),(a',b)}$, and $\|Y\|_1=\operatorname{Re}\operatorname{tr}\sqrt{Y^\dagger Y}$, with the square root given by continuous functional calculus. No positivity, Hermiticity, or trace normalization is required of $\rho$, and no separate positivity condition is imposed on $X$.

QAlgorithms.Papers.Independent.Intakethird100.P83.entanglementGamma

For finite index types $A,B$ and any complex matrix $\rho$ indexed by $A\times B$, this defines the real supremum of all real numbers $r$ for which

$$\lim_{n\to\infty} \sup_{\substack{0\preceq Q\preceq I\\ -2^{-nr}I\preceq Q^\Gamma\preceq2^{-nr}I}} \operatorname{Re}\operatorname{tr}(\rho^{[n]}Q)=1, \qquad (\rho^{[n]})_{(a,b),(a',b')} =\prod_{t=0}^{n-1}\rho_{(a_t,b_t),(a'_t,b'_t)}.$$

The matrices $Q$ are indexed by $A^n\times B^n$, their partial transpose satisfies $(Q^\Gamma)_{(a,b),(a',b')}=Q_{(a,b'),(a',b)}$, and $X\preceq Y$ means $Y-X$ is positive semidefinite. The rates $r$ are not restricted to be nonnegative, and $\rho$ is not assumed to be a density matrix. The sequence includes $n=0$ with empty products equal to $1$. The outer supremum is taken in $\mathbb R$; its totalized value is $0$ if its defining set is empty or unbounded above.

QAlgorithms.Papers.Independent.Intakethird100.P83.entanglementN

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

$$\log_2\!\left(\operatorname{Re}\operatorname{tr} \sqrt{(\rho^\Gamma)^\dagger\rho^\Gamma}\right), \qquad \rho^\Gamma_{(a,b),(a',b')}=\rho_{(a,b'),(a',b)}.$$

The square root is obtained by continuous functional calculus. There is no density-matrix hypothesis. The logarithm is the total real logarithm used in the code; in particular, its value at zero is zero, so this definition gives zero when $\rho=0$.

QAlgorithms.Papers.Independent.Intakethird100.P83.entanglementW

For finite index types $A,B$ and any complex matrix $\rho$ indexed by $A\times B$, this defines

$$\log_2\!\left( \sup_{\substack{R=R^\dagger,\ \|R\|_{\mathrm{op}}\le1\\R^\Gamma\succeq0}} \operatorname{Re}\operatorname{tr}(\rho^\Gamma R) \right),$$

where the operator norm is induced by the Euclidean norm, $\succeq0$ means positive semidefinite, and $(X^\Gamma)_{(a,b),(a',b')}=X_{(a,b'),(a',b)}$. No density-matrix assumption is imposed on $\rho$. The logarithm is the total real function $\log x/\log2$, with $\log0=0$; consequently, a zero value of the inner supremum gives $0$. This includes the case of an empty product index type $A\times B$.

QAlgorithms.Papers.Independent.Intakethird100.P83.exactRegularized

For finite index types $A,B$ with decidable equality and any complex matrix $\rho$ on $A\times B$, define the matrix $\rho^{[m]}$ on $A^m\times B^m$ by

$$\rho^{[m]}_{(a,b),(a',b')}=\prod_{j=0}^{m-1}\rho_{(a_j,b_j),(a'_j,b'_j)}.$$

This declaration defines $\sup_{m\ge1}C(\rho^{[m]})/m$, where $C(\sigma)$ is the real supremum of $\log_2 k$ over positive real $k$ for which

$$\sup_{\substack{Q\ge0,\ I-Q\ge0\\ Q^\Gamma+k^{-1}I\ge0,\ k^{-1}I-Q^\Gamma\ge0}} \operatorname{Re}\operatorname{tr}(\sigma Q)=1.$$

Partial transpose $\Gamma$ exchanges the second-party row and column indices, and every matrix inequality means positive semidefiniteness. Only positive copy numbers occur: the natural-number index zero corresponds to one copy. This is a supremum, with no convergence assertion or density-matrix hypothesis; the real suprema use value zero for empty sets or sets unbounded above.

QAlgorithms.Papers.Independent.Intakethird100.P83.oneCopyExact

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

$$\sup\left\{\log_2 k:k\in\mathbb R,\ k>0,\quad \sup_{\substack{Q\ge0,\ I-Q\ge0\\ Q^\Gamma+k^{-1}I\ge0,\ k^{-1}I-Q^\Gamma\ge0}} \operatorname{Re}\operatorname{tr}(\rho Q)=1\right\}.$$

All matrix inequalities mean that the displayed matrix is positive semidefinite, and $Q^\Gamma_{(a,b),(a',b')}=Q_{(a,b'),(a',b)}$. The parameter $k$ is any positive real number, including values below one, rather than an integer. No density-matrix hypothesis on $\rho$ is imposed. Both suprema are real suprema, whose total convention gives zero for an empty set or a set unbounded above.

