Definitions for Approximate recovery with locality and symmetry constraints

QAlgorithms.Papers.Independent.Intakethird100.P44.ChargeCoefficients

Let $H$ be a finite index type with decidable equality, $J,L$ index types, and $r$ a natural number. Let $P_i\in\operatorname{Mat}(H)$, $0\le i<r$, be nonzero Hermitian projections, mutually orthogonal and summing to $I$, with pairwise distinct real values $v_i$. Given matrices $A_n\in\operatorname{Mat}(H)$, indexed by $n\in J$, and a complex $H\times L$ matrix $W$, the coefficient condition means that there exist complex numbers $c_{ijnm}$, indexed by $i,j<r$ and $n,m\in J$, such that

$$W^\dagger A_n^\dagger P_jA_mW =\sum_{i=0}^{r-1}c_{ijnm}\,W^\dagger P_iW \quad\text{for every }j<r,\ n,m\in J.$$

The coefficients have no positivity, Hermitian symmetry, or normalization requirements. No isometry condition on $W$ or Kraus normalization of the $A_n$ is imposed. If $J$ is empty, the equalities are vacuous; $r=0$ is also permitted when $H$ is empty.

QAlgorithms.Papers.Independent.Intakethird100.P44.ChargeStateCondition

Let $H$ be a finite index type with decidable equality, $J$ a finite index type, $L$ an index type, and $r$ a natural number. Let $P_i\in\operatorname{Mat}(H)$, $0\le i<r$, be nonzero Hermitian projections with $P_iP_j=0$ for $i\ne j$ and $\sum_iP_i=I$, carrying pairwise distinct real values $v_i$. Given matrices $A_n\in\operatorname{Mat}(H)$, indexed by $n\in J$, and a complex $H\times L$ matrix $W$, the condition means that there exist matrices $\sigma_{ij}\in\operatorname{Mat}(J)$, for every $i,j<r$, such that each $\sigma_{ij}$ is positive semidefinite, $\sum_j\operatorname{tr}(\sigma_{ij})=1$ for each $i$, and

$$W^\dagger A_n^\dagger P_jA_mW =\sum_{i=0}^{r-1}(\sigma_{ij})_{nm}\,W^\dagger P_iW \quad\text{for every }j<r,\ n,m\in J.$$

No isometry condition on $W$ or Kraus normalization of $A_n$ is imposed by this definition. If $J$ is empty and $r>0$, the trace normalization is impossible; if $r=0$, all the displayed requirements are vacuous, with the sector data forcing $H$ to be empty.

QAlgorithms.Papers.Independent.Intakethird100.P44.Comparison

For a finite type $H$ with decidable equality, a comparison is data of exactly one of two forms: either a square complex matrix $\rho$ indexed by $H$, together with proofs that $\rho$ is positive semidefinite and $\operatorname{Tr}\rho=1$; or a natural number $k>0$, together with a complex matrix $W$ having rows indexed by $H$ and columns indexed by $\{0,\ldots,k-1\}$, satisfying $W^\dagger W=I_k$. Thus the second form specifies an isometric embedding of a positive-dimensional space. If $H$ is empty, neither form can be supplied; $k=0$ is excluded.

QAlgorithms.Papers.Independent.Intakethird100.P44.DefiniteParity

For complex square matrices $C,X$ on an index type $H$, this predicate means

$$XC=CX\quad\text{or}\quad XC=-CX.$$

The disjunction is inclusive: both equations may hold, as happens when $X=0$ or $C=0$. No normalization, Hermiticity, or invertibility condition is imposed on either matrix.

QAlgorithms.Papers.Independent.Intakethird100.P44.Fermions

For every natural number $n$, this structure consists of $2n$ complex $2^n\times2^n$ matrices $w_0,\ldots,w_{2n-1}$, together with the requirements that every $w_j$ is Hermitian and that, for all $i,j$,

$$w_iw_j+w_jw_i= \begin{cases} 2I,&i=j,\\ 0,&i\ne j. \end{cases}$$

Thus each $w_i^2=I$, and distinct matrices anticommute. The case $n=0$ is included: the underlying matrices have size $1\times1$, but there are no $w_j$, so both requirements are vacuous.

