Definitions for Classical Shadow Tomography with Locally Scrambled Quantum Dynamics

QAlgorithms.Papers.Independent.Intakethird100.P71.CoefficientSystem

For arbitrary natural numbers $n,d$, any measure $\nu$ on complex matrices indexed by $\mathcal C=\{0,\ldots,d-1\}^{\{0,\ldots,n-1\}}$, and any real-valued function $r$ on subsets of $U=\{0,\ldots,n-1\}$, this property requires, for every $B\subseteq U$,

$$\sum_{A,C\subseteq U} r(A) \left(\frac{d^3}{d^2-1}\right)^n \left[ \sum_{\substack{D\subseteq U\\B=A\cap D}} (d^{|D|})^{-1} \left(\frac{-1}{d}\right)^{|C\triangle D|} \right] \left[ \int \operatorname{Re}\operatorname{tr}(R_C(S)^2)\,d\nu(S) \right] = \begin{cases} 1,&B=U,\\ 0,&B\ne U, \end{cases}$$

where $R_C(S)(x,y)=\sum_{z\in\{0,\ldots,d-1\}^{C^c}}S(x\oplus_C z,y\oplus_C z)$, and $\oplus_C$ combines configurations on $C$ and its complement. There are no probability, support, or integrability assumptions on $\nu$; nonintegrable integrals are zero. Division by zero is zero and zeroth powers are one. In particular, when $n>0$ and $d$ is zero or one, every fusion factor vanishes, so no $r$ can satisfy the required equation for $B=U$. When $n=0$, the system is the single equation $r(\varnothing)\int\operatorname{Re}\operatorname{tr}(S^2)\,d\nu(S)=1$.

QAlgorithms.Papers.Independent.Intakethird100.P71.Complete

For arbitrary natural numbers $n,d$ and any measure $\nu$ on complex matrices indexed by $\mathcal C=\{0,\ldots,d-1\}^{\{0,\ldots,n-1\}}$, this property means that the complex-linear map

$$(\mathcal M_\nu(X))_{ij} = \sum_{k,l\in\mathcal C} \left(\int d^n S_{ij}S_{lk}\,d\nu(S)\right)X_{kl}$$

is injective: for every pair of complex matrices $X,Y$, equality $\mathcal M_\nu(X)=\mathcal M_\nu(Y)$ implies $X=Y$. This concerns all matrices, without positivity or trace restrictions. No probability or integrability assumption is imposed on $\nu$, and nonintegrable coefficient integrals are zero. When $d=0<n$, the matrix space has only one element, so injectivity holds vacuously; when $n=0$, the configuration set instead has one element.

QAlgorithms.Papers.Independent.Intakethird100.P71.DualTableau

For any natural number $n$, let $T,D$ each be an $n$-tuple of Pauli strings on $n$ qubits. A string with phase $c\in\mathbb Z/4\mathbb Z$ and factors $P_q\in\{I,X,Y,Z\}$ represents the matrix $i^{\bar c}\bigotimes_{q<n}P_q$, where $\bar c\in\{0,1,2,3\}$, and the single-qubit matrices are $I$, $X=\begin{pmatrix}0&1\\1&0\end{pmatrix}$, $Y=\begin{pmatrix}0&-i\\i&0\end{pmatrix}$, and $Z=\operatorname{diag}(1,-1)$. This predicate requires each of $T,D$ separately to satisfy all of the following: every phase is $0$ or $2$; all represented matrices commute; for every index $j$, every subset $s$ of indices excluding $j$, and every $c\in\mathbb Z/4\mathbb Z$, its $j$-th matrix differs from $i^{\bar c}$ times the product of its matrices indexed by $s$; and no such subset product equals minus the identity. Subset products are taken in increasing index order, with empty product the identity. In addition, for every $i,j<n$, $D_iT_j=-T_jD_i$ if $i=j$, and $D_iT_j=T_jD_i$ otherwise. The definition includes $n=0$, with empty tuples and empty products.

QAlgorithms.Papers.Independent.Intakethird100.P71.IsLocalDesign

For any natural numbers $n,d,t$ and any two measures $\mu,H$ on tuples $V=(V_i)_{i<n}$ of $d\times d$ unitary matrices, this predicate means that, for every complex matrix $X$ indexed by $t$-tuples of configurations in $\mathcal C=(\{0,\ldots,d-1\})^{\{0,\ldots,n-1\}}$,

$$\int B_t(V)^\dagger X B_t(V)\,d\mu(V) = \int B_t(V)^\dagger X B_t(V)\,dH(V), \qquad B_t(V)_{xy}=\prod_{k<t}\prod_{i<n}(V_i)_{x_k(i),y_k(i)}.$$

Neither measure is assumed to be normalized or Haar, and integrability is not part of the predicate; each nonintegrable Bochner integral is zero. The case $t=0$, as well as $n=0$ or $d=0$, is included.

QAlgorithms.Papers.Independent.Intakethird100.P71.LocallyScrambled

For arbitrary natural numbers $n,d$ and any measure $\nu$ on complex matrices indexed by $\mathcal C=\{0,\ldots,d-1\}^{\{0,\ldots,n-1\}}$, this property requires invariance under every family $V=(V_a)_{a=0}^{n-1}$ of unitary $d\times d$ matrices. Explicitly, let

$$L_V(x,y)=\prod_{a=0}^{n-1}(V_a)_{x_a y_a}.$$

Then the pushforward of $\nu$ under $X\mapsto L_V^\dagger X L_V$ must equal $\nu$, for every such family $V$. No probability, positivity, normalization, or injectivity condition is included. The definition allows $n=0$, with the empty product equal to one, and $d=0$; in either case these conjugations are identity maps, so every measure satisfies the property.

QAlgorithms.Papers.Independent.Intakethird100.P71.Permutations

For any natural numbers $n,t$, this abbreviation denotes the set of assignments of a permutation of $\{0,\ldots,t-1\}$ to each site in $\{0,\ldots,n-1\}$. Thus an element is a family $(g_i)_{i=0}^{n-1}$, with each $g_i$ a bijection of the $t$ copy indices. Both parameters may be zero: when $n=0$ there is one empty family, and when $t=0$ each site is assigned the unique permutation of the empty set.

QAlgorithms.Papers.Independent.Intakethird100.P71.ProjectiveEnsemble

For each pair of natural numbers $n,d$, a projective ensemble consists of a measure $\lambda$ on the type $\operatorname{Unitary}(n,d)$, together with a proof that $\lambda$ is a probability measure, meaning that its total mass is $1$. There are no positivity restrictions on $n$ or $d$. The definition of $\operatorname{Unitary}(n,d)$ is not included in the supplied excerpt.

