Definitions for Entanglement manipulation and distillability beyond LOCC

QAlgorithms.Papers.Independent.Intakethird100.P19.ChoiPPT

For arbitrary finite types $A,B,C,D$, possibly empty, and a function $L$ from complex matrices on $A\times B$ to those on $C\times D$, this predicate requires a finite Kraus representation $L(X)=\sum_{\ell=1}^{m}K_\ell X K_\ell^\dagger$ for every $X$, where the $K_\ell$ have output rows $C\times D$ and input columns $A\times B$. It also requires that both $J$ and $J^{T_{C\times A}}$ be positive semidefinite, where

$$J_{((c,a),(d,b)),((c',a'),(d',b'))} =\bigl[L(E_{(a,b),(a',b')})\bigr]_{(c,d),(c',d')}, \qquad (J^{T_{C\times A}})_{(u,v),(u',v')}=J_{(u',v),(u,v')},$$

and $E_{i,j}$ denotes the matrix with a single $1$ at $(i,j)$. The Kraus sum may be empty; neither trace normalization of $J$ nor trace preservation of $L$ is required.

QAlgorithms.Papers.Independent.Intakethird100.P19.Coisometry

For arbitrary natural numbers $k,d$, including zero, and a complex $k\times d$ matrix $P$, this predicate means $PP^\dagger=I_k$, where $P^\dagger$ is the conjugate transpose. Thus the rows of $P$ are orthonormal. When $k=0$, the equality holds for the empty matrix; when $d=0<k$, it cannot hold.

QAlgorithms.Papers.Independent.Intakethird100.P19.CompleteNE

For arbitrary finite types $A,B,C,D$, possibly empty, and a function $L$ from complex matrices on $A\times B$ to those on $C\times D$, this predicate requires the following for every positive integer $k$: $L$ and its amplification $\widetilde L_k$ each admit a finite Kraus representation $T(X)=\sum_{\ell=1}^{m}K_\ell X K_\ell^\dagger$ valid for all $X$, with $m\ge0$, and $\widetilde L_k$ preserves the cone of finite nonnegative sums of tensor products of local positive semidefinite trace-one matrices. Explicitly, with $[k]=\{0,\ldots,k-1\}$,

$$[\widetilde L_k(X)]_{((c,e),(d,f)),((c',e'),(d',f'))} = \left[L\!\left(\bigl(X_{((a,e),(b,f)),((a',e'),(b',f'))}\bigr)_{(a,b),(a',b')}\right)\right]_{(c,d),(c',d')},$$

where $e,e',f,f'\in[k]$; the two input parties are $A\times[k]$ and $B\times[k]$, and the output parties are $C\times[k]$ and $D\times[k]$. In the preserved cone, weights need not sum to $1$, and empty sums are allowed. The quantifier imposes no condition at $k=0$, and no trace preservation is required.

QAlgorithms.Papers.Independent.Intakethird100.P19.CompletePPT

For arbitrary finite types $A,B,C,D$, possibly empty, and a function $L$ from complex matrices on $A\times B$ to those on $C\times D$, this predicate requires the following for every positive integer $k$: both $L$ and its amplification $\widetilde L_k$ have finite Kraus representations $T(X)=\sum_{\ell=1}^{m}K_\ell X K_\ell^\dagger$ for all $X$, with $m\ge0$, and $\widetilde L_k$ maps every matrix positive semidefinite together with its first-party partial transpose to another such matrix. With $[k]=\{0,\ldots,k-1\}$, the amplification is

$$[\widetilde L_k(X)]_{((c,e),(d,f)),((c',e'),(d',f'))} = \left[L\!\left(\bigl(X_{((a,e),(b,f)),((a',e'),(b',f'))}\bigr)_{(a,b),(a',b')}\right)\right]_{(c,d),(c',d')}.$$

The first-party partial transpose acts on $A\times[k]$ at the input and $C\times[k]$ at the output; on any $U\times V$ it is given by $(X^{T_U})_{(u,v),(u',v')}=X_{(u',v),(u,v')}$. No condition is imposed at $k=0$, and no trace normalization or trace preservation is required.

QAlgorithms.Papers.Independent.Intakethird100.P19.DNE

For finite types $A,B,C,D$ and a function $L$ from complex matrices on $A\times B$ to complex matrices on $C\times D$, define the reverse function by

$$L^\flat(Y)_{ij}=\operatorname{tr}\!\bigl(L(E_{ji})Y\bigr), \qquad i,j\in A\times B,$$

where $E_{ji}$ is the corresponding matrix unit. This predicate requires that both $L$ and $L^\flat$ admit finite Kraus representations: $L(X)=\sum_{r<m}K_rXK_r^\dagger$ for all $X$, and $L^\flat(Y)=\sum_{s<\ell}H_sYH_s^\dagger$ for all $Y$, for some natural numbers $m,\ell$ and appropriately sized complex matrices. It also requires that each function preserve finite nonnegative sums of tensor products of density matrices across its respective bipartition: density matrices here mean positive semidefinite matrices with trace $1$, and the weights in these sums need not sum to $1$. Both the Kraus sums and the tensor-product decompositions may be empty, and the four finite types may be empty. Neither function is required to preserve trace or have a normalized Kraus family.

