Definitions for Simulation of non-Pauli Channels

QAlgorithms.Papers.Independent.Intakethird100.P13.BlochCoordinates

Bloch coordinates are a function from $\{0,1,2\}$ to $\mathbb R$, equivalently an arbitrary ordered triple of real numbers. No norm or other constraint is imposed.

QAlgorithms.Papers.Independent.Intakethird100.P13.ClassicalTeleportationLaw

For any real-valued function $p$ of two indices in $\{0,1,2,3\}$, this predicate means that, for each fixed second index $k$, its values over the first index form a probability distribution:

$$p_{lk}\ge0\quad\text{for every }l,k\in\{0,1,2,3\}, \qquad \sum_{l=0}^{3}p_{lk}=1\quad\text{for every }k.$$

Zero entries are allowed; no condition requiring a sum over $k$ to equal one is included.

QAlgorithms.Papers.Independent.Intakethird100.P13.FourOutcomeLaw

For any real-valued function $w$ on $\{0,1,2,3\}$, this predicate means that every value is nonnegative and the four values sum to one:

$$w_k\ge0\quad\text{for every }k\in\{0,1,2,3\}, \qquad \sum_{k=0}^{3}w_k=1.$$

Individual values may be zero.

QAlgorithms.Papers.Independent.Intakethird100.P13.IsPauli

For any function $E$ from complex $2\times2$ matrices to complex $2\times2$ matrices, $\operatorname{IsPauli}(E)$ means that there exist real weights $w_0,w_1,w_2,w_3$ satisfying $w_k\ge0$ for every $k$ and $\sum_{k=0}^3w_k=1$, such that, for every complex $2\times2$ matrix $\rho$,

$$E(\rho)=\sum_{k=0}^3w_kP_k\rho P_k^\dagger,$$

where $(P_0,P_1,P_2,P_3)=(I_2,X,Y,Z)$ are the identity and standard Pauli matrices in the ordered Boolean basis. Zero weights are permitted, and the equality is required on all input matrices.

QAlgorithms.Papers.Independent.Intakethird100.P13.QubitMap

A qubit map is an arbitrary function from complex $2\times2$ matrices indexed by Boolean values to matrices of the same kind. Linearity, positivity, and trace preservation are not required.

QAlgorithms.Papers.Independent.Intakethird100.P13.QubitOp

A qubit operator is an arbitrary complex matrix whose rows and columns are indexed by the two Boolean values, equivalently a $2\times2$ complex matrix. No Hermiticity, positivity, trace, or unitarity condition is imposed.

QAlgorithms.Papers.Independent.Intakethird100.P13.ResourceOp

A resource operator is an arbitrary complex matrix whose rows and columns are indexed by ordered pairs of Boolean values, equivalently a $4\times4$ complex matrix. No Hermiticity, positivity, trace, or unitarity condition is imposed.

QAlgorithms.Papers.Independent.Intakethird100.P13.Simulable

For any complex matrix $\tau$ indexed by $\{0,1\}^2$ and any function $E$ from complex $2\times2$ matrices to complex $2\times2$ matrices, $\operatorname{Simulable}(\tau,E)$ means that there exist real numbers $p_{lk}$, indexed by $k,l\in\{0,1,2,3\}$, such that $p_{lk}\ge0$, $\sum_{l=0}^3p_{lk}=1$ for every $k$, and, for every complex $2\times2$ matrix $\rho$,

$$E(\rho)=\sum_{k=0}^3\sum_{l=0}^3p_{lk}P_lB_k(\tau,\rho)P_l^\dagger.$$

Here $P_0=I_2$, $P_1=X$, $P_2=Y$, and $P_3=Z$, with $X,Y,Z$ the standard Pauli matrices in the ordered Boolean basis $(\mathrm{false},\mathrm{true})$, identified with $(0,1)$; writing $\beta_k(c,a)=(P_k)_{ac}/\sqrt2$, the matrix $B_k$ is defined entrywise by

$$[B_k(\tau,\rho)]_{bb'} =\sum_{c,a,d,a'\in\{0,1\}} \overline{\beta_k(c,a)}\,\rho_{cd}\, \tau_{(a,b),(a',b')}\,\beta_k(d,a').$$

No positivity or trace condition on $\tau$, and no separate channel assumption on $E$, is included in this definition.