QAlgorithms.Papers.Independent.Intakethird100.P71.SnapshotPolynomialIntegrable

For any natural numbers $n,d$, including zero, let $\mathcal C=\{0,\ldots,d-1\}^{\{0,\ldots,n-1\}}$ and let $\nu$ be any measure on the complex matrices indexed by $\mathcal C$. This predicate requires all three of the following: for every natural number $t$, the matrix-valued function $S\mapsto S^{\otimes t}$ is integrable with respect to $\nu$, where its entry at configurations $x,y:\{0,\ldots,t-1\}\to\mathcal C$ is $\prod_{i=0}^{t-1}S_{x_i,y_i}$; for every complex matrix $X$ indexed by $\mathcal C$, the matrix-valued function $S\mapsto\operatorname{tr}(SX)S$ is integrable; and for every pair of such matrices $X,Y$, the real-valued function $S\mapsto(\operatorname{Re}\operatorname{tr}(XS))^2\operatorname{Re}\operatorname{tr}(SY)$ is integrable. The first requirement includes $t=0$, for which the empty product gives the constant one-dimensional matrix $[1]$. No positivity, trace normalization, or probability-measure assumption is imposed.

QAlgorithms.Papers.Independent.Intakethird100.P71.ToyFeature

For any natural number $d$, any measure $\nu$ on complex matrices indexed by pairs in $\{0,\ldots,d-1\}^2$, and any real number $w$, this predicate requires

$$\int \operatorname{Re}\operatorname{tr}\!\left[(\operatorname{Tr}_{A^c}S)^2\right]\,d\nu(S) = \begin{cases} w,&|A|=1,\\ 1,&|A|=0\text{ or }2 \end{cases}$$

for every subset $A\subseteq\{0,1\}$. Here the matrix $\operatorname{Tr}_{A^c}S$, indexed by assignments $x,y:A\to\{0,\ldots,d-1\}$, has entry $\sum_z S_{x\cup z,y\cup z}$, summing over all assignments $z$ on the complementary sites. The square is ordinary matrix multiplication. No probability, positivity, or integrability assumptions are imposed; a nonintegrable real integrand has integral zero. For $d=0$, all the integrands are zero, so the required value $1$ for $A=\varnothing$ cannot hold.

QAlgorithms.Papers.Independent.Intakethird100.P71.averagedSecondMoment

For any natural numbers $n,d$, put $\mathcal C=\{0,\ldots,d-1\}^{\{0,\ldots,n-1\}}$. Let $H$ be any measure on tuples $V=(V_i)_{i=0}^{n-1}$ of unitary $d\times d$ matrices, let $\nu$ be any measure on complex matrices indexed by $\mathcal C$, and assume that the map

$$(\mathcal L_\nu X)_{ij} =\sum_{k,l\in\mathcal C} \left(\int d^n T_{ij}T_{lk}\,d\nu(T)\right)X_{kl}$$

is injective. For arbitrary matrices $\rho,O$, define $U_V$ by $(U_V)_{xy}=\prod_i(V_i)_{x_i,y_i}$ and set $\rho_V=U_V^\dagger\rho U_V$. Let $\nu_{\rho_V}$ be $\nu$ weighted by $\max\{d^n\operatorname{Re}\operatorname{tr}(S\rho_V),0\}$. The averaged second moment is

$$\int\left[ \int\bigl(\operatorname{Re}\operatorname{tr}(O\mathcal L_\nu^{-1}(S))\bigr)^2 \,d\nu_{\rho_V}(S)\right]dH(V).$$

No probability, positivity, normalization, or integrability conditions are imposed beyond injectivity; zero dimensions are included, and all integrals use the total integral convention.

QAlgorithms.Papers.Independent.Intakethird100.P71.channelCoefficient

For arbitrary natural numbers $n,d$, any measure $\nu$ on complex matrices indexed by configurations $\mathcal C=\{0,\ldots,d-1\}^{\{0,\ldots,n-1\}}$, and four configurations $i,j,k,l\in\mathcal C$, this defines the complex number

$$\int d^n S_{ij}S_{lk}\,d\nu(S).$$

The two matrix entries are multiplied without complex conjugation, and the second entry has indices $l,k$, in that order. No normalization, positivity, support, or integrability condition is required of $\nu$; a nonintegrable integrand has integral zero. The parameters include $n=0$, when $\mathcal C$ has one element and $d^0=1$, and $d=0<n$, when no configurations $i,j,k,l$ exist.

QAlgorithms.Papers.Independent.Intakethird100.P71.copyPermutation

For any natural numbers $n,d,t$ and any family $(g_i)_{i=0}^{n-1}$ of permutations of $\{0,\ldots,t-1\}$, this definition gives a complex matrix whose rows and columns are indexed by $t$-tuples of configurations in $\{0,\ldots,d-1\}^{\{0,\ldots,n-1\}}$. Its entry at row $x$ and column $y$ is

$$P_{x,y}= \begin{cases} 1,&x_k(i)=y_{g_i(k)}(i)\text{ for every site }i\text{ and copy }k,\\ 0,&\text{otherwise}. \end{cases}$$

All three parameters may be zero. For $t=0$ the index set is a singleton and the matrix is $[1]$; for $n=0$ it is also $[1]$. If $d=0$ while $n,t>0$, the matrix has an empty index set.

QAlgorithms.Papers.Independent.Intakethird100.P71.efReconstruct

For arbitrary natural numbers $n,d$, any real-valued function $r$ on subsets of $U=\{0,\ldots,n-1\}$, and any complex matrix $X$ indexed by $\mathcal C=\{0,\ldots,d-1\}^{U}$, this defines the matrix

$$\mathcal R_r(X)=d^n\sum_{A\subseteq U}r(A)\,T_A(X), \qquad T_A(X)_{xy}= \begin{cases} \displaystyle \frac{\sum_{z\in\{0,\ldots,d-1\}^{A^c}} X((x|_A)\oplus_A z,(y|_A)\oplus_A z)} {d^{\,n-|A|}}, &x|_{A^c}=y|_{A^c},\\[6pt] 0,&\text{otherwise}. \end{cases}$$

Here $\oplus_A$ combines configurations on $A$ and its complement, and real scalars are regarded as complex scalars. No equation or other condition on $r$, and no positivity or normalization condition on $X$, is required. All subsets, including the empty and full subsets, are used. Zero parameters are allowed, division by zero is zero, and zeroth powers are one; when $n=0$, the formula reduces to $r(\varnothing)X$.