QAlgorithms.Papers.Independent.Intakethird100.P44.FixesObservables

For a finite index type $H$ with decidable equality, a set $\mathcal A\subseteq\operatorname{Mat}(H)$, and an arbitrary map $N:\operatorname{Mat}(H)\to\operatorname{Mat}(H)$, this predicate means that for every $X\in\mathcal A$ and every $a,b\in H$,

$$\operatorname{Tr}\bigl(N(E_{ba})X\bigr)=X_{ab},$$

where $E_{ba}$ is the matrix with entry $1$ at row $b$, column $a$, and zero elsewhere. No channel or linearity assumption on $N$ is part of the predicate. The set $\mathcal A$ may be empty, in which case the condition is vacuous.

QAlgorithms.Papers.Independent.Intakethird100.P44.HasIntervalSupport

For every natural number $n$, a family $f$ of Hermitian complex $2^n\times2^n$ matrices $w_0,\ldots,w_{2n-1}$ satisfying $w_iw_j+w_jw_i=2I$ when $i=j$ and $0$ otherwise, a complex $2^n\times2^n$ matrix $X$, and a natural number $\ell$, this predicate says that there exists a starting index $s\in\{0,\ldots,2n-1\}$ such that $X$ belongs to the complex linear span of all products

$$w_{j_1}\cdots w_{j_{2h}} \quad\text{with}\quad (j_a+2n-s)\bmod 2n<\ell \ \text{for every }a.$$

Here $h$ is any natural number, repeated indices and arbitrary ordering are permitted, and the empty product is $I$. Thus $\ell=0$ permits precisely scalar multiples of $I$ when a starting index exists; for $n>0$ and $\ell\ge 2n$, every mode index is permitted. For $n=0$, the predicate is false because there is no starting index.

QAlgorithms.Papers.Independent.Intakethird100.P44.IsChannelDilation

For finite index types $H,K,E$ with decidable equality, an arbitrary map $N:\operatorname{Mat}(H)\to\operatorname{Mat}(K)$, and a complex matrix $V$ with rows indexed by $K\times E$ and columns indexed by $H$, this predicate means

$$V^\dagger V=I_H \quad\text{and}\quad (N(X))_{ab} =\sum_{e\in E}(VXV^\dagger)_{(a,e),(b,e)}$$

for every matrix $X\in\operatorname{Mat}(H)$ and all $a,b\in K$. The equality concerns every input matrix. Empty index types are permitted, subject to the displayed isometry equation.

QAlgorithms.Papers.Independent.Intakethird100.P44.IsGeneralizedComplement

For finite index types $H,K,E$ with decidable equality and maps $N:\operatorname{Mat}(H)\to\operatorname{Mat}(K)$, $C:\operatorname{Mat}(H)\to\operatorname{Mat}(E)$, this predicate asserts the existence of a natural number $m$, matrices $A_j:H\to K$ for $0\le j<m$, and maps $R:\operatorname{Mat}(\mathrm{Fin}\,m)\to\operatorname{Mat}(E)$, $S:\operatorname{Mat}(E)\to\operatorname{Mat}(\mathrm{Fin}\,m)$, such that

$$\sum_{j<m}A_j^\dagger A_j=I_H,\qquad N(X)=\sum_{j<m}A_jXA_j^\dagger,\qquad C=R\circ C_A,\qquad C_A=S\circ C,$$

where $C_A(X)_{ij}=\operatorname{Tr}(A_iXA_j^\dagger)$. Both $R$ and $S$ must be channels: each has a finite Kraus representation $T(X)=\sum_\ell B_\ell XB_\ell^\dagger$ with $\sum_\ell B_\ell^\dagger B_\ell=I$ on its input space. All map equalities hold on every input matrix. The definition allows $m=0$ and empty ambient index types whenever these conditions can hold.

QAlgorithms.Papers.Independent.Intakethird100.P44.IsPhysicalChannel

For finite index types $H,K$ with decidable equality and arbitrary maps $P:\operatorname{Mat}(H)\to\operatorname{Mat}(H)$, $Q:\operatorname{Mat}(K)\to\operatorname{Mat}(K)$, and $N:\operatorname{Mat}(H)\to\operatorname{Mat}(K)$, this predicate means

$$Q(N(P(X)))=Q(N(X)) \quad\text{for every }X\in\operatorname{Mat}(H),$$

where $\operatorname{Mat}(I)$ denotes the square complex matrices indexed by $I$. It imposes no channel, linearity, positivity, or normalization requirement on these maps, and $X$ ranges over all matrices, not merely density matrices; empty index types are included.