QAlgorithms.Papers.Independent.Intakethird100.P13.attainableRegion

For every pair of real numbers $\gamma,\eta$, this defines the real convex hull of the following eight triples:

$$(a,ab,-a^2b),\ (a,-ab,a^2b),\ (ab,a,-a^2b),\ (-ab,a,a^2b),\ (-a,ab,a^2b),\ (-a,-ab,-a^2b),\ (ab,-a,a^2b),\ (-ab,-a,-a^2b),$$

where $a=\sqrt{(1-\gamma)/(1-\eta)}$ and $b=1-\eta/\gamma$. Equivalently, the resulting set consists of all sums $\sum_{j=0}^7 t_jv_j$ of these triples with real coefficients $t_j\ge0$ and $\sum_{j=0}^7t_j=1$. No restrictions are placed on the parameters; division by zero gives zero and the real square root of a nonpositive number is zero, so repeated or collapsed triples are included.

QAlgorithms.Papers.Independent.Intakethird100.P13.attainableVertices

For every pair of real numbers $\gamma,\eta$, define

$$a=\sqrt{\frac{1-\gamma}{1-\eta}},\qquad b=1-\frac{\eta}{\gamma}.$$

This declaration defines a function from $\{0,\ldots,7\}$ to real triples, assigning, in that order,

$$\begin{aligned} 0&\mapsto(a,ab,-a^2b),& 1&\mapsto(a,-ab,a^2b),\\ 2&\mapsto(ab,a,-a^2b),& 3&\mapsto(-ab,a,a^2b),\\ 4&\mapsto(-a,ab,a^2b),& 5&\mapsto(-a,-ab,-a^2b),\\ 6&\mapsto(ab,-a,a^2b),& 7&\mapsto(-ab,-a,-a^2b). \end{aligned}$$

The parameters are unrestricted, division by zero is defined to be zero, and the real square root is zero for nonpositive arguments. In particular, $\eta=1$ gives $a=0$, and $\gamma=0$ gives $b=1$; the eight listed triples need not be distinct.

QAlgorithms.Papers.Independent.Intakethird100.P13.augmentedMoment

For any complex matrix $\tau$ indexed by $\{0,1\}^2$, any $j\in\{0,1,2,3\}$, and any $i\in\{0,1,2\}$, the real number called the augmented moment is

$$\operatorname{augmentedMoment}(\tau,j,i) =\operatorname{Re}\operatorname{Tr}\!\left[(P_j\otimes P_{i+1})\tau\right],$$

where $(P_0,P_1,P_2,P_3)=(I_2,X,Y,Z)$, with $X,Y,Z$ the standard Pauli matrices in the Boolean basis ordered as $(\mathrm{false},\mathrm{true})$. The matrix $\tau$ is arbitrary; it is not required to be a density matrix.

QAlgorithms.Papers.Independent.Intakethird100.P13.axialReplacement

For every real number $z$, this defines a function on all complex $2\times2$ matrices by

$$\rho\longmapsto\frac{\operatorname{tr}\rho}{2} \begin{pmatrix}1+z&0\\0&1-z\end{pmatrix},$$

where rows and columns are ordered false, true. There is no restriction on $z$ or on the input matrix $\rho$.

QAlgorithms.Papers.Independent.Intakethird100.P13.bellVertex

The function $\operatorname{bellVertex}$ assigns a real three-component vector, with coordinates ordered $0,1,2$, to each label in $\{0,1,2,3\}$ as follows:

$$\operatorname{bellVertex}(0)=(1,-1,1),\qquad \operatorname{bellVertex}(1)=(1,1,-1),\qquad \operatorname{bellVertex}(2)=(-1,-1,-1),\qquad \operatorname{bellVertex}(3)=(-1,1,1).$$