QAlgorithms.Papers.Independent.Intakethird100.P71.feature

For arbitrary natural numbers $n,d$, any measure $\nu$ on the space of complex matrices indexed by $\mathcal C=\{0,\ldots,d-1\}^{\{0,\ldots,n-1\}}$, and any subset $A\subseteq\{0,\ldots,n-1\}$, this defines

$$\int \operatorname{Re}\operatorname{tr}(R_A(S)^2)\,d\nu(S), \qquad R_A(S)(x,y)= \sum_{z\in\{0,\ldots,d-1\}^{A^c}} S(x\oplus_A z,y\oplus_A z),$$

where $x,y$ are configurations on $A$, and $x\oplus_A z$ combines the indicated configurations on $A$ and $A^c$. The measure need not be a probability measure or be supported on density matrices, and no integrability hypothesis is imposed; the integral is zero when its integrand is not integrable. Empty regions and zero values of $n$ or $d$ are allowed.

QAlgorithms.Papers.Independent.Intakethird100.P71.featureThree

For any natural numbers $n,d$, let $C=\{0,\ldots,d-1\}^{\{0,\ldots,n-1\}}$. Given any measure $\nu$ on the complex matrices indexed by $C$, and a family $h_i$ of permutations of $\{0,1,2\}$, this defines

$$\int \operatorname{tr}\!\left(S^{\otimes3}P_h\right)\,d\nu(S)\in\mathbb C,$$

where both matrices are indexed by triples $x=(x_0,x_1,x_2)\in C^3$, with entries

$$(S^{\otimes3})_{x,y}=\prod_{k=0}^{2}S_{x_k,y_k}, \qquad (P_h)_{x,y}= \begin{cases} 1,&\forall i,k,\ x_k(i)=y_{h_i(k)}(i),\\ 0,&\text{otherwise}. \end{cases}$$

No probability, positivity, normalization, or integrability hypothesis is imposed on $\nu$ or its matrices; the complex integral is defined to be zero when its integrand is not integrable. If $n=0$, $C$ is a singleton and the conditions defining $P_h$ are vacuous; if $d=0<n$, the matrix index set is empty and the result is zero.

QAlgorithms.Papers.Independent.Intakethird100.P71.frameGap

For any natural numbers $n,d$, any measure $H$ on tuples $V=(V_i)_{i<n}$ of $d\times d$ unitary matrices, and any measure $\nu$ on complex operators indexed by $\mathcal C=(\{0,\ldots,d-1\})^{\{0,\ldots,n-1\}}$, let $\widetilde\nu$ be the pushforward of $H\otimes\nu$ by $(V,S)\mapsto L(V)^\dagger S L(V)$, with $L(V)_{xy}=\prod_{i<n}(V_i)_{x_i y_i}$. This defines

$$\int\bigl(\operatorname{Re}\operatorname{tr}(ST)\bigr)^2\,d(\nu\otimes\nu)(S,T) - \int\bigl(\operatorname{Re}\operatorname{tr}(ST)\bigr)^2\,d(\widetilde\nu\otimes\widetilde\nu)(S,T).$$

No normalization, integrability, or positivity hypotheses are imposed, and no sign for this difference is asserted. Each nonintegrable Bochner integral is assigned zero. The definition includes $n=0$ and $d=0$.

QAlgorithms.Papers.Independent.Intakethird100.P71.framePotential

For any natural numbers $n,d$ and any measure $\nu$ on complex matrices indexed by $\mathcal C=(\{0,\ldots,d-1\})^{\{0,\ldots,n-1\}}$, this defines the real number

$$\int \bigl(\operatorname{Re}\operatorname{tr}(ST)\bigr)^2\,d(\nu\otimes\nu)(S,T).$$

There is no assumption that $\nu$ is normalized or that $S,T$ are Hermitian, positive, or trace one; the real part is taken before squaring. This is the total real-valued Bochner integral, which is defined to be zero when its integrand is not integrable. Zero values of $n$ or $d$ are included.

QAlgorithms.Papers.Independent.Intakethird100.P71.fusion

For arbitrary natural numbers $n,d$ and subsets $A,B,C\subseteq\{0,\ldots,n-1\}$, this defines

$$\left(\frac{d^3}{d^2-1}\right)^n \sum_{\substack{D\subseteq\{0,\ldots,n-1\}\\B=A\cap D}} (d^{|D|})^{-1} \left(\frac{-1}{d}\right)^{|C\triangle D|}.$$

The sum ranges over every subset $D$ satisfying the displayed equality; if none exists, it is zero. No positive lower bound on $d$ is assumed, so all zero denominators use the value zero, and zeroth powers have value one. For $n=0$, all three subsets are empty and the value is one for every $d$. For $n>0$ and $d=0$ or $d=1$, the value is zero.

QAlgorithms.Papers.Independent.Intakethird100.P71.fusionSite

For every natural number $d$ and every three Boolean values $a,b,c$, this defines the following real number, with $d$ interpreted as a real number in the formulas:

$$\begin{array}{c|c} (a,b,c)&\text{value}\\ \hline (\mathrm{true},\mathrm{true},\mathrm{true})&d^2/(d^2-1)\\ (\mathrm{true},\mathrm{true},\mathrm{false})&-d/(d^2-1)\\ (\mathrm{true},\mathrm{false},\mathrm{true})&-d^2/(d^2-1)\\ (\mathrm{true},\mathrm{false},\mathrm{false})&d^3/(d^2-1)\\ (\mathrm{false},\mathrm{false},\mathrm{false})&d\\ (\mathrm{false},\mathrm{true},\mathrm{true})&0\\ (\mathrm{false},\mathrm{true},\mathrm{false})&0\\ (\mathrm{false},\mathrm{false},\mathrm{true})&0 \end{array}$$

There is no lower bound on $d$. Division by zero is defined to be zero, so at $d=1$ every branch with $a=\mathrm{true}$ has value zero; at $d=0$, every branch has value zero.

QAlgorithms.Papers.Independent.Intakethird100.P71.gueCoordinates

For every natural number $d$, including zero, let $C=\{0,\ldots,d-1\}^2$. This defines the product measure on real arrays

$$z:C\times C\times\{\mathrm{false},\mathrm{true}\}\to\mathbb R$$

whose coordinate measures are all Gaussian with mean $0$ and variance $1$. Thus the $2d^4$ coordinates are independent standard real Gaussian variables. When $d=0$, the coordinate set is empty and the measure assigns probability one to the unique empty array.

QAlgorithms.Papers.Independent.Intakethird100.P71.gueOperator

For any natural number $d$, let $C=\{0,\ldots,d-1\}^2$. Given any real array $z:C\times C\times\{\mathrm{false},\mathrm{true}\}\to\mathbb R$, form

$$Z_{ij}=z(i,j,\mathrm{false})+\mathrm i\,z(i,j,\mathrm{true}), \qquad G=\frac{Z+Z^\dagger}{2}.$$

This defines the complex $C\times C$ matrix

$$G-\frac{\operatorname{tr}(G)}{d^2}I_C,$$

where $Z^\dagger$ is the conjugate transpose. Equivalently, its entries are

$$\frac{z(i,j,\mathrm{false})+z(j,i,\mathrm{false})}{2} +\frac{\mathrm i}{2}\bigl(z(i,j,\mathrm{true})-z(j,i,\mathrm{true})\bigr) -\frac{\sum_{k\in C}z(k,k,\mathrm{false})}{d^2}\delta_{ij}.$$

The input array is unrestricted. Division by zero is defined to be zero; for $d=0$, $C$ is empty and the output is the unique empty matrix.