QAlgorithms.Papers.Independent.Intakethird100.P44.ObservableAlgebra

For a finite type $H$ with decidable equality, an observable algebra consists of a unital complex star subalgebra $\mathcal A$ of the square complex matrices indexed by $H$, together with a map $P$ on those matrices satisfying three conditions: $P$ is a channel, meaning that there are $m\in\mathbb N$ and matrices $E_1,\ldots,E_m$ with $\sum_j E_j^\dagger E_j=I$ and $P(X)=\sum_j E_jXE_j^\dagger$ for every matrix $X$; $P(X)\in\mathcal A$ for every $X$; and

$$\operatorname{Tr}\bigl((X-P(X))^\dagger Y\bigr)=0 \quad\text{for every matrix }X\text{ and every }Y\in\mathcal A.$$

The quantification over $H$ permits the empty type, and the Kraus family may be empty when its completeness equation holds.

QAlgorithms.Papers.Independent.Intakethird100.P44.PhysicallyCorrectable

Let $H$ be a finite index type with decidable equality, $L$ a finite index type, $P,N:\operatorname{Mat}(H)\to\operatorname{Mat}(H)$ arbitrary maps, and $W$ an arbitrary complex matrix with rows indexed by $H$ and columns by $L$. Physical correctability means that there exists a map $R:\operatorname{Mat}(H)\to\operatorname{Mat}(H)$ admitting a finite Kraus representation $R(Y)=\sum_aV_aYV_a^\dagger$ with $\sum_aV_a^\dagger V_a=I$, such that

$$P(R(P(Y)))=P(R(Y))\quad\text{for every }Y\in\operatorname{Mat}(H), \qquad R(N(WXW^\dagger))=P(WXW^\dagger)\quad\text{for every }X\in\operatorname{Mat}(L).$$

There is no requirement here that $W$ be an isometry or that $P,N$ themselves be channels; the second equality is required for all complex matrices $X$, including matrices that are not density matrices. Empty index types are permitted.

QAlgorithms.Papers.Independent.Intakethird100.P44.RegionalCoefficients

For finite index types $H,L,J$ with decidable equality, a set $A$ of complex $H\times H$ matrices, matrices $C\in\mathbb C^{H\times H}$ and $W\in\mathbb C^{H\times L}$, and a family $(N_i)_{i\in J}$ of complex $H\times H$ matrices, this condition means that there exist complex scalars $\lambda_{ijk}$, indexed by $i,j\in J$ and Boolean $k$, such that, for every such $i,j,k$ and every complex $H\times H$ matrix $Y$ commuting with every element of $A$,

$$W^\dagger N_i^\dagger N_j(P_kYP_k)W =\lambda_{ijk}\,W^\dagger(P_kYP_k)W, \qquad P_{\mathrm{false}}=\frac{I+C}{2},\quad P_{\mathrm{true}}=\frac{I-C}{2}.$$

No positivity, normalization, or compatibility conditions on the scalars are required. In particular, when $J$ is empty, the condition holds vacuously.

QAlgorithms.Papers.Independent.Intakethird100.P44.RegionalStateCondition

For finite index types $H,L,J$ with decidable equality, a set $A$ of complex $H\times H$ matrices, matrices $C\in\mathbb C^{H\times H}$ and $W\in\mathbb C^{H\times L}$, and a family $(N_i)_{i\in J}$ of complex $H\times H$ matrices, this condition means that there exist two positive semidefinite $J\times J$ matrices $\rho_{\mathrm{false}},\rho_{\mathrm{true}}$, each of trace $1$, such that, for every $H\times H$ matrix $Y$ commuting with every element of $A$ and every complex $J\times J$ matrix $X$,

$$\sum_{i,j\in J}X_{ij}\,W^\dagger N_i^\dagger YN_jW = \sum_{k\in\{\mathrm{false},\mathrm{true}\}} \operatorname{tr}(\rho_kX)\,W^\dagger P_kYP_kW, \qquad P_{\mathrm{false}}=\frac{I+C}{2},\quad P_{\mathrm{true}}=\frac{I-C}{2}.$$

The matrices $X,Y$ need not be states or Hermitian. There are no normalization assumptions on the noise family or $W$; if $J$ is empty, the required trace-one matrices cannot exist.