QAlgorithms.Papers.Independent.Intakethird100.P19.KNE

For arbitrary finite types $A,B,C,D$, possibly empty, a natural number $k$, and a function $L$ from complex matrices on $A\times B$ to those on $C\times D$, this predicate requires $k>0$, a finite Kraus representation of $L$, and a finite Kraus representation of the amplified function $\widetilde L$. Writing $[k]=\{0,\ldots,k-1\}$, the latter function acts from matrices on $(A\times[k])\times(B\times[k])$ to those on $(C\times[k])\times(D\times[k])$, with entries

$$[\widetilde L(X)]_{((c,e),(d,f)),((c',e'),(d',f'))} = \left[L\!\left(\bigl(X_{((a,e),(b,f)),((a',e'),(b',f'))}\bigr)_{(a,b),(a',b')}\right)\right]_{(c,d),(c',d')}.$$

A finite Kraus representation of a function $T$ means $T(X)=\sum_{\ell=1}^{m}K_\ell X K_\ell^\dagger$ for all $X$, for some $m\ge0$. Finally, $\widetilde L$ must carry every finite sum $\sum_j w_j R_j\otimes S_j$, with $w_j\ge0$ and $R_j,S_j$ positive semidefinite of trace $1$ on the two input parties, to a sum of the same form on the two output parties. The weights need not sum to $1$, and empty sums are allowed. At $k=0$ the predicate is false.

QAlgorithms.Papers.Independent.Intakethird100.P19.KPPT

For arbitrary finite types $A,B,C,D$, possibly empty, a natural number $k$, and a function $L$ from complex matrices on $A\times B$ to those on $C\times D$, this predicate requires $k>0$ and finite Kraus representations of both $L$ and its amplification $\widetilde L$. Each such representation has the form $T(X)=\sum_{\ell=1}^{m}K_\ell X K_\ell^\dagger$ for every $X$, with $m\ge0$. With $[k]=\{0,\ldots,k-1\}$, amplification is defined by

$$[\widetilde L(X)]_{((c,e),(d,f)),((c',e'),(d',f'))} = \left[L\!\left(\bigl(X_{((a,e),(b,f)),((a',e'),(b',f'))}\bigr)_{(a,b),(a',b')}\right)\right]_{(c,d),(c',d')}.$$

Furthermore, whenever $X$ and its partial transpose on the first party $A\times[k]$ are positive semidefinite, $\widetilde L(X)$ and its partial transpose on $C\times[k]$ must also be positive semidefinite. For a matrix on $U\times V$, this partial transpose has entries $(X^{T_U})_{(u,v),(u',v')}=X_{(u',v),(u,v')}$. No trace condition is imposed. At $k=0$ the predicate is false.

QAlgorithms.Papers.Independent.Intakethird100.P19.LOCCDistillable

For arbitrary natural numbers $a,b$ and a complex matrix $\rho$ on $\{0,\ldots,a-1\}\times\{0,\ldots,b-1\}$, this predicate requires that $\rho$ be positive semidefinite with trace $1$, and that there exist maps $L_n$, one for every $n\ge0$, taking matrices on the two parties’ sequence spaces of length $n+1$ to matrices on $\{0,1\}\times\{0,1\}$, such that every $L_n$ is both a channel and a finite local-operation protocol. Here a channel has a finite Kraus representation $L_n(X)=\sum_iK_iXK_i^\dagger$ with $\sum_iK_i^\dagger K_i=I$. A finite local-operation protocol is generated recursively from identity maps by allowing either party to apply a finite local Kraus family satisfying this same completeness equality and then choose another such finite protocol according to the Kraus outcome, summing the resulting branches; intermediate dimensions and finite branch counts are natural numbers without additional positivity restrictions. The required convergence is

$$\lim_{n\to\infty} \left\|L_n\!\left(\rho^{[n+1]}\right)-\Omega_2\right\|_1=0,$$

where $\rho^{[m]}$ has entries given by products of $m$ entries of $\rho$, with each party’s sequence indices grouped together through the specified finite-set bijections, $(\Omega_2)_{(i,j),(r,s)}=\frac12$ when $i=j$ and $r=s$ and is zero otherwise, and $\|Z\|_1=\operatorname{Re}\operatorname{tr}\sqrt{Z^\dagger Z}$. Thus only positive copy counts occur. If $a=0$ or $b=0$, the initial trace-one requirement is impossible.

QAlgorithms.Papers.Independent.Intakethird100.P19.NE

For finite types $A,B,C,D$ and a function $L$ from complex matrices on $A\times B$ to complex matrices on $C\times D$, this predicate requires both a finite Kraus representation and preservation of the following class of matrices. The representation requirement is that there exist a natural number $m$ and matrices $K_j$, with rows indexed by $C\times D$ and columns by $A\times B$, such that for every matrix $X$,

$$L(X)=\sum_{j<m}K_jXK_j^\dagger.$$

The preservation requirement is that whenever $X$ has a finite decomposition $\sum_{r<q}w_r(R_r\otimes S_r)$, where $w_r\ge0$ and each $R_r,S_r$ is positive semidefinite with trace $1$ on $A,B$, respectively, $L(X)$ has a decomposition of the same form using density matrices on $C,D$. Neither decomposition requires its weights to sum to $1$. Empty sums and empty finite types are included. No equation $\sum_jK_j^\dagger K_j=1$ or trace-preservation requirement is imposed.