QAlgorithms.Papers.Independent.Intakethird100.P71.hsSquare

For arbitrary natural numbers $n,d$, including zero, and any complex matrix $O$ indexed by $C=\{0,\ldots,d-1\}^{\{0,\ldots,n-1\}}$, this defines

$$\operatorname{Re}\operatorname{tr}(O^2) =\operatorname{Re}\sum_{x,y\in C}O_{x,y}O_{y,x}.$$

No Hermiticity or positivity hypothesis is required. If $d=0<n$, the value is zero; if $n=0$, $C$ has one element and the value is the real part of the square of the sole matrix entry.

QAlgorithms.Papers.Independent.Intakethird100.P71.localConj

For all natural numbers $n,d$, every $V\in\operatorname{Local}(n,d)$, and every $X\in\operatorname{Op}(n,d)$, define

$$\operatorname{localConj}(V,X)=L_V^\dagger X L_V, \qquad (L_V)_{x,y}=\prod_i(V_i)_{x_i,y_i},$$

where $\dagger$ denotes conjugate transpose and each $V_i$ is viewed as a complex $d\times d$ matrix. No positivity, trace, or normalization hypothesis is imposed on $X$. The types $\operatorname{Local}(n,d)$ and $\operatorname{Op}(n,d)$, and the product’s site index type, are not defined in the supplied excerpt; an empty product equals $1$.

QAlgorithms.Papers.Independent.Intakethird100.P71.localMatrix

For all natural numbers $n,d$ and every $V$ of type $\operatorname{Local}(n,d)$, define an operator $L_V\in\operatorname{Op}(n,d)$ by the entry formula

$$(L_V)_{x,y}=\prod_i (V_i)_{x_i,y_i},$$

where each $V_i$ is viewed as a complex matrix indexed by $\operatorname{Fin}(d)\times\operatorname{Fin}(d)$. An empty product equals $1$. The supplied excerpt does not define $\operatorname{Local}(n,d)$, $\operatorname{Op}(n,d)$, or their configuration and site index types.

QAlgorithms.Papers.Independent.Intakethird100.P71.localMomentAction

For any natural numbers $n,d,t$, any measure $\mu$ on tuples $V=(V_i)_{i<n}$ of $d\times d$ unitary matrices, and any complex matrix $X$ indexed by $t$-tuples of configurations from $\mathcal C=(\{0,\ldots,d-1\})^{\{0,\ldots,n-1\}}$, this defines the matrix-valued Bochner integral

$$\int B_t(V)^\dagger X B_t(V)\,d\mu(V), \qquad B_t(V)_{xy}=\prod_{k<t}\prod_{i<n}(V_i)_{x_k(i),y_k(i)}.$$

No normalization of $\mu$, integrability hypothesis, or restriction on $X$ is required; the integral is zero if the matrix-valued integrand is not integrable. All three natural numbers may be zero. In particular, for $t=0$ the matrix $B_t(V)$ is the $1\times1$ identity.

QAlgorithms.Papers.Independent.Intakethird100.P71.localObservable

For any natural number $d$, including zero, and any complex $d\times d$ matrix $A$, this defines the matrix indexed by pairs from $\{0,\ldots,d-1\}^2$ with entries

$$O_{(a,b),(c,e)}=A_{a,c}\delta_{b,e}.$$

Equivalently, it is $A\otimes I_d$, with the first coordinate carrying $A$. No assumptions are imposed on $A$. For $d=0$, the matrix is empty.

QAlgorithms.Papers.Independent.Intakethird100.P71.localityNumerator

For any natural number $d$, including zero, and any complex matrix $O$ indexed by pairs from $\{0,\ldots,d-1\}^2$, define $d\times d$ matrices

$$(R_0)_{a,b}=\sum_c O_{(a,c),(b,c)}, \qquad (R_1)_{a,b}=\sum_c O_{(c,a),(c,b)}.$$

This defines the real number

$$\operatorname{Re}\operatorname{tr}(R_0^2) +\operatorname{Re}\operatorname{tr}(R_1^2).$$

The squares are ordinary matrix products, and no assumptions are imposed on $O$. For $d=0$, both traces are empty sums and the value is zero.

QAlgorithms.Papers.Independent.Intakethird100.P71.observableFeature

For any natural numbers $n,d$, let $\mathcal C=\{0,\ldots,d-1\}^{\{0,\ldots,n-1\}}$, let $H$ be any measure on tuples $V=(V_i)_{i=0}^{n-1}$ of unitary $d\times d$ matrices, let $O$ be any complex matrix indexed by $\mathcal C$, and let $A\subseteq\{0,\ldots,n-1\}$. Define $(U_V)_{xy}=\prod_i(V_i)_{x_i,y_i}$ and $X_V=U_V^\dagger O U_V$. For assignments $x,y$ on $A$, define the matrix $R_A(X_V)$ by

$$R_A(X_V)_{xy}=\sum_{z:A^c\to\{0,\ldots,d-1\}}(X_V)_{x\cup z,\,y\cup z},$$

where $x\cup z$ denotes the configuration taking its coordinates from $x$ on $A$ and from $z$ elsewhere. The observable feature is defined as $\int\operatorname{Re}\operatorname{tr}(R_A(X_V)^2)\,dH(V)$. The square is ordinary matrix multiplication. No Hermiticity, positivity, normalization, or integrability assumption is imposed; zero dimensions, empty $A$, and full $A$ are included, and a nonintegrable integrand has integral zero.

QAlgorithms.Papers.Independent.Intakethird100.P71.observableThree

For any natural numbers $n,d$, put $C=\{0,\ldots,d-1\}^{\{0,\ldots,n-1\}}$. Let $\nu$ be any measure on complex $C\times C$ matrices, subject to the hypothesis that the complex linear map

$$(\mathcal M_\nu X)_{ij} =\sum_{k,l\in C} \left(\int d^n S_{ij}S_{lk}\,d\nu(S)\right)X_{kl}$$

is injective. For arbitrary complex $C\times C$ matrices $\rho,O$, and a family $g_i$ of permutations of $\{0,1,2\}$, this defines

$$d^n\operatorname{tr}(TP_g)\in\mathbb C, \qquad T_{x,y}=R_{x_0,y_0}R_{x_1,y_1}\rho_{x_2,y_2},$$

where $R$ is the unique matrix satisfying $\mathcal M_\nu R=O$, and

$$(P_g)_{x,y}= \begin{cases} 1,&\forall i,k,\ x_k(i)=y_{g_i(k)}(i),\\ 0,&\text{otherwise} \end{cases} \quad(x,y\in C^3).$$

The inverse exists because the injective linear map acts on a finite dimensional space of equal input and output dimension. There are no density matrix or Hermiticity assumptions on $\rho,O$, and nonintegrable coefficient integrands have integral zero. Zero values of $n,d$ are included, with $d^0=1$; when $d=0<n$, all these matrices are empty and the result is zero.