QAlgorithms.Papers.Independent.Intakethird100.P44.RingCode

For natural numbers $n,k$, a family $(w_a)_{a<2n}$ of Hermitian complex $2^n\times2^n$ matrices satisfying $w_aw_b+w_bw_a=2I$ for $a=b$ and $0$ otherwise, and a ring layout with $n>0,\ k>0$, a strictly increasing function $f:\{0,\ldots,2k-1\}\to\{0,\ldots,2n-1\}$, and even gaps $g_j=((f(j^+)+2n)\mathbin{\dotminus}f(j)\mathbin{\dotminus}1)\bmod(2n)$, a ring code consists of a complex $2^n\times2^k$ matrix $W$ and proofs that

$$W^\dagger W=I,\qquad WW^\dagger= \prod_{j=0}^{2k-1}\ \prod_{t=0}^{g_j/2-1} \frac12\left( I-\mathrm{i}\, w_{(f(j)+1+2t)\bmod(2n)} w_{(f(j)+2+2t)\bmod(2n)} \right).$$

Here $j^+$ is the cyclic successor, $\dotminus$ denotes subtraction truncated at zero, and the product follows increasing $t$ within increasing $j$, with empty products equal to $I$. Thus the required matrix is an isometry whose range projection equals exactly the displayed ordered product.

QAlgorithms.Papers.Independent.Intakethird100.P44.RingLayout

For natural numbers $n,k$, a ring layout consists of proofs that $n>0$ and $k>0$, a strictly increasing function $f:\{0,\ldots,2k-1\}\to\{0,\ldots,2n-1\}$, and a proof that every number

$$g_f(j)=\bigl((f(j^+)+2n)\mathbin{\dotminus}f(j)\mathbin{\dotminus}1\bigr)\bmod(2n)$$

is even, where $j^+$ is the cyclic successor and $\dotminus$ denotes subtraction truncated at zero. Strict increase implies injectivity and hence $k\le n$. There are no layouts with $n=0$, $k=0$, or $k>n$.

QAlgorithms.Papers.Independent.Intakethird100.P44.SameMaximum

For arbitrary types $X,Y$, predicates $P:X\to\mathrm{Prop}$, $Q:Y\to\mathrm{Prop}$, and real-valued functions $f:X\to\mathbb R$, $g:Y\to\mathbb R$, this predicate asserts that there exist $x\in X$ and $y\in Y$ such that $P(x)$, $Q(y)$, $f(x)=g(y)$, every $x'$ satisfying $P(x')$ has $f(x')\le f(x)$, and every $y'$ satisfying $Q(y')$ has $g(y')\le g(y)$. Thus both maxima must be attained and equal; uniqueness is not required. If either allowed domain is empty, the predicate is false.

QAlgorithms.Papers.Independent.Intakethird100.P44.Sectors

For a finite index type $H$ with decidable equality and a natural number $r$, sector data consist of complex matrices $P_i\in\operatorname{Mat}(H)$ and real numbers $v_i$, indexed by $0\le i<r$, such that every $P_i$ is Hermitian and nonzero,

$$P_iP_j= \begin{cases} P_i,&i=j,\\ 0,&i\ne j, \end{cases} \qquad \sum_{i=0}^{r-1}P_i=I,$$

and the values $v_i$ are pairwise distinct. In particular, the matrices are mutually orthogonal projections. The definition permits $r=0$; then completeness requires $0=I$, which holds precisely when $H$ is empty. If $H$ is empty, the nonzero requirement prevents $r>0$.