QAlgorithms.Papers.Independent.Intakethird100.P13.blochAmplitudeDamping

For every real number $\gamma$, this defines a function $\mathcal D_\gamma$ from complex $2\times2$ matrices to complex $2\times2$ matrices by

$$\mathcal D_\gamma(\rho)=\frac{\operatorname{tr}\rho}{2}(I+\gamma Z) +\frac{\sqrt{1-\gamma}}{2}\operatorname{tr}(X\rho)X +\frac{\sqrt{1-\gamma}}{2}\operatorname{tr}(Y\rho)Y +\frac{1-\gamma}{2}\operatorname{tr}(Z\rho)Z,$$

where, with rows and columns ordered as false, true,

$$X=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad Y=\begin{pmatrix}0&-i\\i&0\end{pmatrix},\qquad Z=\begin{pmatrix}1&0\\0&-1\end{pmatrix},$$

and $I$ is the identity matrix. There is no restriction on $\gamma$ or $\rho$; the real square root is zero for nonpositive arguments, so $\sqrt{1-\gamma}=0$ when $\gamma\ge1$.

QAlgorithms.Papers.Independent.Intakethird100.P13.bobVector

For any complex matrix $\tau$ indexed by $\{0,1\}^2$, its defined vector $\operatorname{bobVector}(\tau)$ is the real three-component vector whose coordinates are

$$[\operatorname{bobVector}(\tau)]_i =\operatorname{Re}\operatorname{Tr}\!\left[(I_2\otimes \sigma_i)\tau\right], \qquad i\in\{0,1,2\},$$

where $(\sigma_0,\sigma_1,\sigma_2)=(X,Y,Z)$ are the standard Pauli matrices in the ordered Boolean basis. No positivity or normalization condition on $\tau$ is assumed.

QAlgorithms.Papers.Independent.Intakethird100.P13.channelDiamondDistance

For any two functions $E,F$ from complex $2\times2$ matrices to complex $2\times2$ matrices, define their blockwise extensions to matrices indexed by Boolean pairs by

$$[\widehat E(\rho)]_{(a,b),(c,d)} =\left[E\!\left((\rho_{(a,i),(c,j)})_{i,j}\right)\right]_{bd},$$

and define $\widehat F$ in the same way; every index here is Boolean. This declaration defines

$$\sup_{\substack{\rho\succeq0\\\operatorname{tr}\rho=1}} \operatorname{Re}\operatorname{tr} \sqrt{(\widehat E(\rho)-\widehat F(\rho))^\dagger (\widehat E(\rho)-\widehat F(\rho))},$$

where $\rho$ ranges over positive semidefinite complex $4\times4$ matrices of trace one, $\dagger$ is conjugate transpose, and the square root is the positive semidefinite matrix square root obtained by functional calculus. No linearity or channel assumptions are imposed on $E,F$; the auxiliary index has exactly two values and there is no factor of $1/2$. The real supremum convention returns zero if the set of values is unbounded above.

QAlgorithms.Papers.Independent.Intakethird100.P13.channelTraceDistance

For any two functions $E,F$ from complex $2\times2$ matrices to complex $2\times2$ matrices, this defines the real supremum

$$\sup_{\substack{\rho\succeq0\\\operatorname{tr}\rho=1}} \operatorname{Re}\operatorname{tr} \sqrt{(E(\rho)-F(\rho))^\dagger(E(\rho)-F(\rho))},$$

where $\rho$ ranges over all positive semidefinite complex $2\times2$ matrices of trace one, $\dagger$ denotes conjugate transpose, and the square root is the positive semidefinite matrix square root obtained by functional calculus. No linearity or channel assumptions are imposed on $E$ or $F$, and there is no factor of $1/2$. The real supremum convention returns zero if the set of values is unbounded above.