QAlgorithms.Papers.Independent.Intakethird100.P71.oneSiteInverse

For every natural number $d$, including zero, and every complex $d\times d$ matrix $X$, this defines the matrix

$$(d+1)X-\operatorname{tr}(X)I_d.$$

Equivalently, its $(i,j)$ entry is $(d+1)X_{ij}-\operatorname{tr}(X)\delta_{ij}$. No positivity, Hermiticity, trace, or invertibility hypothesis is required. For $d=0$, it is the unique empty matrix.

QAlgorithms.Papers.Independent.Intakethird100.P71.operatorBorel

For every pair of natural numbers $n,d$, including zero, the measurable structure on $\operatorname{Op}(n,d)$ coincides with the Borel measurable structure generated by its topology. The definition of $\operatorname{Op}(n,d)$ is not included in the supplied excerpt.

QAlgorithms.Papers.Independent.Intakethird100.P71.operatorMeasurable

For every pair of natural numbers $n,d$, including zero, the type $\operatorname{Op}(n,d)$ is equipped with the Borel measurable structure generated by its topology. The definition of the operator type $\operatorname{Op}(n,d)$ is not included in the supplied excerpt.

QAlgorithms.Papers.Independent.Intakethird100.P71.partialPurity

For arbitrary natural numbers $n,d$, a subset $A\subseteq\{0,\ldots,n-1\}$, and any complex matrix $X$ indexed by configurations $\mathcal C=\{0,\ldots,d-1\}^{\{0,\ldots,n-1\}}$, define a matrix $R_A$ indexed by configurations on $A$ by

$$R_A(x,y)=\sum_{z\in\{0,\ldots,d-1\}^{A^c}}X(x\oplus_A z,y\oplus_A z),$$

where $x\oplus_A z$ agrees with $x$ on $A$ and with $z$ on its complement. The defined quantity is

$$\operatorname{Re}\operatorname{tr}(R_A^2) =\operatorname{Re}\sum_{x,y}R_A(x,y)R_A(y,x).$$

There is no assumption that $X$ is Hermitian, positive, or normalized; the square is matrix multiplication, with no conjugate transpose. Empty $A$, $n=0$, and $d=0$ are included, with empty sums equal to zero.

QAlgorithms.Papers.Independent.Intakethird100.P71.partialTrace

For every natural number $n$, every finite type $\alpha$ equipped with decidable equality, every subset $A\subseteq\{0,\ldots,n-1\}$, and every complex matrix $X$ indexed by functions $\{0,\ldots,n-1\}\to\alpha$, define a matrix indexed by functions $A\to\alpha$ through

$$(\operatorname{partialTrace}_A X)_{x,y} =\sum_{z:A^{\complement}\to\alpha} X_{x\cup z,\;y\cup z},$$

where $x\cup z$ is the function equal to $x$ on $A$ and to $z$ on its complement. No condition is imposed on $X$. Empty index sets and an empty $\alpha$ are allowed: an empty sum is zero, and there is exactly one function from an empty domain to $\alpha$; in particular, $A=\varnothing$ gives the trace as the sole matrix entry, while $A=\{0,\ldots,n-1\}$ gives $X$ under restriction of indices.

QAlgorithms.Papers.Independent.Intakethird100.P71.pauliFactor

For any natural number $n$, any Pauli string $O$ consisting of a phase in $\mathbb Z/4\mathbb Z$ and one factor from $\{I,X,Y,Z\}$ at each of the $n$ qubits, any index $i<n$, and any Boolean $a$, this defines the complex number $1$ if the factor of $O$ at $i$ is $I$ or $a$ is true, and $0$ otherwise. The phase of $O$ does not affect the value. When $n=0$, no index $i$ exists.

QAlgorithms.Papers.Independent.Intakethird100.P71.posteriorOf

For all natural numbers $n,d$, every measure $\nu$ on $\operatorname{Op}(n,d)$, and every operator $\rho\in\operatorname{Op}(n,d)$, define a measure by weighting $\nu$ with the nonnegative extended-real density

$$S\longmapsto \operatorname{ofReal}\!\left(d^n\operatorname{Re}\operatorname{tr}(S\rho)\right).$$

Here $\operatorname{ofReal}$ replaces negative real values by zero. Neither $\nu$ nor $\rho$ is required to be normalized, and $\rho$ is not required to be positive or Hermitian. Both $n$ and $d$ may be zero; when $d=0<n$, the density is identically zero. The operator type $\operatorname{Op}(n,d)$ is not defined in the supplied excerpt.

QAlgorithms.Papers.Independent.Intakethird100.P71.prior

For all natural numbers $n,d$ and every projective ensemble $E$, whose law $\lambda$ is a probability measure on $\operatorname{Unitary}(n,d)$, define a measure on $\operatorname{Op}(n,d)$ by

$$\nu_E=\operatorname{ofReal}\bigl((d^n)^{-1}\bigr) \sum_{b\in\operatorname{Config}(n,d)} \bigl(U\mapsto U^\dagger E_{b,b}U\bigr)_*\lambda,$$

where $E_{b,b}$ has its sole nonzero entry $1$ at $(b,b)$, the subscript $*$ denotes pushforward of a measure, and $\operatorname{ofReal}(r)=\max(r,0)$ viewed as a nonnegative extended real. The inverse is taken in $\mathbb R$, where $0^{-1}=0$; consequently the scaling factor is zero when $d=0<n$, whereas $d^0=1$, including when $d=0$. The types $\operatorname{Config}$, $\operatorname{Unitary}$, and $\operatorname{Op}$ are not defined in the supplied excerpt.

QAlgorithms.Papers.Independent.Intakethird100.P71.productWgThree

For any natural numbers $n,d$, including zero, and families $g=(g_i)_{i=0}^{n-1}$ and $h=(h_i)_{i=0}^{n-1}$ of permutations of $\{0,1,2\}$, this defines the real number

$$\prod_{i=0}^{n-1} W_d(g_i h_i^{-1}),$$

where permutation multiplication is composition and, writing $a=d$ as a real number and $\chi(\pi)=|\{j\in\{0,1,2\}:\pi(j)=j\}|-1$,

$$W_d(\pi)=\frac16\left( \frac1{a(a+1)(a+2)} +\frac{2\chi(\pi)}{a(a+1)(a-1)} +\begin{cases} \dfrac{\operatorname{sgn}(\pi)}{a(a-1)(a-2)},&d\ge3,\\ 0,&d<3. \end{cases} \right).$$

Every division is total, with division by zero equal to zero. For $n=0$, the product is $1$.