QAlgorithms.Papers.Independent.Intakethird100.P19.PPTPositive

For finite types $A,B$ and a complex matrix $X$ indexed by $A\times B$, this predicate requires that both $X$ and the matrix $X^{T_A}$ be positive semidefinite, where

$$X^{T_A}\bigl((a,b),(a',b')\bigr) =X\bigl((a',b),(a,b')\bigr).$$

No trace or normalization requirement is imposed. The zero matrix satisfies the predicate, and empty finite types are allowed.

QAlgorithms.Papers.Independent.Intakethird100.P19.PPTPreserving

For arbitrary finite types $A,B,C,D$, possibly empty, and a function $L$ from complex matrices on $A\times B$ to complex matrices on $C\times D$, this predicate requires both a finite Kraus representation and preservation of positivity under partial transpose. Explicitly, there must exist $m\in\mathbb N$ and matrices $K_1,\ldots,K_m$, with rows indexed by $C\times D$ and columns by $A\times B$, such that $L(X)=\sum_{\ell=1}^m K_\ell X K_\ell^\dagger$ for every $X$. Moreover, every matrix $X$ for which both $X$ and $X^{T_A}$ are positive semidefinite must have both $L(X)$ and $L(X)^{T_C}$ positive semidefinite, where $(X^{T_A})_{(a,b),(a',b')}=X_{(a',b),(a,b')}$, and the output partial transpose is defined analogously on $C$. The Kraus family may be empty, and no trace normalization or trace preservation is required.

QAlgorithms.Papers.Independent.Intakethird100.P19.RenyiOrder

This abbreviation denotes the extended nonnegative real numbers $[0,\infty]$, including both $0$ and $+\infty$, with no additional restriction on the allowed values.

QAlgorithms.Papers.Independent.Intakethird100.P19.SEP

For arbitrary finite types $A,B,C,D$, possibly empty, and a function $L$ from complex matrices on $A\times B$ to those on $C\times D$, this predicate requires a finite Kraus representation $L(X)=\sum_{\ell=1}^{m}K_\ell X K_\ell^\dagger$ for every $X$, with each $K_\ell$ having output rows $C\times D$ and input columns $A\times B$, together with the following condition on the matrix $J$ indexed by $(C\times A)\times(D\times B)$:

$$J_{((c,a),(d,b)),((c',a'),(d',b'))} =\bigl[L(E_{(a,b),(a',b')})\bigr]_{(c,d),(c',d')}.$$

Here $E_{u,v}$ has entry $1$ at $(u,v)$ and zero elsewhere. There must exist $q\in\mathbb N$, nonnegative real weights $w_j$, and positive semidefinite matrices $R_j$ on $C\times A$ and $S_j$ on $D\times B$, each of trace $1$, such that $J=\sum_{j=1}^{q}w_j(R_j\otimes S_j)$. The weights need not sum to $1$; empty sums, and hence the zero matrix, are allowed. No trace preservation of $L$ is required.

QAlgorithms.Papers.Independent.Intakethird100.P19.SeparableState

For finite types $A,B$ and a complex matrix $X$ indexed by $A\times B$, this predicate requires that $X$ be positive semidefinite with complex trace $1$, and that there exist a natural number $q$, real weights $w_0,\ldots,w_{q-1}\ge0$, and matrices $R_j$ on $A$ and $S_j$ on $B$, each positive semidefinite with trace $1$, such that

$$X=\sum_{j<q}w_j(R_j\otimes S_j).$$

There is no separately imposed equation $\sum_jw_j=1$, although it follows from the trace requirements. The finite sum is allowed to be empty, but then $X=0$, contradicting its required trace $1$. The types $A,B$ are not assumed nonempty; if either is empty, the predicate cannot hold.

QAlgorithms.Papers.Independent.Intakethird100.P19.Undetected

For arbitrary natural numbers $a,b,c,d,n$, a family $W_0,\ldots,W_{n-1}$ of complex square matrices indexed by $(\{0,\ldots,c-1\}\times\{0,\ldots,a-1\})\times(\{0,\ldots,d-1\}\times\{0,\ldots,b-1\})$, and a function $L$ from matrices indexed by $\{0,\ldots,a-1\}\times\{0,\ldots,b-1\}$ to matrices indexed by $\{0,\ldots,c-1\}\times\{0,\ldots,d-1\}$, this predicate requires a finite Kraus representation $L(X)=\sum_{j=0}^{m-1}K_jXK_j^\dagger$ for some $m\in\mathbb N$ and rectangular complex matrices $K_j$, and requires $\operatorname{Re}\operatorname{tr}(W_iC_L)\ge0$ for every $i<n$. The matrix $C_L$ is defined explicitly by

$$(C_L)_{((u,r),(v,s)),((u',r'),(v',s'))} =\bigl[L(E_{(r,s),(r',s')})\bigr]_{(u,v),(u',v')},$$

where $E_{x,y}$ has entry $1$ at $(x,y)$ and zero elsewhere. There is no Kraus completeness condition, no trace-preservation requirement, and no witness or Hermitian hypothesis on the $W_i$. All dimensions may be zero; when $n=0$, only the finite Kraus representation is required.