QAlgorithms.Papers.Independent.Intakethird100.P44.StrongCorrectable

For finite index types $H,L$ with decidable equality, a set $A$ of complex $H\times H$ matrices, a complex $H\times H$ matrix $C$, a function $N$ from complex $H\times H$ matrices to themselves, and a complex $H\times L$ matrix $W$, this condition means that there exists a function $R$ on complex $H\times H$ matrices with a finite Kraus representation $R(Z)=\sum_a E_aZE_a^\dagger$, where $\sum_aE_a^\dagger E_a=I$, such that its trace dual

$$[R^\vee(Y)]_{ab}=\operatorname{tr}\!\bigl(R(E_{ba})Y\bigr)$$

maps every $Y\in A$ into $A$ and fixes every matrix commuting with all elements of $A$, and such that, for every complex $L\times L$ matrix $X$,

$$R\!\left(N(WXW^\dagger)\right) =\frac12\left(WXW^\dagger+C\,WXW^\dagger C\right).$$

Here $E_{ba}$ denotes the matrix unit. The quantifier over $X$ includes arbitrary matrices, and the definition imposes no channel assumption on $N$, no isometry assumption on $W$, and no Hermiticity or involution assumption on $C$.

QAlgorithms.Papers.Independent.Intakethird100.P44.StrongLocal

For a finite index type $H$ with decidable equality, an arbitrary set $A$ of complex matrices on $H$, and a map $N$ on those matrices, this predicate requires that $N$ be a channel, that its trace dual

$$N^\vee(X)_{ab}=\operatorname{tr}\!\bigl(N(E_{ba})X\bigr)$$

belong to $A$ whenever $X\in A$, and that $N^\vee(Y)=Y$ for every matrix $Y$ commuting with every member of $A$. Here $E_{ba}$ is the matrix unit, and being a channel means admitting a finite representation $N(X)=\sum_jK_jXK_j^\dagger$ with $\sum_jK_j^\dagger K_j=I$. The set $A$ is not required to be an algebra or nonempty; if $A=\varnothing$, the preservation condition is vacuous and the fixed-observable condition applies to every matrix.

QAlgorithms.Papers.Independent.Intakethird100.P44.TensorCorrectable

For finite index types $H,B,L$, with decidable equality, a map $N:\mathbb C^{H\times H}\to\mathbb C^{H\times H}$ and a complex matrix $W:\mathbb C^L\to\mathbb C^H\otimes\mathbb C^B$, this predicate means that there exists a channel $R$ on $H$ such that, for every complex $L\times L$ matrix $X$,

$$((R\circ N)\otimes\mathrm{id}_B)(WXW^\dagger)=WXW^\dagger.$$

Here applying a map $F$ to the first factor means that the $((h,b),(h',b'))$ entry of its result on $Z$ is $F((Z_{(a,b),(a',b')})_{a,a'})_{h,h'}$. A channel means a map admitting a finite representation $R(Y)=\sum_j E_jYE_j^\dagger$ with $\sum_jE_j^\dagger E_j=I$. The predicate imposes no channel assumption on $N$, no isometry assumption on $W$, and no nonemptiness assumption on the index types; its equality concerns all $X$, including matrices that are not density matrices.

QAlgorithms.Papers.Independent.Intakethird100.P44.codeSlice

For index types $H,B,L$, a complex matrix $W$ with rows indexed by $H\times B$ and columns indexed by $L$, and any $n\in B$, the slice at $n$ is the matrix with rows indexed by $H$ and columns indexed by $L$ whose entries are

$$(W_n)_{a\ell}=W_{(a,n),\ell}.$$

No normalization, positivity, or isometry condition on $W$ is required.

QAlgorithms.Papers.Independent.Intakethird100.P44.comparisonRootFidelity

For finite index types $H,K$ with decidable equality, a comparison $c$ on $H$, and arbitrary maps $N,M:\operatorname{Mat}(H)\to\operatorname{Mat}(K)$, let $S(Z)$ denote square-root continuous functional calculus, put $(\Pi_\rho)_{(a,b),(c,d)}=S(\rho)_{ba}\overline{S(\rho)_{dc}}$, and define $F_\rho(T,U)=\operatorname{Re}\operatorname{Tr}S(S(A)BS(A))$, where $A=(\mathrm{id}\otimes T)(\Pi_\rho)$, $B=(\mathrm{id}\otimes U)(\Pi_\rho)$, and the reference extension applies the map separately to each matrix block indexed by the reference coordinates. If $c$ is the fixed form with positive semidefinite trace-one matrix $\rho$, the declared value is $F_\rho(N,M)$. If $c$ is the code form with $k>0$ and $W^\dagger W=I_k$, its value is the real infimum

$$\inf_{\substack{\rho\in\operatorname{Mat}(\{0,\ldots,k-1\})\\ \rho\succeq0,\ \operatorname{Tr}\rho=1}} F_\rho\!\left(N\circ(X\mapsto WXW^\dagger), M\circ(X\mapsto WXW^\dagger)\right).$$

No channel assumption on $N,M$, and no attainment of this infimum, is included.