QAlgorithms.Papers.Independent.Intakethird100.P71.raw

For all natural numbers $n,d$, every $U\in\operatorname{Unitary}(n,d)$, and every $b\in\operatorname{Config}(n,d)$, define

$$\operatorname{raw}(U,b)=U^\dagger E_{b,b}U,$$

where $U$ is viewed as an operator in $\operatorname{Op}(n,d)$, and $E_{b,b}$ has entry $1$ at $(b,b)$ and entry $0$ everywhere else. Equivalently, its $(x,y)$-entry is $\overline{U_{b,x}}U_{b,y}$. The definitions of the three named types are not supplied, and no positive lower bound on $n$ or $d$ is assumed.

QAlgorithms.Papers.Independent.Intakethird100.P71.reducedSnapshot

For every natural number $n$, every finite type $\alpha$ equipped with decidable equality, every subset $A\subseteq\{0,\ldots,n-1\}$, and every complex matrix $X$ indexed by functions $\{0,\ldots,n-1\}\to\alpha$, define a matrix on the same index set by

$$(\operatorname{reducedSnapshot}_A X)_{x,y} = \begin{cases} \displaystyle \frac{\sum_{z:A^\complement\to\alpha} X_{(x|_A)\cup z,\;(y|_A)\cup z}} {|\alpha|^{\,n-|A|}}, & x_i=y_i\text{ for every }i\notin A,\\[6pt] 0,&\text{otherwise}. \end{cases}$$

The unions here combine functions on complementary domains. No positivity, Hermiticity, or normalization is assumed for $X$. The exponent uses natural-number subtraction, and $|A|\le n$; empty sums equal zero, $0^0=1$, and complex division by zero is defined to be zero. The definition includes $n=0$, empty $A$, and empty $\alpha$.

QAlgorithms.Papers.Independent.Intakethird100.P71.regionalWg

For arbitrary natural numbers $n,d$ and subsets $A,B\subseteq\{0,\ldots,n-1\}$, this defines the real number

$$(d^2-1)^{-n}\left(\frac{-1}{d}\right)^{|A\triangle B|},$$

where $A\triangle B$ is their symmetric difference and the first factor means $\bigl((d^2-1)^{-1}\bigr)^n$. There is no restriction $d>1$: reciprocals of zero and division by zero have value zero, and a zeroth power, including $0^0$, has value one. Consequently the value is one when $n=0$, and it is zero when $d=1$ and $n>0$.

QAlgorithms.Papers.Independent.Intakethird100.P71.sandwichPrior

For any natural numbers $n,d$, any unitary $U_0$ on configurations $\mathcal C=(\{0,\ldots,d-1\})^{\{0,\ldots,n-1\}}$, and any measures $\mu_L,\mu_R$ on tuples of $d\times d$ unitary matrices, this defines the measure

$$\operatorname{ofReal}\!\bigl((d^n)^{-1}\bigr) \sum_{b\in\mathcal C}(F_b)_*(\mu_L\otimes\mu_R), \qquad F_b(L,R)=U^\dagger|b\rangle\langle b|U, \quad U=L_{\mathrm{all}}U_0R_{\mathrm{all}},$$

where $(L_{\mathrm{all}})_{xy}=\prod_{i<n}(L_i)_{x_i y_i}$, similarly for $R_{\mathrm{all}}$, and $\operatorname{ofReal}$ converts a nonnegative real scalar to an extended nonnegative real scalar. Neither input measure is required to be a probability measure. If $d=0<n$, the configuration set is empty and the resulting measure is zero; if $n=0$, there is one configuration and the scalar is $1$, including when $d=0$.

QAlgorithms.Papers.Independent.Intakethird100.P71.sandwichRaw

For any natural numbers $n,d$, any unitary matrix $U_0$ indexed by $\mathcal C=(\{0,\ldots,d-1\})^{\{0,\ldots,n-1\}}$, any two tuples $L=(L_i)_{i<n}$ and $R=(R_i)_{i<n}$ of $d\times d$ unitary matrices, and any configuration $b\in\mathcal C$, this defines

$$U^\dagger |b\rangle\langle b|U, \qquad U=L_{\mathrm{all}}U_0R_{\mathrm{all}}, \quad (L_{\mathrm{all}})_{xy}=\prod_{i<n}(L_i)_{x_i y_i}, \quad (R_{\mathrm{all}})_{xy}=\prod_{i<n}(R_i)_{x_i y_i}.$$

Here $|b\rangle\langle b|$ has its only nonzero entry, equal to $1$, at $(b,b)$. For $n=0$, configurations form a singleton and both local products are $1$; for $d=0<n$, there is no configuration $b$ to supply.

QAlgorithms.Papers.Independent.Intakethird100.P71.scalarCode

For every complex number $z$, this defines a natural number equal to $1$ when $z=1$, equal to $2$ when $z=-1$, and equal to $0$ for every other complex number.

QAlgorithms.Papers.Independent.Intakethird100.P71.secondMoment

For any natural numbers $n,d$, let $\mathcal C=\{0,\ldots,d-1\}^{\{0,\ldots,n-1\}}$, let $\nu$ be any measure on complex matrices indexed by $\mathcal C$, and assume that the map

$$(\mathcal L_\nu X)_{ij} =\sum_{k,l\in\mathcal C} \left(\int d^n T_{ij}T_{lk}\,d\nu(T)\right)X_{kl}$$

is injective, so that its inverse exists. For arbitrary matrices $\rho,O$, define the measure $\nu_\rho$ by weighting $\nu$ with the nonnegative extended-real density $\max\{d^n\operatorname{Re}\operatorname{tr}(S\rho),0\}$. The second moment is defined by

$$\int \bigl(\operatorname{Re}\operatorname{tr}(O\mathcal L_\nu^{-1}(S))\bigr)^2\,d\nu_\rho(S).$$

Neither $\rho$ nor $O$ is required to be a density matrix or Hermitian, $\nu_\rho$ need not be a probability measure, and no integrability assumption is imposed. All integrals are total integrals, returning zero for nonintegrable integrands; zero dimensions are included.