QAlgorithms.Papers.Independent.Intakethird100.P83.pptDistillationFidelity

For finite index types $A,B$, any complex matrix $\rho$ indexed by $A\times B$, and any real $k$, this defines

$$\sup\left\{\operatorname{Re}\operatorname{tr}(\rho Q): k>0,\ Q\succeq0,\ I-Q\succeq0,\ Q^\Gamma+k^{-1}I\succeq0,\ k^{-1}I-Q^\Gamma\succeq0\right\},$$

where $I$ is the identity and $(Q^\Gamma)_{(a,b),(a',b')}=Q_{(a,b'),(a',b)}$. No density-matrix assumption is imposed on $\rho$, and $k$ need not be an integer. For $k\le0$, the set is empty and the real-valued supremum used here is $0$. If $A\times B$ is empty, the value is also $0$.

QAlgorithms.Papers.Independent.Intakethird100.P83.qutritAntisymmetricState

This defines the complex matrix

$$\frac16(I-S)$$

on the coordinate space indexed by $\{0,1,2\}\times\{0,1,2\}$, where $I$ is the identity matrix and $S_{(i,j),(k,l)}=1$ exactly when $i=l$ and $j=k$, and is $0$ otherwise. Equivalently, its $((i,j),(k,l))$-entry is $\frac16(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk})$.

QAlgorithms.Papers.Independent.Intakethird100.P83.qutritKet

For any $a,b\in\{0,1,2\}$, this defines the complex-valued function on $\{0,1,2\}\times\{0,1,2\}$ that equals $1$ at $(a,b)$ and $0$ at every other pair.

QAlgorithms.Papers.Independent.Intakethird100.P83.qutritPsi

For every real number $\alpha$, this defines three complex-valued functions on $\{0,1,2\}\times\{0,1,2\}$. Writing $e_{ab}(i,j)=1$ when $(i,j)=(a,b)$, and $e_{ab}(i,j)=0$ otherwise, they are

$$\psi_0(\alpha)=\sqrt{\alpha}\,e_{01}+\sqrt{1-\alpha}\,e_{10},\qquad \psi_1(\alpha)=\sqrt{\alpha}\,e_{02}+\sqrt{1-\alpha}\,e_{20},\qquad \psi_2(\alpha)=\sqrt{\alpha}\,e_{12}+\sqrt{1-\alpha}\,e_{21}.$$

The square roots are real square roots, subsequently regarded as complex numbers. There is no restriction on $\alpha$; the real square-root function used here returns $0$ on negative arguments.

QAlgorithms.Papers.Independent.Intakethird100.P83.regroupedPower

For arbitrary types $A,B$, any complex matrix $\rho$ indexed by $A\times B$, and any natural number $n$, this defines a matrix indexed by pairs of functions $a:\{0,\ldots,n-1\}\to A$ and $b:\{0,\ldots,n-1\}\to B$, whose entries are

$$(\rho^{[n]})_{(a,b),(a',b')} =\prod_{t=0}^{n-1}\rho_{(a_t,b_t),(a'_t,b'_t)}.$$

There are no positivity or normalization assumptions. The definition includes $n=0$: both function spaces then have exactly one element, and the resulting one-by-one matrix has entry $1$, by the empty-product convention.

QAlgorithms.Papers.Independent.Intakethird100.P83.regroupedTensor

For arbitrary types $A,B,C,D$, a complex matrix $\rho$ indexed by $A\times B$, and a complex matrix $\sigma$ indexed by $C\times D$, this defines a matrix indexed by $(A\times C)\times(B\times D)$, with entries

$$\bigl(\operatorname{regroupedTensor}(\rho,\sigma)\bigr)_{((a,c),(b,d)),((a',c'),(b',d'))} =\rho_{(a,b),(a',b')}\,\sigma_{(c,d),(c',d')}.$$

No finiteness, nonemptiness, positivity, or normalization assumption is imposed.