QAlgorithms.Papers.Independent.Intakethird100.P44.constantChannel

For a finite input index type $H$, an output index type $K$, and any complex matrix $\sigma\in\operatorname{Mat}(K)$, this defines the map

$$X\longmapsto\operatorname{tr}(X)\,\sigma \qquad(X\in\operatorname{Mat}(H)).$$

No positivity or trace normalization of $\sigma$ is assumed, and no channel property is asserted. If $H$ is empty, its trace is zero and the map is identically zero.

QAlgorithms.Papers.Independent.Intakethird100.P44.evenRegion

For every natural number $n$, a family of Hermitian $2^n\times2^n$ matrices $w_j$, $0\leq j<2n$, satisfying $w_iw_j+w_jw_i=2\delta_{ij}I$, and a finite subset $S\subseteq\{0,\ldots,2n-1\}$, this definition gives the complex linear span of all matrices

$$w_{j_1}w_{j_2}\cdots w_{j_{2r}}, \qquad r\geq0,\quad j_1,\ldots,j_{2r}\in S.$$

The indices may repeat, and their order is retained. The empty word is allowed and contributes $I$; consequently, when $S=\varnothing$, the defined space is the span of $I$. This also describes the case $n=0$.

QAlgorithms.Papers.Independent.Intakethird100.P44.fixedFidelity

For finite index types $H,K$ with decidable equality, any matrix $\rho\in\operatorname{Mat}(H)$, and arbitrary maps $N,M:\operatorname{Mat}(H)\to\operatorname{Mat}(K)$, define $S(Z)$ to be the continuous functional calculus of the real square-root function applied to $Z$, and define the matrix $\Pi_\rho$ on $H\times H$ by

$$(\Pi_\rho)_{(a,b),(c,d)} =S(\rho)_{ba}\,\overline{S(\rho)_{dc}}.$$

For any map $T$, its reference extension is defined entrywise by

$$[(\mathrm{id}_H\otimes T)(X)]_{(a,u),(b,v)} =\left[T\!\left((X_{(a,i),(b,j)})_{i,j\in H}\right)\right]_{uv}.$$

The declared real number is

$$\operatorname{Re}\operatorname{Tr} S\!\left(S(A)BS(A)\right), \qquad A=(\mathrm{id}_H\otimes N)(\Pi_\rho),\quad B=(\mathrm{id}_H\otimes M)(\Pi_\rho).$$

There is no density-matrix assumption on $\rho$, no channel assumption on $N,M$, and no nonemptiness assumption on the index types; the functional calculus is used with its total definition for all matrices.

QAlgorithms.Papers.Independent.Intakethird100.P44.krausComplement

For finite index types $H,K,J$ with decidable equality and any family of complex matrices $A_j$ with rows indexed by $K$ and columns indexed by $H$, this definition gives the map $C:\operatorname{Mat}(H)\to\operatorname{Mat}(J)$ whose entries are

$$C(X)_{ij}=\operatorname{Tr}(A_iXA_j^\dagger).$$

No equation $\sum_jA_j^\dagger A_j=I_H$, channel assumption, or restriction on the input matrix $X$ is imposed. The indexing family may be empty.

QAlgorithms.Papers.Independent.Intakethird100.P44.localComplement

For finite index types $H,E$ with decidable equality, let $A$ consist of a unital complex star subalgebra $\mathcal A\subseteq\operatorname{Mat}(H)$ and a map $P$ admitting a finite Kraus representation $P(X)=\sum_jE_jXE_j^\dagger$, $\sum_jE_j^\dagger E_j=I_H$, such that $P(X)\in\mathcal A$ and $\operatorname{Tr}((X-P(X))^\dagger Y)=0$ for every matrix $X$ and every $Y\in\mathcal A$. For any matrix $V$ with rows indexed by $H\times E$ and columns indexed by $H$, the declared map $L_{A,V}:\operatorname{Mat}(H)\to\operatorname{Mat}(H\times E)$ is

$$L_{A,V}(X)_{(a,e),(b,f)} = \left[P\!\left(((VXV^\dagger)_{(i,e),(j,f)})_{i,j\in H}\right)\right]_{ab}.$$

No isometry condition on $V$ is imposed here, and empty index types are included.