QAlgorithms.Papers.Independent.Intakethird100.P71.shadowEstimate

For any natural numbers $n,d$, let $\mathcal C=\{0,\ldots,d-1\}^{\{0,\ldots,n-1\}}$, let $\nu$ be any measure on complex matrices indexed by $\mathcal C$, and suppose the complex-linear map $\mathcal L_\nu$ defined by

$$(\mathcal L_\nu X)_{ij} =\sum_{k,l\in\mathcal C} \left(\int d^n T_{ij}T_{lk}\,d\nu(T)\right)X_{kl}$$

is injective. Since its domain and codomain are the same finite-dimensional space, it has an inverse. For arbitrary matrices $O,S$ indexed by $\mathcal C$, the shadow estimate is defined as $\operatorname{Re}\operatorname{tr}(O\mathcal L_\nu^{-1}(S))$. The dimensions may be zero, and no Hermiticity, positivity, normalization, or integrability hypotheses are added; the coefficient integrals use the total integral, which is zero for a nonintegrable integrand.

QAlgorithms.Papers.Independent.Intakethird100.P71.shadowSampleMean

For any natural numbers $n,d,m$, let $\mathcal C=\{0,\ldots,d-1\}^{\{0,\ldots,n-1\}}$, let $\nu$ be any measure on complex matrices indexed by $\mathcal C$, and assume the complex-linear map

$$(\mathcal L_\nu X)_{ij} =\sum_{k,l\in\mathcal C} \left(\int d^n T_{ij}T_{lk}\,d\nu(T)\right)X_{kl}$$

is injective. For any matrix $O$ and any prescribed tuple of matrices $(S_0,\ldots,S_{m-1})$, the shadow sample mean is defined as

$$m^{-1}\sum_{k=0}^{m-1} \operatorname{Re}\operatorname{tr}(O\mathcal L_\nu^{-1}(S_k)).$$

The tuple is arbitrary and is not assumed to have been sampled from a measure. When $m=0$, the empty sum and the convention $0^{-1}=0$ give value zero. Zero dimensions are allowed, and the coefficient integrals use the total integral convention.

QAlgorithms.Papers.Independent.Intakethird100.P71.shotVariance

For any natural numbers $n,d$, put $\mathcal C=\{0,\ldots,d-1\}^{\{0,\ldots,n-1\}}$. Let $\nu$ be any measure on complex matrices indexed by $\mathcal C$, and suppose the map

$$(\mathcal L_\nu X)_{ij} =\sum_{k,l\in\mathcal C} \left(\int d^n T_{ij}T_{lk}\,d\nu(T)\right)X_{kl}$$

is injective. For arbitrary matrices $\rho,O$, let $\nu_\rho$ be $\nu$ weighted by $\max\{d^n\operatorname{Re}\operatorname{tr}(S\rho),0\}$ and write $a(S)=\operatorname{Re}\operatorname{tr}(O\mathcal L_\nu^{-1}(S))$. The shot variance is defined as

$$\int a(S)^2\,d\nu_\rho(S) -\left(\int a(S)\,d\nu_\rho(S)\right)^2.$$

There are no positivity, normalization, or integrability hypotheses beyond the stated injectivity condition, and this definition does not assert nonnegativity. Zero dimensions are allowed, and each integral uses the total convention that assigns zero to a nonintegrable integrand.

QAlgorithms.Papers.Independent.Intakethird100.P71.snapshotInverse

For arbitrary natural numbers $n,d$, any measure $\nu$ on complex matrices indexed by $\mathcal C=\{0,\ldots,d-1\}^{\{0,\ldots,n-1\}}$, and the hypothesis that

$$(\mathcal M_\nu(X))_{ij} = \sum_{k,l\in\mathcal C} \left(\int d^n S_{ij}S_{lk}\,d\nu(S)\right)X_{kl}$$

is injective on all complex matrices, this defines the inverse complex-linear equivalence $\mathcal M_\nu^{-1}$. Surjectivity is obtained from injectivity because the domain and codomain are the same finite-dimensional space, so the defined inverse satisfies both $\mathcal M_\nu^{-1}(\mathcal M_\nu(X))=X$ and $\mathcal M_\nu(\mathcal M_\nu^{-1}(X))=X$ for every matrix $X$. No additional probability or integrability hypothesis is required; nonintegrable coefficient integrals are zero. The construction includes the zero-dimensional matrix space arising when $d=0<n$, as well as the one-dimensional matrix space arising when $n=0$.

QAlgorithms.Papers.Independent.Intakethird100.P71.snapshotMeasurementMap

For arbitrary natural numbers $n,d$ and any measure $\nu$ on complex matrices indexed by $\mathcal C=\{0,\ldots,d-1\}^{\{0,\ldots,n-1\}}$, this defines a complex-linear map $\mathcal M_\nu$ on that entire matrix space by

$$(\mathcal M_\nu(X))_{ij} = \sum_{k,l\in\mathcal C} \left(\int d^n S_{ij}S_{lk}\,d\nu(S)\right)X_{kl}.$$

Thus its coefficients are separately defined integrals, with no conjugation of the entries of $S$. No assumption says that $\nu$ is a probability measure or supported on positive or normalized matrices, or that these integrands are integrable; each nonintegrable coefficient integral is zero. The map acts on arbitrary complex matrices. Zero parameters are included: $n=0$ gives a one-dimensional matrix space, while $d=0<n$ gives the zero-dimensional matrix space.

QAlgorithms.Papers.Independent.Intakethird100.P71.snapshotMoment

For arbitrary natural numbers $n,d$, any measure $\nu$ on complex matrices indexed by $\mathcal C=\{0,\ldots,d-1\}^{\{0,\ldots,n-1\}}$, and any natural number $t$, this defines the matrix-valued integral

$$\int S^{\otimes t}\,d\nu(S), \qquad (S^{\otimes t})_{ab} = \prod_{q=0}^{t-1}S_{a(q),b(q)},$$

whose row and column indices $a,b$ are functions $\{0,\ldots,t-1\}\to\mathcal C$. Thus coordinates are grouped by copy, with one full configuration per copy. No probability, positivity, normalization, or integrability hypothesis is imposed; if the matrix-valued integrand is not integrable, its integral is the zero matrix. The case $t=0$ is included: its index set has one element and its integrand is the $1\times1$ matrix with entry one, so its integral is not stipulated to equal one for an arbitrary measure.

QAlgorithms.Papers.Independent.Intakethird100.P71.snapshotSampleLaw

For any natural numbers $n,d,m$, let $\mathcal C=\{0,\ldots,d-1\}^{\{0,\ldots,n-1\}}$, let $\nu$ be any measure on complex matrices indexed by $\mathcal C$, and let $\rho$ be any such matrix. Weight $\nu$ by the nonnegative extended-real density $\max\{d^n\operatorname{Re}\operatorname{tr}(S\rho),0\}$ to obtain a measure $\nu_\rho$. The snapshot sample law is defined as the finite product measure $\bigotimes_{k=0}^{m-1}\nu_\rho$ on $m$-tuples of matrices. No density-matrix or probability-measure assumption is imposed. In particular, $m=0$ is included and gives the empty product measure on the singleton set of empty tuples; $n$ and $d$ may also be zero.