QAlgorithms.Papers.Independent.Intakethird100.P83.rhoAlpha

For every real number $\alpha$, this defines a complex matrix indexed by pairs from $\{0,1,2\}$. Let $e_{ab}$ be the coordinate vector with value $1$ at $(a,b)$ and $0$ elsewhere, and set $\psi_0=\sqrt{\alpha}\,e_{01}+\sqrt{1-\alpha}\,e_{10}$, $\psi_1=\sqrt{\alpha}\,e_{02}+\sqrt{1-\alpha}\,e_{20}$, and $\psi_2=\sqrt{\alpha}\,e_{12}+\sqrt{1-\alpha}\,e_{21}$, using real square roots embedded in $\mathbb C$. The matrix is

$$\rho_\alpha=\frac13\sum_{m=0}^{2}\psi_m\psi_m^\dagger, \qquad (\rho_\alpha)_{x,y}=\frac13\sum_{m=0}^{2}\psi_m(x)\overline{\psi_m(y)}.$$

No normalization of the vectors is performed in this definition, and no restriction $\alpha\in[0,1]$ is imposed; real square roots of negative arguments are $0$.

QAlgorithms.Papers.Independent.Intakethird100.P83.supportNormMinimum

For finite index types $A,B$ with decidable equality and any complex matrix $\rho$ on $A\times B$, let $P_\rho$ be the orthogonal projection onto the range of the Euclidean linear map $x\mapsto\rho x$. This defines

$$\inf\left\{\|R^\Gamma\|_{\mathrm{op}}: R-P_\rho\ \text{and}\ I-R\ \text{are positive semidefinite}\right\},$$

where $R$ ranges over complex matrices on $A\times B$, $\Gamma$ exchanges the two $B$-indices, and the operator norm is induced by the Euclidean norm. No positivity or Hermiticity of $\rho$ is assumed: $P_\rho$ is defined from its range. The feasible matrices include $P_\rho$; when $\rho=0$, the value is zero.

QAlgorithms.Papers.Independent.Intakethird100.P83.supportSpace

For any finite type $I$ with decidable equality and any complex matrix $\rho$ on $I$, this defines the complex linear subspace

$$\left\{y:I\to\mathbb C:\exists x:I\to\mathbb C,\quad y_i=\sum_{j\in I}\rho_{ij}x_j\ \text{for every }i\in I\right\}.$$

Thus it is the range of matrix multiplication by $\rho$. No positivity, Hermiticity, or normalization is required; the zero matrix has range $\{0\}$, including when $I$ is empty.

QAlgorithms.Papers.Independent.Intakethird100.P83.symmetricSupport

This defines the complex linear subspace of all functions $\{0,1,2\}\times\{0,1,2\}\to\mathbb C$ spanned by

$$e_{01}+e_{10},\qquad e_{02}+e_{20},\qquad e_{12}+e_{21},$$

where $e_{ab}(i,j)$ equals $1$ if $(i,j)=(a,b)$ and $0$ otherwise. Thus its elements are exactly the complex linear combinations of these three specified functions.

QAlgorithms.Papers.Independent.Intakethird100.P83.transposeConjugate

For index types $A,B,C,D$ and any function $N$ from complex matrices on $A\times B$ to complex matrices on $C\times D$, this defines the function

$$X\longmapsto \bigl(N(X^\Gamma)\bigr)^\Gamma,$$

where the input partial transpose exchanges the two $B$-indices and the output partial transpose exchanges the two $D$-indices: $X^\Gamma_{(a,b),(a',b')}=X_{(a,b'),(a',b)}$, and likewise on the output. No linearity, positivity, or normalization condition is imposed on $N$.

QAlgorithms.Papers.Independent.Intakethird100.P83.wValue

For finite index types $A,B$ and any complex matrix $\rho$ indexed by $A\times B$, this defines

$$\sup\left\{ \operatorname{Re}\operatorname{tr}(\rho^\Gamma R): R=R^\dagger,\ \|R\|_{\mathrm{op}}\le1,\ R^\Gamma\succeq0 \right\},$$

where $\|\cdot\|_{\mathrm{op}}$ is the operator norm induced by the Euclidean norm, $\succeq0$ means positive semidefinite, and $(X^\Gamma)_{(a,b),(a',b')}=X_{(a,b'),(a',b)}$. No positivity or normalization assumption is imposed on $\rho$. The zero matrix is among the admissible $R$. If $A\times B$ is empty, this value is $0$.

QAlgorithms.Papers.Independent.Intakethird100.P83.wZeroValue

For finite index types $A,B$ with decidable equality and any complex matrix $\rho$ on $A\times B$, this defines the real supremum of $q=\operatorname{Re}\operatorname{tr}(\rho R)$ over all complex matrices $R$ such that

$$R\ge0,\qquad qI-R\ge0,\qquad R^\Gamma+I\ge0,\qquad I-R^\Gamma\ge0.$$

Here every inequality denotes positive semidefiniteness, and $\Gamma$ exchanges the two $B$-indices. No positivity or normalization condition is imposed on $\rho$. The zero matrix $R$ is included and contributes zero; the real supremum convention also gives zero if the set of values is unbounded above.