QAlgorithms.Papers.Independent.Intakethird100.P19.UnitBipartiteKet

For natural numbers $a,b$, a value of this structure consists of a complex vector $\psi$ indexed by $\{0,\ldots,a-1\}\times\{0,\ldots,b-1\}$, together with the requirement $\|\psi\|_2=1$. Equivalently, $\sum_{i<a}\sum_{j<b}|\psi_{ij}|^2=1$. Zero dimensions are permitted as parameters, but if $a=0$ or $b=0$, the vector space contains only the zero vector, so no value of this structure exists.

QAlgorithms.Papers.Independent.Intakethird100.P19.UnitBipartiteKet.rho

For any natural numbers $a,b$ and any structured unit vector $\psi$ indexed by $\{0,\ldots,a-1\}\times\{0,\ldots,b-1\}$, this definition returns the complex matrix

$$\rho_\psi\bigl((i,j),(k,\ell)\bigr)=\psi_{ij}\,\overline{\psi_{k\ell}}.$$

The input includes the requirement $\|\psi\|_2=1$; consequently, there are no such inputs when either dimension is zero.

QAlgorithms.Papers.Independent.Intakethird100.P19.Witness

For any finite types $A,B$ and complex square matrix $W$ indexed by $A\times B$, this predicate means that $W$ is Hermitian and $\operatorname{Re}\operatorname{tr}(WX)\ge0$ for every matrix $X$ admitting a representation

$$X=\sum_{j=0}^{q-1}w_j(A_j\otimes B_j),$$

where $q$ is any natural number, each $w_j$ is a nonnegative real number, and each $A_j$ and $B_j$ is a positive semidefinite matrix of trace $1$ indexed by $A$ and $B$, respectively. The weights need not sum to $1$, and $q=0$ gives $X=0$. The predicate imposes neither positive semidefiniteness nor a normalization condition on $W$. Empty index types are allowed.

QAlgorithms.Papers.Independent.Intakethird100.P19.antisymmetricDensity

For every natural number $d$, this defines

$$\bigl(d\,\max\{d-1,0\}\bigr)^{-1}(I_{d^2}-F_d),$$

as a complex matrix indexed by pairs from $\{0,\ldots,d-1\}$, where $(F_d)_{(i,j),(k,l)}=1$ exactly when $i=l$ and $j=k$, and is zero otherwise. The subtraction $d-1$ is natural-number subtraction, so it is truncated at zero, and inversion satisfies $0^{-1}=0$. Thus $d=1$ gives the zero matrix, and $d=0$ gives the matrix on the empty index set; no positive-dimension hypothesis is required.

QAlgorithms.Papers.Independent.Intakethird100.P19.betaWernerState

For every natural number $d$ and arbitrary real number $\beta$, this defines the complex matrix

$$\bigl(d^2-(\beta+1)\bigr)^{-1} \left(I_{d^2}-\frac{\beta+1}{d}F_d\right),$$

indexed by pairs from $\{0,\ldots,d-1\}$, where $(F_d)_{(i,j),(k,l)}=1$ exactly when $i=l$ and $j=k$, and is zero otherwise. No restriction on $\beta$, positivity hypothesis, or density condition is imposed. Division and inversion are total, with $0^{-1}=0$; in particular, when $d^2=\beta+1$, the matrix is zero. For $d=0$, the real quotient $(\beta+1)/d$ is $0$, and the resulting matrix has an empty index set.

QAlgorithms.Papers.Independent.Intakethird100.P19.binaryMeasurePrepare

For any finite type $I$, any type $J$, arbitrary complex square matrices $A$ indexed by $I$ and $\rho,\sigma$ indexed by $J$, this defines the function on complex $I\times I$ matrices given by

$$X\longmapsto \operatorname{tr}(AX)\,\rho+\operatorname{tr}((I_I-A)X)\,\sigma,$$

where $I_I$ is the identity matrix and the trace coefficients are complex numbers. There are no positivity, effect, density, or trace-preservation hypotheses, and $J$ need not be finite. If $I$ is empty, the function is identically zero.

QAlgorithms.Papers.Independent.Intakethird100.P19.compressChoi

For arbitrary natural numbers $d,k$, an arbitrary function $L$ from complex matrices on $\{0,\ldots,d-1\}^2$ to matrices on the same index set, and arbitrary complex $k\times d$ matrices $P,Q$, define a matrix $C_L$, indexed by $((c,a),(e,b))$ with all four indices less than $d$, by

$$(C_L)_{((c,a),(e,b)),\,((c',a'),(e',b'))} =\bigl[L(E_{(a,b),(a',b')})\bigr]_{(c,e),(c',e')},$$

where $E_{u,v}$ has a single entry $1$ at $(u,v)$ and zero elsewhere. The defined output is

$$TC_LT^\dagger,\qquad T=(I_d\otimes P)\otimes(I_d\otimes Q),$$

indexed by $(\{0,\ldots,d-1\}\times\{0,\ldots,k-1\})^2$. No linearity, positivity, channel condition, or coisometry condition is imposed on $L,P,Q$; zero dimensions are included.