QAlgorithms.Papers.Independent.Intakethird100.P71.tableauInput

For any natural number $n$, any two $n$-tuples $T,D$ of Pauli strings on $n$ qubits, and any Pauli string $O$ on $n$ qubits, this defines a list of natural numbers beginning with the single entry $n$, followed by the encodings of $T$, then $D$, then $O$, with every Boolean encoding bit converted to $0$ for false or $1$ for true. A string of phase $c\in\mathbb Z/4\mathbb Z$, represented by $\bar c\in\{0,1,2,3\}$, is encoded by the two bits $[\bar c\ge2,\ \bar c\bmod2=1]$, followed by its Pauli factors in increasing qubit order, using $I\mapsto[0,0]$, $X\mapsto[1,0]$, $Y\mapsto[1,1]$, and $Z\mapsto[0,1]$. A tableau encoding concatenates its string encodings in increasing row order. No validity or commutation conditions are required. For $n=0$, both tableau encodings are empty, so the output is $0$ followed by the two phase bits of $O$.

QAlgorithms.Papers.Independent.Intakethird100.P71.tableauRecord

For every natural number $n$, every list $T_0,\ldots,T_{n-1}$ of $n$-qubit Pauli strings, and every $n$-qubit Pauli string $O$, this defines a list of $2n+1$ natural numbers. A Pauli string $P$ consists of a phase $c_P\in\mathbb Z/4\mathbb Z$ and factors $P_i\in\{I,X,Y,Z\}$, and represents the matrix $A_P=i^{c_P}\bigotimes_{i=0}^{n-1}P_i$, using the representative $c_P\in\{0,1,2,3\}$. Put $\Pi_T=\prod_{j=0}^{n-1}(I+A_{T_j})/2$, with factors in increasing index order, and $\tau=\operatorname{tr}(A_O\Pi_T)$. The first list entry is $1$ if $\tau=1$, $2$ if $\tau=-1$, and $0$ for every other complex value of $\tau$. It is followed, in increasing order of $i$, by the pair $(1,1)$ when $O_i=I$, and $(0,1)$ otherwise. There are no validity, commutativity, or phase restrictions on $T$ or $O$. The case $n=0$ is included: the empty product $\Pi_T$ is the identity on the one-dimensional space, and the output has just the first entry.

QAlgorithms.Papers.Independent.Intakethird100.P71.toyCoefficients

For any natural number $d$, real number $w$, and subset $A\subseteq\{0,1\}$, writing $a=d$ as a real number, this defines

$$\begin{cases} \dfrac{a^3w-3a^2+3aw-2w^2+1} {(aw-1)(a^2-2aw+1)},&|A|=0,\\[6pt] \dfrac{-a^4w+2a^3-2a+w} {a(aw-1)(a^2-2aw+1)},&|A|=1,\\[6pt] \dfrac{(a^2-1)^2} {a^2(a^2-2aw+1)},&|A|=2. \end{cases}$$

There are no restrictions on $d,w$ and no nonzero denominator hypotheses. Each fraction whose denominator is zero has value zero.

QAlgorithms.Papers.Independent.Intakethird100.P71.twirled

For any natural numbers $n,d$, let $\mathcal C=(\{0,\ldots,d-1\})^{\{0,\ldots,n-1\}}$, and let operators be complex matrices indexed by $\mathcal C$. Given any measure $H$ on tuples $V=(V_i)_{i<n}$ of $d\times d$ unitary matrices and any measure $\nu$ on operators, this defines the pushforward of $H\otimes\nu$ under $(V,S)\mapsto L(V)^\dagger S L(V)$, where $L(V)_{xy}=\prod_{i<n}(V_i)_{x_i y_i}$. Neither measure is required to be a probability measure, and $S$ is not required to be a density matrix. The definition includes $n=0$, when $\mathcal C$ is a singleton and the empty product defining $L(V)$ is $1$, and $d=0$.

QAlgorithms.Papers.Independent.Intakethird100.P71.twoSiteProduct

For any natural number $d$, including zero, and arbitrary complex $d\times d$ matrices $X,Y$, this defines a matrix indexed by functions $x,y:\{0,1\}\to\{0,\ldots,d-1\}$, whose entries are

$$Z_{x,y}=X_{x(0),y(0)}Y_{x(1),y(1)}.$$

Thus, identifying each function with its ordered pair of values, $Z_{(a,b),(c,e)}=X_{a,c}Y_{b,e}$. For $d=0$, the index set is empty.

QAlgorithms.Papers.Independent.Intakethird100.P71.unitaryBorel

For every finite type $\iota$ equipped with decidable equality, the measurable structure on the group of complex unitary matrices indexed by $\iota\times\iota$ coincides with the Borel measurable structure generated by its topology. This includes the case where $\iota$ is empty.

QAlgorithms.Papers.Independent.Intakethird100.P71.unitaryMeasurable

For every finite type $\iota$ equipped with decidable equality, the group of complex unitary matrices indexed by $\iota\times\iota$ is equipped with the Borel measurable structure generated by its topology. The index type $\iota$ is allowed to be empty.

QAlgorithms.Papers.Independent.Intakethird100.P71.varianceCoefficient

For any natural numbers $n,d$, any real-valued function $r$ on the subsets of $\{0,\ldots,n-1\}$, and any four such subsets $A,B,C,D$, the variance coefficient is defined as

$$r(A)r(B) \left(\frac{d^2}{d^2-1}\right)^n \frac{d^{|A\cap B\cap C|}}{d^{|C|}} \left(-\frac1d\right)^{|C\triangle D|},$$

with all arithmetic performed in the real numbers and $C\triangle D$ denoting symmetric difference. There are no restrictions on $r$ or the subsets, and $n=0$, $d=0$, and $d=1$ are included. Division by zero has value zero, and every zeroth power, including $0^0$, has value one.

QAlgorithms.Papers.Independent.Intakethird100.P71.wgThree

For every natural number $d$, including zero, and every permutation $p$ of $\{0,1,2\}$, let $a=d$ as a real number and let $\chi=|\{i\in\{0,1,2\}:p(i)=i\}|-1$. This definition assigns the real number

$$\frac16\left[ \frac1{a(a+1)(a+2)} +\frac{2\chi}{a(a+1)(a-1)} + \begin{cases} \displaystyle\frac{\operatorname{sgn}(p)}{a(a-1)(a-2)},&d\ge3,\\ 0,&d<3, \end{cases} \right],$$

where $\operatorname{sgn}(p)\in\{-1,1\}$ is regarded as real. All divisions use the convention that division by zero is zero; consequently the definition also applies at $d=0,1,2$, and its value is zero at $d=0$ and $d=1$.