QAlgorithms.Papers.Independent.Intakethird100.P44.minimumGap

For natural numbers $n,k$ and a ring layout $r$, let $u_j=r.\mathrm{free}(j)$ for $j=0,\ldots,2k-1$. A ring layout requires $n>0$, $k>0$, the indices $u_j\in\{0,\ldots,2n-1\}$ to be strictly increasing, and every cyclic gap

$$g_j=(u_{(j+1)\bmod 2k}+2n-u_j-1)\bmod 2n$$

to be even. The defined natural number is

$$\min(2n,g_0,\ldots,g_{2k-1}),$$

computed by starting at $2n$ and successively taking the minimum with each gap in increasing index order. Zero gaps are allowed; the gap list is nonempty because $k>0$.

QAlgorithms.Papers.Independent.Intakethird100.P44.modeAt

For every natural number $n$ with a supplied proof that $n>0$, every index $j\in\{0,\ldots,2n-1\}$, and every natural-number offset $a$, this defines the index

$$(j+a)\bmod(2n)\in\{0,\ldots,2n-1\}.$$

Offsets may be zero or arbitrarily large; they wrap around modulo $2n$. The supplied positivity proof ensures the modulus is nonzero and the result is a valid index.

QAlgorithms.Papers.Independent.Intakethird100.P44.pairingProjector

For natural numbers $n,k$, a family $(w_a)_{a<2n}$ of Hermitian complex $2^n\times2^n$ matrices satisfying $w_aw_b+w_bw_a=2I$ for $a=b$ and $0$ otherwise, and a ring layout consisting of $n>0,\ k>0$, a strictly increasing function $f:\{0,\ldots,2k-1\}\to\{0,\ldots,2n-1\}$, and even gaps $g_j=((f(j^+)+2n)\mathbin{\dotminus}f(j)\mathbin{\dotminus}1)\bmod(2n)$, this defines the matrix

$$\prod_{j=0}^{2k-1}\ \prod_{t=0}^{g_j/2-1} \frac12\left( I-\mathrm{i}\, w_{(f(j)+1+2t)\bmod(2n)} w_{(f(j)+2+2t)\bmod(2n)} \right).$$

Here $\dotminus$ is natural subtraction truncated at zero, $j^+$ is the cyclic successor, and the factors are multiplied first in increasing $t$ within each gap and then in increasing $j$. A zero gap contributes no factors, and if all gaps are zero, the result is $I$.

QAlgorithms.Papers.Independent.Intakethird100.P44.parityProject

For a finite index type $H$ with decidable equality and any complex matrix $C$ on $H$, this definition assigns the map

$$P_C(X)=\frac12(X+CXC)$$

on all complex matrices $X$ on $H$. It assumes no Hermiticity, unitarity, or involution property of $C$; the rightmost factor is $C$ itself.

QAlgorithms.Papers.Independent.Intakethird100.P44.paritySector

For a finite index type $H$ with decidable equality, every complex matrix $C$ indexed by $H\times H$, and every Boolean $k$, this defines

$$P_k=\frac12\bigl(I+(-1)^kC\bigr),$$

where false corresponds to $k=0$ and true to $k=1$. Thus $P_{\mathrm{false}}=(I+C)/2$ and $P_{\mathrm{true}}=(I-C)/2$. No Hermiticity or involution assumption on $C$ is imposed.

QAlgorithms.Papers.Independent.Intakethird100.P44.poisoning

For natural numbers $n,k$, let $f$ consist of Hermitian complex $2^n\times2^n$ matrices $w_j$, indexed by $0\le j<2n$, with $w_iw_j+w_jw_i=2\delta_{ij}I$, and let $r$ be a ring layout: $n,k>0$, its free indices $u_j=r.\mathrm{free}(j)$ are strictly increasing for $0\le j<2k$, and all cyclic gaps $(u_{(j+1)\bmod 2k}+2n-u_j-1)\bmod 2n$ are even. The defined map sends every complex $2^n\times2^n$ matrix $X$ to

$$\mathcal N(X)=\frac12\left(w_{u_0}Xw_{u_0}^{\dagger} +w_{u_1}Xw_{u_1}^{\dagger}\right).$$

The two selected free indices have zero-based positions $0$ and $1$; their existence follows from $k>0$. No positivity or normalization condition is imposed on the argument $X$.

