Dependencies

Let \(T_1,T_2:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be CP maps with \(T_1\le T_2\). Then there exist ancilla dimensions \(r_1,m\), Heisenberg-form Stinespring matrices \(V_1:\mathbb {C}^D\to \mathbb {C}^D\otimes \mathbb {C}^{r_1}\) and \(V_2:\mathbb {C}^D\to \mathbb {C}^D\otimes \mathbb {C}^m\) realizing \(T_1,T_2\) via \(T_i(A)=V_i^\dagger (A\otimes \mathbb {1})V_i\), and a rectangular contraction \(\widetilde C:\mathbb {C}^m\to \mathbb {C}^{r_1}\) with \(\widetilde C^\dagger \widetilde C\le \mathbb {1}_m\) such that

This is the explicit square corollary obtained by constructing both dilation matrices, rather than the supplied-dilation statement of Theorem 3.11.7.

Let \(I\) be a nonempty finite set, and let \(T_i,T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be completely positive linear maps such that \(\sum _{i\in I}T_i=T\). If \(V:\mathbb {C}^D\to \mathbb {C}^D\otimes \mathbb {C}^r\) satisfies \(T(A)=V^\dagger (A\otimes \mathbb {1}_r)V\), then there are positive semidefinite operators \(P_i\in M_{r}(\mathbb {C})\) satisfying \(\sum _{i\in I}P_i=\mathbb {1}_r\) and \(T_i(A)=V^\dagger (A\otimes P_i)V\) for every \(i\in I\) and \(A\in M_{D}(\mathbb {C})\).

The following conditions are equivalent:

1. \(\tau _{E^m}{\gt}0\) for every sufficiently large \(m\);

2. \(\mathcal S_m=M_{D}(\mathbb {C})\) for every sufficiently large \(m\).

Let \(\{ \Phi _i\} \) be a quantum instrument on \(M_{d}(\mathbb {C})\) with \(d\geq 1\) whose total channel is the identity. Then for every outcome \(i\) the probability \(p_i(\rho )\) is a non-negative constant, the same for every \(\rho \) of unit trace.

For \(r,s\), let \(P_{\mathrm{top}}=C_{r,s}^\dagger C_{r,s}\) and \(P_{\mathrm{bot}}=\mathbb {1}-P_{\mathrm{top}}\). Both are PSD (for \(P_{\mathrm{bot}}\), by Lemma 3.11.10) and \(P_{\mathrm{top}}+P_{\mathrm{bot}}=\mathbb {1}\).

For natural numbers \(r,s\), let \(C=C_{r,s}\in \mathbb {C}^{r\times (r+s)}\) be the rectangular \(r\times (r+s)\) matrix whose rows are the first \(r\) rows of the identity on \(\mathbb {C}^{r+s}\): \(C_{ij}=1\) if \(j=i{\lt}r\) and \(0\) otherwise.

A linear map \(E : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) is a quantum channel if it is both completely positive and trace-preserving (CPTP), following  [ Wol12 ] .

For a linear map \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\), the channel determinant \(\det T\) is the determinant of the matrix of \(T\) with respect to the standard matrix-unit basis of \(M_{D}(\mathbb {C})\). This is the quantity studied in [ Wol12 , Section 6.1 ] .

The Choi matrix of a linear map \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) is

Concretely, \(\tau _{(i_1,i_2),(j_1,j_2)} = \frac{1}{D}\, (T(E_{i_2 j_2}))_{i_1 j_1}\) where \(E_{i_2 j_2}\) is the matrix unit. This is the Choi–Jamiol\- kowski convention of [ Wol12 , Proposition 2.1 ] .

The Choi matrix of a linear map \(T:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) is

an operator on \(\mathbb {C}^{d'}\otimes \mathbb {C}^{d}\). This is the correspondence of [ Wol12 , Proposition 2.1 ] .

For a linear map \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) with Choi–Jamiołkowski operator \(\tau \), the purity of \(\tau \) is \(\operatorname{tr}[\tau ^\dagger \tau ]\), the sum

of the squared moduli of the entries of \(\tau \); in particular it is non-negative.

For linear maps \(S,T:M_{d_{\rm in}}(\mathbb {C})\to M_{d_{\rm out}}(\mathbb {C})\), write \(T\le S\) (in the CP order) when \(S-T\) is completely positive.