QAlgorithms.Papers.Independent.Intakethird100.P19.finiteSchmidtRank

For arbitrary natural numbers $a,b$ and an arbitrary complex vector $\psi$ indexed by $\{0,\ldots,a-1\}\times\{0,\ldots,b-1\}$, form $R_{rs}=\sum_{t<b}\psi_{rt}\overline{\psi_{st}}$. This definition returns the number of strictly positive eigenvalues of $R$, counting multiplicities: explicitly, it counts those indices $i<a$ for which the eigenvalue at ascending position $a-1-i$ is greater than zero. No normalization of $\psi$ is assumed. Zero eigenvalues are excluded, and the count is zero if either dimension is zero or $\psi=0$.

QAlgorithms.Papers.Independent.Intakethird100.P19.frobenius

For any finite type $I$ and arbitrary complex square matrix $W$ indexed by $I$, this definition returns

$$\sqrt{\sum_{i\in I}\sum_{j\in I}|W_{ij}|^2}.$$

No Hermitian, positivity, or normalization assumption is imposed. If $I$ is empty, both sums are empty and the value is $0$.

QAlgorithms.Papers.Independent.Intakethird100.P19.localCoisometryCompression

For arbitrary natural numbers $k,d$, complex $k\times d$ matrices $P,Q$, and an arbitrary complex matrix $X$ indexed by pairs in $\{0,\ldots,d-1\}^2$, this defines the matrix indexed by $\{0,\ldots,k-1\}^2$ given by

$$(P\otimes Q)X(P\otimes Q)^\dagger.$$

Neither $PP^\dagger=I_k$ nor $QQ^\dagger=I_k$ is assumed, and $X$ need not be positive or have trace one. Zero dimensions are allowed.

QAlgorithms.Papers.Independent.Intakethird100.P19.negativityMap

For every natural number $d$, let $F_d$ have entries $(F_d)_{(i,j),(k,l)}=1$ when $i=l$ and $j=k$, and zero otherwise, and set $A_d=(dI_{d^2}+F_d)/(d+1)$. This defines the function on complex matrices indexed by $\{0,\ldots,d-1\}\times\{0,\ldots,d-1\}$ given by

$$X\longmapsto \operatorname{tr}(A_dX)\,(d^2)^{-1}I_{d^2} +\operatorname{tr}((I_{d^2}-A_d)X)\,E_d,$$

where $(E_d)_{(i,j),(k,l)}=d^{-1}$ when $i=j$ and $k=l$, and is zero otherwise. The trace coefficients are complex, and the input $X$ has no positivity or normalization requirement. The definition includes $d=0$, with every inverse of zero interpreted as zero and the matrices indexed by the empty set.

QAlgorithms.Papers.Independent.Intakethird100.P19.noiseRobustness

For arbitrary finite types $A,B$, possibly empty, and arbitrary complex matrices $\rho,\sigma$ on $A\times B$, this definition is the infimum in $[0,\infty]$ of all nonnegative real numbers $t$, embedded into $[0,\infty]$, such that $\sigma$ is positive semidefinite of trace $1$, $\sigma$ is a finite sum of nonnegative real multiples of tensor products of local positive semidefinite trace-one matrices, and $\rho+t\sigma$ is also a finite sum of that form. Explicitly, such a sum is $\sum_{j=1}^{q}w_jR_j\otimes S_j$, where $q\in\mathbb N$, $w_j\ge0$, $\operatorname{tr}R_j=\operatorname{tr}S_j=1$, and $R_j,S_j$ are positive semidefinite on $A,B$, respectively; no condition requires the weights to sum to $1$, and the empty sum is allowed. The matrix $\rho$ is not assumed to be a density matrix. If no admissible $t$ exists, including when $\sigma$ fails the stated conditions, the value is $+\infty$.

QAlgorithms.Papers.Independent.Intakethird100.P19.partyAmplification

For arbitrary types $A,B,C,D,E,F$, with no finiteness or nonemptiness assumptions, and any function $L$ from complex matrices on $A\times B$ to those on $C\times D$, this definition gives a function $\widetilde L$ from complex matrices on $(A\times E)\times(B\times F)$ to those on $(C\times E)\times(D\times F)$. For every input matrix $X$ and every output row and column $((c,e),(d,f))$ and $((c',e'),(d',f'))$, it is defined by

$$[\widetilde L(X)]_{((c,e),(d,f)),((c',e'),(d',f'))} =[L(Y)]_{(c,d),(c',d')}, \qquad Y_{(a,b),(a',b')}= X_{((a,e),(b,f)),((a',e'),(b',f'))}.$$

No linearity, positivity, or normalization condition is imposed on $L$.