QAlgorithms.Papers.Independent.Intakethird100.P13.closestPauli

For every Boolean $u$, every real number $\eta$, and every real triple $q=(q_0,q_1,q_2)$, set $a=\sqrt{1-\eta}$ and let $s=-1$ if $u$ is true and $s=1$ if $u$ is false. This defines the function on complex $2\times2$ matrices

$$\rho\longmapsto \frac{\operatorname{tr}\rho}{2}I+ \frac12\left( aq_0\operatorname{tr}(X\rho)X -sa q_1\operatorname{tr}(Y\rho)Y +s(1-\eta)q_2\operatorname{tr}(Z\rho)Z \right),$$

where $I$ is the identity and, in false, true index order,

$$X=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad Y=\begin{pmatrix}0&-i\\i&0\end{pmatrix},\qquad Z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$

There are no restrictions on $\eta,q,\rho$, and the real square root is zero for nonpositive arguments.

QAlgorithms.Papers.Independent.Intakethird100.P13.closestPoint

For every Boolean $u$, every real number $\eta$, and every real triple $q=(q_0,q_1,q_2)$, let $a=\sqrt{1-\eta}$. This defines the real triple

$$\begin{cases} (aq_0,-aq_1,-(1-\eta)q_2),&u=\mathrm{true},\\ (aq_0,aq_1,(1-\eta)q_2),&u=\mathrm{false}. \end{cases}$$

The inputs are unrestricted; the real square root is zero for nonpositive arguments, so $a=0$ whenever $\eta\ge1$.

QAlgorithms.Papers.Independent.Intakethird100.P13.contractPoint

For every real number $\alpha$ and every vector $q=(q_0,q_1,q_2)\in\mathbb R^3$, the definition gives

$$\operatorname{contractPoint}(\alpha,q) =(\sqrt{\alpha}\,q_0,\sqrt{\alpha}\,q_1,\alpha q_2).$$

There is no restriction on $\alpha$ or $q$; the real square root used here is $0$ when $\alpha<0$.

QAlgorithms.Papers.Independent.Intakethird100.P13.discardClassical

This defines a real-valued function of two indices $l,k\in\{0,1,2,3\}$ by

$$p(l,k)= \begin{cases} 1,&l=0,\\ 0,&l\ne0. \end{cases}$$

Its value is independent of the second index $k$.

QAlgorithms.Papers.Independent.Intakethird100.P13.fCoordinate

For any complex matrix $\tau$ indexed by $\{0,1\}^2$, any real array $p:\{0,1,2,3\}^2\to\mathbb R$, any $i\in\{0,1,2\}$, and any $j\in\{0,1,2,3\}$, the defined real coordinate is

$$f_{ij}(\tau,p)= \operatorname{Re}\operatorname{Tr}\!\left[(P_j\otimes P_{i+1})\tau\right] \cdot\frac14\sum_{k=0}^3\sum_{l=0}^3 (-1)^{\delta_{k0}+\delta_{j2}+\delta_{j0}+\delta_{kj} +\delta_{i+1,l}+\delta_{0l}}\,p_{lk},$$

where $(P_0,P_1,P_2,P_3)=(I_2,X,Y,Z)$, with the standard Pauli matrices in the ordered Boolean basis, and $\delta_{ab}$ is $1$ if $a=b$ and $0$ otherwise. No positivity or normalization condition on $\tau$ or $p$ is imposed.

QAlgorithms.Papers.Independent.Intakethird100.P13.labelKroneckerDelta

For every pair of labels $a,b\in\{0,1,2,3\}$, $\operatorname{labelKroneckerDelta}(a,b)$ is the natural number

$$\delta_{ab}= \begin{cases} 1,&a=b,\\ 0,&a\ne b. \end{cases}$$

QAlgorithms.Papers.Independent.Intakethird100.P13.noisyTeleport

For any complex matrix $\tau$ indexed by pairs of Boolean values and any real-valued function $p$ on $\{0,1,2,3\}^2$, this defines a function $E_{\tau,p}$ from Boolean-indexed complex matrices to matrices of the same kind by

$$E_{\tau,p}(\rho)= \sum_{k=0}^{3}\sum_{l=0}^{3}p_{lk}\, P_l C_{\tau,k}(\rho)P_l^\dagger, \qquad [C_{\tau,k}(\rho)]_{b,b'}= \frac12\sum_{c,a,d,a'\in\{\mathrm{false},\mathrm{true}\}} \overline{(P_k)_{a,c}}\, \rho_{c,d}\, \tau_{(a,b),(a',b')}\, (P_k)_{a',d}.$$