A linear map \(E : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) is completely positive (CP) if it admits a Kraus representation: there exist operators \(\{ K_i\} _{i=0}^{r-1}\) with \(K_i \in M_{D}(\mathbb {C})\) such that, for every \(X \in M_{D}(\mathbb {C})\),

The Kraus representation also gives entrywise positivity on every positive block matrix by Theorem 2.2.1, and hence the associated completely positive map between matrix \(C^*\)-algebras in Theorem 2.2.2.

A linear map \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) is doubly-stochastic if \(T(\mathbb {1})\propto \mathbb {1}\) and the reduced density matrix \(\operatorname{tr}_{1}[\tau ]\) of its Choi matrix \(\tau =(T\otimes \operatorname{id})(|\Omega \rangle \! \langle \Omega |)\) is proportional to the identity. This is the normal form in [ Wol12 , Proposition 2.9 ] .

A linear map \(T : M_{d_1}(\mathbb {C}) \to M_{d_2}(\mathbb {C})\) between matrix algebras of possibly different dimensions is doubly-stochastic if \(T(\mathbb {1})\propto \mathbb {1}\) and \(T^{*}(\mathbb {1})\propto \mathbb {1}\), where \(T^{*}\) is the trace-pairing adjoint. This is the normal-form condition of [ Wol12 , Proposition 2.9 ] ; by the rectangular Choi–Jamiolkowski correspondence it is equivalent to both partial traces of the Choi matrix being proportional to the identity.

In \(\mathbb {C}^2 \otimes \mathbb {C}^2\), let \(v = |01\rangle - |10\rangle \). Equivalently, \(v_{(0,1)}=1\), \(v_{(1,0)}=-1\), and all other coefficients are zero.

For \(r\geq 1\), let \(W_0:\mathbb {C}^D\to \mathbb {C}^D\otimes \mathbb {C}^r\) be the isometry \(W_0x=x\otimes e_0\).

The flip operator \(F\) on \(\mathbb {C}^d \otimes \mathbb {C}^d\) is

Its matrix entries are

Let

A real matrix \(C\in M_4(\mathbb R)\) is \(M\)-selfadjoint when \(C^{\mathsf T}M=MC\).

For a complex-linear map \(E:\mathbb C^{I\times I}\to \mathbb C^{J\times J}\), define its Frobenius transport by

For finite index sets \(I\) and \(J\), define

This is a complex-linear isometric equivalence when the matrix space is equipped with the Frobenius norm.

A family \((\sigma _\alpha )\) in \(M_{d}(\mathbb {C})\) is Hilbert–Schmidt orthonormal when

This is the pairing \(\operatorname{tr}(PA^\dagger B)\) of [ Wol12 , Equation (2.18) ] in the case \(P=\mathbb {1}\).

A quantum instrument with \(n\) outcomes is a family \(\{ \Phi _i\} _{i=0}^{n-1}\) of completely positive maps on \(M_{D}(\mathbb {C})\) whose sum \(\sum _i\Phi _i\) is trace-preserving.

Associated to an instrument are the total channel \(\sum _i\Phi _i\), the unnormalized update \(\rho \mapsto \Phi _i(\rho )\) for each outcome \(i\), the outcome probability \(p_i(\rho )=\operatorname{tr}(\Phi _i(\rho ))\), and the normalized posterior state \(\Phi _i(\rho )/p_i(\rho )\) whenever \(p_i(\rho )\neq 0\).

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a bijective complex-linear map. Write \(T^{-1}\) for the inverse linear map supplied by the associated linear equivalence; it is both a left and a right inverse of \(T\).

An invertible filtering operation on \(M_{D}(\mathbb {C})\) is a completely positive map

If \(T:M_{d_1}(\mathbb {C})\to M_{d_2}(\mathbb {C})\), pre- and postfiltering are written in Wolf’s order as \(\Phi _2\circ T\circ \Phi _1\). The matrices \(X\) are retained as part of the data; in particular, matrices which differ by a phase are not identified.

Let \(I\) be a finite index set and let \(C\in \mathbb C^{I\times I}\) be invertible. Congruence by \(C\) is the complex-linear equivalence

whose inverse is

A linear map \(E : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) is irreducible if whenever \(P\) is an orthogonal projection satisfying \(E(P M_{D}(\mathbb {C}) P) \subseteq P M_{D}(\mathbb {C}) P\), then \(P = 0\) or \(P = \mathbb {1}\). This is [ Wol12 , Theorem 6.2(1) ] . The definition applies to any linear map; complete positivity is not required.