QAlgorithms.Papers.Independent.Intakethird100.P19.partyCopies

For arbitrary natural numbers $a,b,n$ and an arbitrary complex matrix $\rho$ indexed by $\{0,\ldots,a-1\}\times\{0,\ldots,b-1\}$, choose the particular bijections used in the definition from length-$n$ sequences over these two index sets to finite index sets of their respective cardinalities $a^n,b^n$. Writing $x_r$ and $y_s$ for the sequences represented by indices $r<a^n$ and $s<b^n$, this defines a matrix $\rho^{[n]}$ by

$$\rho^{[n]}_{(r,s),(r',s')} =\prod_{j=0}^{n-1}\rho_{(x_r(j),y_s(j)),(x_{r'}(j),y_{s'}(j))}.$$

No positivity or normalization of $\rho$ is assumed. When $n=0$, both sequence sets have one element, even if $a=0$ or $b=0$, and the empty product makes the output the $1\times1$ matrix with entry $1$.

QAlgorithms.Papers.Independent.Intakethird100.P19.partyGroupedChoi

For finite types $A,B$, arbitrary types $C,D$, and any function $L$ from complex matrices on $A\times B$ to complex matrices on $C\times D$, this definition returns the matrix indexed by $(C\times A)\times(D\times B)$ with entries

$$J_L\bigl(((c,a),(d,b)),((c',a'),(d',b'))\bigr) = L\!\left(E_{(a,b),(a',b')}\right)_{(c,d),(c',d')},$$

where the matrix unit on the right has one entry equal to $1$ at the indicated row and column and zeros elsewhere. There are no finiteness assumptions on $C,D$ and no linearity, positivity, or normalization assumptions on $L$. Empty types are allowed.

QAlgorithms.Papers.Independent.Intakethird100.P19.petzStateDivergence

For any finite type $I$, real number $\alpha$, and complex square matrices $\rho,\sigma$ indexed by $I$, this definition returns an element of $[0,\infty]$. It returns $\infty$ unless both matrices are positive semidefinite and have complex trace $1$. For such matrices, write $K$ for the condition $\ker\sigma\subseteq\ker\rho$, and define $P_t(H)$ by spectral functional calculus using the function that is $0$ at $0$ and $x^t$ at nonzero $x$. If $\alpha=1$, the value is $\infty$ when $K$ fails, and otherwise is

$$\frac{\max\!\left\{0,\operatorname{Re}\operatorname{tr}\!\left[f(\rho)-\rho\,g(\sigma)\right]\right\}}{\log2}, \qquad f(x)=x\log x,\quad g(x)=\log x,$$

where $f(\rho)$ and $g(\sigma)$ use spectral functional calculus and $\log0=0$. If $\alpha\ne1$, the value is $\infty$ when $\alpha>1$ and $K$ fails; in every other case, set $z=\operatorname{Re}\operatorname{tr}(P_\alpha(\rho)P_{1-\alpha}(\sigma))$, and return $\max\{0,\log_2z/(\alpha-1)\}$ if $z>0$, and $\infty$ otherwise. There is no restriction on $\alpha$; in particular, $P_0$ is zero on the kernel rather than the identity there. If $I$ is empty, the density conditions fail and the value is $\infty$.

QAlgorithms.Papers.Independent.Intakethird100.P19.powerDividedDifference

For arbitrary real numbers $\alpha,x,y$, this defines

$$D_\alpha(x,y)= \begin{cases} (1-\alpha)x^{-\alpha},&x=y,\\[2mm] \dfrac{x^{1-\alpha}-y^{1-\alpha}}{x-y},&x\ne y. \end{cases}$$

Every real power here uses the total real-power operation appearing in the code. There are no positivity assumptions on $x,y$ and no restriction on $\alpha$; in particular, the equal-input branch also applies at $x=y=0$, using that operation’s defined value there.

QAlgorithms.Papers.Independent.Intakethird100.P19.rankThreeKet

For every natural number $d$, this defines the complex Euclidean vector $v_d$ indexed by $0\le i,j<d$ through

$$v_d(i,j)= \begin{cases} 1/\sqrt3,&i=j\text{ and }i<3,\\ 0,&\text{otherwise}. \end{cases}$$

Equivalently, it is $\frac1{\sqrt3}\sum_{i<\min(d,3)}|i,i\rangle$, with $|i,i\rangle$ the corresponding coordinate unit vector. The definition includes $d<3$: its squared norm is $\min(d,3)/3$, so it is zero for $d=0$ and has norm one precisely when $d\ge3$.

QAlgorithms.Papers.Independent.Intakethird100.P19.rankThreeMap

For every natural number $d$, let $v_d(i,j)=1/\sqrt3$ when $i=j<3$, and $0$ otherwise, for $0\le i,j<d$. Form $W$ by transposing the first subsystem of $v_dv_d^\dagger$, explicitly

$$W_{(i,j),(r,s)}=v_d(r,j)\overline{v_d(i,s)}, \qquad A=\frac{W+2I_{d^2}}3.$$

This defines the function from arbitrary complex $d^2\times d^2$ matrices to complex $4\times4$ matrices by

$$X\longmapsto \operatorname{tr}(AX)\frac{I_4}{4} +\operatorname{tr}((I_{d^2}-A)X)\,\Omega_2,$$

where $(\Omega_2)_{(i,j),(r,s)}=\frac12$ if $i=j$ and $r=s$, and is zero otherwise, with $i,j,r,s\in\{0,1\}$. These are complex traces, without taking real parts. No hypothesis $d\ge3$ is imposed; the definition also includes $d=0,1,2$, for which the specified $v_d$ is not a unit vector.