Here $\rho$ is any Boolean-indexed complex matrix, $b,b'$ are Boolean values, a bar denotes complex conjugation, $\dagger$ denotes conjugate transpose, and, in Boolean order $\mathrm{false},\mathrm{true}$,

$$P_0=\begin{pmatrix}1&0\\0&1\end{pmatrix},\quad P_1=\begin{pmatrix}0&1\\1&0\end{pmatrix},\quad P_2=\begin{pmatrix}0&-\mathrm i\\ \mathrm i&0\end{pmatrix},\quad P_3=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$

No density-matrix condition on $\tau$ or $\rho$, and no nonnegativity or normalization condition on $p$, is assumed.

QAlgorithms.Papers.Independent.Intakethird100.P13.normalizedQubitChoi

For every function $E$ from complex $2\times2$ matrices to complex $2\times2$ matrices, this defines a complex $4\times4$ matrix $C_E$, indexed by pairs of Boolean values, by

$$(C_E)_{(a,b),(c,d)} =\left[E\!\left(\tfrac12 e_{ac}\right)\right]_{bd} \qquad(a,b,c,d\in\{\mathrm{false},\mathrm{true}\}),$$

where $e_{ac}$ is the $2\times2$ matrix whose only nonzero entry is a $1$ in row $a$, column $c$. This is the blockwise application of $E$ to the second factor of the matrix $|\Phi\rangle\langle\Phi|$, where $|\Phi\rangle=(|\mathrm{false},\mathrm{false}\rangle+|\mathrm{true},\mathrm{true}\rangle)/\sqrt2$. No linearity, positivity, or trace preservation is assumed of $E$.

QAlgorithms.Papers.Independent.Intakethird100.P13.pauliDamping

For every Boolean $u$, every real number $\eta$, and every real triple $q=(q_0,q_1,q_2)$, this defines a function on all complex $2\times2$ matrices $\rho$. Write

$$X=\begin{pmatrix}0&1\\1&0\end{pmatrix},\quad Y=\begin{pmatrix}0&-i\\i&0\end{pmatrix},\quad Z=\begin{pmatrix}1&0\\0&-1\end{pmatrix},$$

with Boolean indices ordered false, true, and set

$$A_q(\rho)=\frac{\operatorname{tr}\rho}{2}I+ \frac12\bigl(q_0\operatorname{tr}(X\rho)X-q_1\operatorname{tr}(Y\rho)Y +q_2\operatorname{tr}(Z\rho)Z\bigr).$$

For any matrix $B$, set

$$D_\eta(B)=\frac{\operatorname{tr}B}{2}(I+\eta Z) +\frac{\sqrt{1-\eta}}2\bigl(\operatorname{tr}(XB)X+\operatorname{tr}(YB)Y\bigr) +\frac{1-\eta}{2}\operatorname{tr}(ZB)Z.$$

The output is $XD_\eta(A_q(\rho))X^\dagger$ if $u$ is true, and $D_\eta(A_q(\rho))$ if $u$ is false, where $\dagger$ denotes conjugate transpose. There are no restrictions on $\eta$ or $q$; the real square root is zero when its argument is nonpositive.

QAlgorithms.Papers.Independent.Intakethird100.P13.pauliMixture

For any function $w:\{0,1,2,3\}\to\mathbb R$, $\operatorname{pauliMixture}(w)$ is the function on all complex $2\times2$ matrices given by

$$\rho\longmapsto\sum_{k=0}^3w_kP_k\rho P_k^\dagger,$$

where $(P_0,P_1,P_2,P_3)=(I_2,X,Y,Z)$, with $X,Y,Z$ the standard Pauli matrices in the Boolean basis ordered as $(\mathrm{false},\mathrm{true})$, and ${}^\dagger$ denotes conjugate transpose. No nonnegativity or normalization condition on $w$, and no density-matrix condition on $\rho$, is required.