QAlgorithms.Papers.Independent.Intakethird100.P44.regionParity

For every natural number $n$, every family $f=(w_j)_{0\le j<2n}$ of complex $2^n\times2^n$ matrices satisfying $w_j^\dagger=w_j$ and $w_iw_j+w_jw_i=2I$ when $i=j$, and $0$ otherwise, and every finite set $\omega\subseteq\{0,\ldots,n-1\}$, this defines

$$C_\omega=\mathrm{i}^{|\omega|} \prod_{\substack{0\le j<2n\\ \exists s\in\omega,\ \lfloor j/2\rfloor=s}}^{\text{in increasing order of }j}w_j.$$

The multiplication order is the increasing order of the selected mode indices. The empty product is $I$, so $\omega=\varnothing$, including the case $n=0$, gives $C_\omega=I$.

QAlgorithms.Papers.Independent.Intakethird100.P44.ringGap

For natural numbers $n,k$, any function $f:\{0,\ldots,2k-1\}\to\{0,\ldots,2n-1\}$, and any $j<2k$, this defines

$$g_f(j)=\bigl((f(j^+)+2n)\mathbin{\dotminus}f(j)\mathbin{\dotminus}1\bigr)\bmod(2n),$$

where $j^+$ is the cyclic successor of $j$, and $a\mathbin{\dotminus}b=\max(a-b,0)$ is natural-number subtraction, performed in the displayed order. No injectivity, ordering, or positivity hypotheses are imposed. When $k=0$, there is no index $j$; when $n=0$, a function $f$ together with such a $j$ cannot exist.

QAlgorithms.Papers.Independent.Intakethird100.P44.sectorChargeObservable

Let $H$ be a finite index type with decidable equality, $r$ a natural number, and suppose the sector data comprise nonzero Hermitian matrices $P_i\in\operatorname{Mat}(H)$, mutually orthogonal with $P_i^2=P_i$ and $\sum_iP_i=I$, and pairwise distinct real numbers $v_i$, for $0\le i<r$. The associated charge observable is the complex matrix

$$C=\sum_{i=0}^{r-1}v_iP_i.$$

For $r=0$, this is the empty sum $0$, and the sector assumptions require $H$ to be empty.

QAlgorithms.Papers.Independent.Intakethird100.P44.sectorProject

Let $H$ be a finite index type with decidable equality, $r$ a natural number, and let $P_i\in\operatorname{Mat}(H)$ be nonzero Hermitian matrices satisfying $P_iP_j=P_i$ when $i=j$, $P_iP_j=0$ otherwise, and $\sum_iP_i=I$, together with pairwise distinct real values $v_i$. The associated map on all complex matrices on $H$ is

$$X\longmapsto\sum_{i=0}^{r-1}P_iXP_i.$$

The values $v_i$ do not enter this formula. For $r=0$, the sum is zero and the sector assumptions force $H$ to be empty.

QAlgorithms.Papers.Independent.Intakethird100.P44.siteModes

For every natural number $n$, including $n=0$, and every subset $\omega$ of $\{0,\ldots,n-1\}$, this defines the finite set

$$S_\omega=\{j\in\{0,\ldots,2n-1\}:\exists s\in\omega,\ \lfloor j/2\rfloor=s\}.$$

Thus each site $s\in\omega$ contributes the two indices $2s$ and $2s+1$. If $\omega$ is empty, including necessarily when $n=0$, the result is empty.

QAlgorithms.Papers.Independent.Intakethird100.P44.totalParity

For any natural number $n$ and any family $f$ of Hermitian $2^n\times2^n$ matrices $w_j$, indexed by $0\leq j<2n$, satisfying $w_iw_j+w_jw_i=2\delta_{ij}I$, this definition assigns the matrix

$$C_f=\mathrm{i}^{\,n}w_0w_1\cdots w_{2n-1},$$

with the factors multiplied in increasing index order. For $n=0$, the product is empty and $C_f=I$ on the one-dimensional space.