QAlgorithms.Papers.Independent.Intakethird100.P19.relativeRenyi

For any finite types $A,B$, real number $\alpha$, and arbitrary complex matrix $\rho$ indexed by $A\times B$, this is the infimum in $[0,\infty]$ of $D_\alpha(\rho,\sigma)$ over matrices $\sigma$ that are positive semidefinite with trace $1$ and admit a finite representation $\sigma=\sum_{j=0}^{q-1}w_j(A_j\otimes B_j)$, where $q\in\mathbb N$, $w_j\ge0$, and every $A_j,B_j$ is positive semidefinite with trace $1$; no separate condition $\sum_jw_j=1$ is imposed in this representation. Here $D_\alpha$ is $\infty$ unless $\rho,\sigma$ are both positive semidefinite with trace $1$. For such matrices, let $K$ mean $\ker\sigma\subseteq\ker\rho$, and let $P_t(H)$ apply spectrally the function $0\mapsto0$, $x\mapsto x^t$ for $x\ne0$. At $\alpha=1$, $D_\alpha$ is $\infty$ if $K$ fails, and otherwise equals $\max\{0,\operatorname{Re}\operatorname{tr}(f(\rho)-\rho g(\sigma))\}/\log2$, with $f(x)=x\log x$, $g(x)=\log x$, and $\log0=0$. For $\alpha\ne1$, it is $\infty$ if $\alpha>1$ and $K$ fails; otherwise, writing $z=\operatorname{Re}\operatorname{tr}(P_\alpha(\rho)P_{1-\alpha}(\sigma))$, it is $\max\{0,\log_2z/(\alpha-1)\}$ when $z>0$, and $\infty$ when $z\le0$. No density assumption is made on the input $\rho$; an empty set of candidate values has infimum $\infty$, including when either $A$ or $B$ is empty.

QAlgorithms.Papers.Independent.Intakethird100.P19.schmidtAt

For arbitrary natural numbers $a,b,j$ and an arbitrary complex vector $\psi$ indexed by $\{0,\ldots,a-1\}\times\{0,\ldots,b-1\}$, let $R$ be the matrix with entries $R_{rs}=\sum_{t<b}\psi_{rt}\overline{\psi_{st}}$. This definition returns the eigenvalue at position $a-1-j$ in the ascending eigenvalue list of $R$, counting multiplicities, when $j<a$, and returns $0$ when $j\ge a$. Thus it lists these eigenvalues in descending order and extends the list by zeros. No unit-norm assumption is imposed; in particular, the function is identically zero if $a=0$, $b=0$, or $\psi=0$.

QAlgorithms.Papers.Independent.Intakethird100.P19.schmidtRenyi

For arbitrary natural numbers $a,b$, an order $\alpha\in[0,\infty]$, and any vector $\psi\in\mathbb C^{\{0,\ldots,a-1\}\times\{0,\ldots,b-1\}}$, with no normalization assumption, form the matrix $R$ with entries $R_{ij}=\sum_{k=0}^{b-1}\psi_{ik}\overline{\psi_{jk}}$, and list its eigenvalues with multiplicity in decreasing order as $\lambda_0,\ldots,\lambda_{a-1}$. Let $r=\#\{i:\lambda_i>0\}$, and set $\lambda_0^{*}=\lambda_0$ if $a>0$, and $\lambda_0^{*}=0$ otherwise. The definition returns

$$\begin{cases} -\log_2\lambda_0^{*},&\alpha=\infty,\\ \log_2 r,&\alpha=0,\\ -\sum_{i=0}^{a-1}\lambda_i\log_2\lambda_i,&\alpha=1,\\ \displaystyle\frac{\log_2\!\left(\sum_{i=0}^{a-1}\lambda_i^{\alpha}\right)}{1-\alpha},&0<\alpha<\infty,\ \alpha\ne1. \end{cases}$$

Here $\log_2 x=\log x/\log2$ uses the total real logarithm, in particular $\log_2 0=0$. Zero dimensions and the zero vector are included; if all the $\lambda_i$ vanish, every branch returns $0$.

QAlgorithms.Papers.Independent.Intakethird100.P19.sourceOutputChoi

For a finite type $I$, an arbitrary type $J$, and any function $L$ from complex matrices on $I$ to complex matrices on $J$, this definition returns the matrix indexed by $J\times I$ whose entries are

$$C_L\bigl((u,i),(v,j)\bigr)=L(E_{ij})_{uv},$$

where $E_{ij}$ is the matrix on $I$ with its sole nonzero entry equal to $1$ in row $i$, column $j$. No finiteness assumption is imposed on $J$, and no linearity, positivity, or normalization assumption is imposed on $L$. If either indexing type is empty, the resulting matrix has an empty index set.