QAlgorithms.Papers.Independent.Intakethird100.P13.pauliPointChannel

For any real three-component vector $q=(q_0,q_1,q_2)$, $\operatorname{pauliPointChannel}(q)$ is the function on all complex $2\times2$ matrices defined by

$$\rho\longmapsto \frac{\operatorname{Tr}\rho}{2}I_2+ \frac12\left( q_0\operatorname{Tr}(X\rho)X -q_1\operatorname{Tr}(Y\rho)Y +q_2\operatorname{Tr}(Z\rho)Z \right),$$

where $X,Y,Z$ are the standard Pauli matrices in the ordered Boolean basis and the traces are complex-valued. The vector $q$ is arbitrary; membership in any specified region or a channel-positivity condition is not part of this definition.

QAlgorithms.Papers.Independent.Intakethird100.P13.pauliTetrahedron

The defined set $\operatorname{pauliTetrahedron}$ consists of all real three-component vectors expressible as

$$\lambda_0(1,-1,1)+\lambda_1(1,1,-1) +\lambda_2(-1,-1,-1)+\lambda_3(-1,1,1), \qquad \lambda_k\ge0,\quad \sum_{k=0}^3\lambda_k=1.$$

Thus it is the real convex hull of these four explicitly listed vectors, including their boundary points.

QAlgorithms.Papers.Independent.Intakethird100.P13.qubitBellBranch

For any complex matrix $\tau$ indexed by pairs of Boolean values, any $k\in\{0,1,2,3\}$, and any complex matrix $\rho$ indexed by Boolean values, this defines a Boolean-indexed complex matrix $C_{\tau,k}(\rho)$ whose entries, for Boolean $b,b'$, are

$$[C_{\tau,k}(\rho)]_{b,b'}= \frac12\sum_{c,a,d,a'\in\{\mathrm{false},\mathrm{true}\}} \overline{(P_k)_{a,c}}\, \rho_{c,d}\, \tau_{(a,b),(a',b')}\, (P_k)_{a',d},$$

where a bar denotes complex conjugation and, in Boolean order $\mathrm{false},\mathrm{true}$,

$$P_0=\begin{pmatrix}1&0\\0&1\end{pmatrix},\quad P_1=\begin{pmatrix}0&1\\1&0\end{pmatrix},\quad P_2=\begin{pmatrix}0&-\mathrm i\\ \mathrm i&0\end{pmatrix},\quad P_3=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$

Neither $\tau$ nor $\rho$ is required to be a density matrix.

QAlgorithms.Papers.Independent.Intakethird100.P13.regularPauliPoint

For all real numbers $\gamma,s_{30}$ and all vectors $s=(s_0,s_1,s_2)\in\mathbb R^3$, set $r=(1-\gamma)/(1-\gamma s_{30})$. The definition gives

$$\operatorname{regularPauliPoint}(\gamma,s_{30},s) =(\sqrt r\,s_0,\sqrt r\,s_1,rs_2).$$

No bounds or nonzero-denominator hypothesis are imposed: division by zero has value $0$, so the resulting vector is zero when $1-\gamma s_{30}=0$, and $\sqrt r=0$ when $r<0$.

QAlgorithms.Papers.Independent.Intakethird100.P13.rescaling

For every pair of real numbers $\gamma,s_{30}$, this defines

$$\frac{1-\gamma}{1-\gamma s_{30}}.$$

No nonzero-denominator or parameter-range hypothesis is imposed; when $1-\gamma s_{30}=0$, the value is zero by the convention for real division by zero.

QAlgorithms.Papers.Independent.Intakethird100.P13.resourcePauliCorrelation

For any complex matrix $\tau$ indexed by $\{0,1\}^2$, the defined real $3\times3$ matrix $\operatorname{resourcePauliCorrelation}(\tau)$ has row $j$, column $i$ entry

$$[\operatorname{resourcePauliCorrelation}(\tau)]_{ji} =\operatorname{Re}\operatorname{Tr}\!\left[(\sigma_j\otimes\sigma_i)\tau\right], \qquad j,i\in\{0,1,2\},$$

where $(\sigma_0,\sigma_1,\sigma_2)=(X,Y,Z)$ are the standard Pauli matrices in the ordered Boolean basis. No density-matrix assumption on $\tau$ is included.

QAlgorithms.Papers.Independent.Intakethird100.P13.sCoordinate

For any real array $p:\{0,1,2,3\}^2\to\mathbb R$, any $i\in\{0,1,2\}$, and any $j\in\{0,1,2,3\}$, the defined real coordinate is

$$s_{ij}(p)=\frac14\sum_{k=0}^3\sum_{l=0}^3 (-1)^{\delta_{k0}+\delta_{j2}+\delta_{j0}+\delta_{kj} +\delta_{i+1,l}+\delta_{0l}}\,p_{lk},$$

where $\delta_{ab}$ is $1$ when $a=b$ and $0$ otherwise, and the exponent is a sum of natural numbers. The array $p$ is arbitrary; nonnegativity and column-sum conditions are not required.

QAlgorithms.Papers.Independent.Intakethird100.P13.teleportationBellAmplitude

For every $k\in\{0,1,2,3\}$ and Boolean values $c,a$, this defines the complex number

$$\beta_k(c,a)=\frac{1}{\sqrt2}(P_k)_{a,c},$$

where, in Boolean order $\mathrm{false},\mathrm{true}$,

$$P_0=\begin{pmatrix}1&0\\0&1\end{pmatrix},\quad P_1=\begin{pmatrix}0&1\\1&0\end{pmatrix},\quad P_2=\begin{pmatrix}0&-\mathrm i\\ \mathrm i&0\end{pmatrix},\quad P_3=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$

Thus $a$ is the row index and $c$ is the column index of the selected matrix.

QAlgorithms.Papers.Independent.Intakethird100.P13.teleportationBellMixture

For any real-valued function $w$ on $\{0,1,2,3\}$, this defines the complex matrix $T_w$ indexed by pairs of Boolean values through

$$(T_w)_{(c,a),(d,a')}= \frac12\sum_{k=0}^{3}w_k(P_k)_{a,c}\,\overline{(P_k)_{a',d}},$$

for all Boolean $c,a,d,a'$, where complex conjugation is denoted by a bar and, in Boolean order $\mathrm{false},\mathrm{true}$,

$$P_0=\begin{pmatrix}1&0\\0&1\end{pmatrix},\quad P_1=\begin{pmatrix}0&1\\1&0\end{pmatrix},\quad P_2=\begin{pmatrix}0&-\mathrm i\\ \mathrm i&0\end{pmatrix},\quad P_3=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$

The weights are arbitrary real numbers: nonnegativity and a sum of one are not assumed.

QAlgorithms.Papers.Independent.Intakethird100.P13.teleportationBellProjector

For every $k\in\{0,1,2,3\}$, this defines a complex matrix $B_k$ indexed by pairs of Boolean values: for $x=(c,a)$ and $y=(d,a')$,

$$(B_k)_{(c,a),(d,a')}=\frac12(P_k)_{a,c}\,\overline{(P_k)_{a',d}},$$

where the bar denotes complex conjugation and, in Boolean order $\mathrm{false},\mathrm{true}$,

$$P_0=\begin{pmatrix}1&0\\0&1\end{pmatrix},\quad P_1=\begin{pmatrix}0&1\\1&0\end{pmatrix},\quad P_2=\begin{pmatrix}0&-\mathrm i\\ \mathrm i&0\end{pmatrix},\quad P_3=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$

QAlgorithms.Papers.Independent.Intakethird100.P13.teleportationPauli

This defines four complex $2\times2$ matrices $P_k$, indexed by $k\in\{0,1,2,3\}$. With rows and columns ordered as $\mathrm{false},\mathrm{true}$, they are

$$P_0=\begin{pmatrix}1&0\\0&1\end{pmatrix},\qquad P_1=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad P_2=\begin{pmatrix}0&-\mathrm i\\ \mathrm i&0\end{pmatrix},\qquad P_3=\begin{pmatrix}1&0\\0&-1\end{pmatrix},$$

where $\mathrm i^2=-1$.