QAlgorithms.Papers.Independent.Intakethird100.P19.squaredSchmidtCoefficient

For arbitrary natural numbers $a,b$, an arbitrary complex vector $\psi$ indexed by $\{0,\ldots,a-1\}\times\{0,\ldots,b-1\}$, and an index $i<a$, form the matrix

$$R_{rs}=\sum_{t<b}\psi_{rt}\,\overline{\psi_{st}}.$$

This definition returns the eigenvalue of $R$ at position $a-1-i$ in its ascending eigenvalue list, counting multiplicities; thus $i=0$ selects the largest eigenvalue. Here the ascending list is obtained by reversing the Hermitian matrix’s descending eigenvalue list; the custom eigenvalue function would return zeros for a non-Hermitian matrix, but $R$ is always positive semidefinite. No normalization of $\psi$ is required. If $b=0$ or $\psi=0$, every available value is zero; if $a=0$, there is no admissible index $i$.

QAlgorithms.Papers.Independent.Intakethird100.P19.standardStateRobustness

For arbitrary finite types $A,B$, possibly empty, and any complex matrix $\rho$ on $A\times B$, this definition is the real infimum

$$\inf\left\{s\in\mathbb R:\ s\ge0,\ \exists\sigma,\ \sigma\text{ is positive semidefinite of trace }1,\ \sigma\text{ and }\rho+s\sigma\text{ are separable positive sums}\right\}.$$

A separable positive sum means a matrix $\sum_{j=1}^{q}w_jR_j\otimes S_j$, with $q\in\mathbb N$, nonnegative real $w_j$, and positive semidefinite trace-one local matrices $R_j$ on $A$ and $S_j$ on $B$; the weights need not sum to $1$, and $q=0$ gives zero. There is no density-matrix assumption on $\rho$. If the set whose infimum is taken is empty, this real-valued definition returns $0$; in particular this occurs when either $A$ or $B$ is empty.

QAlgorithms.Papers.Independent.Intakethird100.P19.swapOperator

For every natural number $d$, this defines the complex square matrix $F_d$ indexed by pairs from $\{0,\ldots,d-1\}$, with

$$(F_d)_{(i,j),(k,l)} =\begin{cases}1,&i=l\text{ and }j=k,\\0,&\text{otherwise}.\end{cases}$$

Thus its action exchanges the two coordinates. The definition includes $d=0$, when the index set is empty.

QAlgorithms.Papers.Independent.Intakethird100.P19.tensorMaps

For arbitrary types $A,B,C,D,E,F,G,H$, assuming only that $A,B,E,F$ are finite, and arbitrary functions $L$ from complex matrices on $A\times B$ to those on $C\times D$ and $M$ from matrices on $E\times F$ to those on $G\times H$, this definition gives a function $\mathcal T$ from matrices on $(A\times E)\times(B\times F)$ to those on $(C\times G)\times(D\times H)$, defined by

$$\begin{aligned} [\mathcal T(X)]_{((c,g),(d,h)),((c',g'),(d',h'))} ={}&\sum_{i,j\in A\times B}\sum_{s,t\in E\times F} X_{((i_A,s_E),(i_B,s_F)),((j_A,t_E),(j_B,t_F))}\\ &\quad\cdot[L(E_{i,j})]_{(c,d),(c',d')} \cdot[M(E_{s,t})]_{(g,h),(g',h')}. \end{aligned}$$

Here subscripts indicate the components of each pair, and $E_{u,v}$ denotes the matrix with entry $1$ at $(u,v)$ and zero elsewhere. Neither function is assumed linear or positive. Empty input indexing sets give empty sums, hence zero output entries.

QAlgorithms.Papers.Independent.Intakethird100.P19.traceBilinearDual

For finite types $I,J$ and any function $L$ from complex matrices on $I$ to complex matrices on $J$, this definition produces a function $L^\flat$ in the reverse direction, given for every matrix $Y$ on $J$ and indices $i,j\in I$ by

$$L^\flat(Y)_{ij} =\operatorname{tr}\!\bigl(L(E_{ji})Y\bigr) =\sum_{u,v\in J}L(E_{ji})_{uv}Y_{vu},$$

where $E_{ji}$ has entry $1$ in row $j$, column $i$, and zeros elsewhere. The definition assumes neither linearity nor positivity nor trace preservation of $L$, and uses no complex conjugation in this trace pairing. Empty finite types are included; in particular, if $J$ is empty, every displayed trace is zero.

QAlgorithms.Papers.Independent.Intakethird100.P19.witnessMap

For arbitrary natural numbers $a,b$ and any complex square matrix $W$ indexed by $\{0,\ldots,a-1\}\times\{0,\ldots,b-1\}$, let $A_W=(W+2I_{ab})/3$. This defines the function from matrices on that index set to matrices indexed by $\{0,1\}\times\{0,1\}$ given by

$$X\longmapsto \operatorname{tr}(A_WX)\frac{I_4}{4} +\operatorname{tr}((I_{ab}-A_W)X)\,E_2,$$

where $(E_2)_{(i,j),(k,l)}=\tfrac12$ when $i=j$ and $k=l$, and is $0$ otherwise. No witness, Hermitian, positivity, or normalization condition is required of $W$. If $a=0$ or $b=0$, the function is identically zero.