A linear map \(\mathcal S\) between matrix algebras is completely positive in rectangular Kraus form if there are finitely many operators \(A_i:H\to K\) such that

No trace-preservation normalization is imposed.

A linear map \(\mathcal{S}\) between matrix algebras is trace-preserving completely positive if it has a Kraus form \(\mathcal{S}(X)=\sum _i A_iXA_i^\dagger \) with \(\sum _i A_i^\dagger A_i=I\). The Kraus operators may be rectangular, so the input and output dimensions need not agree.

A Naimark dilation of a POVM \(\{ E_i\} \) consists of an isometry \(V\) into a larger Hilbert space together with a projective measurement \(\{ P_i\} \) there such that \(E_i=V^\dagger P_iV\) for all \(i\).

For \(\psi \in \mathbb {C}^D\) and \(m\in \mathbb N\), define

The Kraus family has eventually full vector spread if, for every sufficiently large \(m\) and every nonzero \(\psi \in \mathbb {C}^D\), \(H_m(K,\psi )=\mathbb {C}^D\). This is [ Wol12 , Theorem 6.8(2) ] .

A finite Kraus family has eventually full word span when \(S_N(K)=M_{D}(\mathbb {C})\) for all sufficiently large \(N\).

For a finite Kraus family \(K_0,\ldots ,K_{d-1}\in M_{D}(\mathbb {C})\) and \(N\in \mathbb N\), define

The empty product is the identity matrix.

A map \(T:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) has Kraus cardinality \(r\) if it has an exact \(r\)-operator Kraus representation. Its Kraus rank (Choi rank) is the rank of its Choi matrix, \(r=\operatorname {rank}(\tau )\); following [ Wol12 , Theorem 2.1, footnote ] , this is distinguished from the rank of \(T\) as a linear map.

Let \(\mathcal L:M_{d}(\mathbb {C})\to M_{e}(\mathbb {C})\) be complex-linear. Its coefficient matrix \(C(\mathcal L)\in M_{e^2\times d^2}(\mathbb {C})\) in the matrix-unit bases is defined by

If a Stinespring matrix \(V:\mathbb {C}^{d_{\mathrm{in}}}\to \mathbb {C}^{d_{\mathrm{out}}}\) acts on the first factor while a finite-dimensional space \(R\) is left unchanged, the corresponding local Stinespring matrix is \(W=V\otimes \mathbb {1}_R\).

For a completely positive, trace-preserving qubit channel \(T' : M_{2}(\mathbb {C}) \to M_{2}(\mathbb {C})\), write \(\widehat{T'}\) for its matrix in the normalized Pauli basis \(\{ \sigma _{0}/\sqrt{2},\sigma _{1}/\sqrt{2}, \sigma _{2}/\sqrt{2},\sigma _{3}/\sqrt{2}\} \). The channel is in diagonal Lorentz normal form when it is unital and every off-diagonal entry of \(\widehat{T'}\) is zero.

For a completely positive, trace-preserving qubit channel \(T' : M_{2}(\mathbb {C}) \to M_{2}(\mathbb {C})\), write \(\widehat{T'}\) in the normalized Pauli basis. The channel is in non-diagonal Lorentz normal form when, for some \(x \in [0,1]\),

Trace preservation supplies the first row of the displayed matrix.

For a completely positive, trace-preserving qubit channel \(T' : M_{2}(\mathbb {C}) \to M_{2}(\mathbb {C})\), write \(\widehat{T'}\) in the normalized Pauli basis. The channel is in singular Lorentz normal form when

equivalently, every input state is mapped to \((1+\sigma _{3})/2\). Trace preservation supplies the first row of the displayed matrix.

A linear map \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) maps the positive semidefinite cone onto itself if \(T\) is positive and every positive semidefinite matrix is the image under \(T\) of a positive semidefinite matrix. This is condition (1) of [ Wol12 , Proposition 3.6 ] .

For every \(d\in \mathbb {N}\), define \(|\Omega \rangle \! \langle \Omega |\) on \(\mathbb {C}^d\otimes \mathbb {C}^d\) using

When \(d\geq 1\), this is the maximally entangled state: a rank-one projector with \((|\Omega \rangle \! \langle \Omega |)_{(i_1,i_2),(j_1,j_2)} = \frac{1}{d}\delta _{i_1 i_2}\delta _{j_1 j_2}\) and \(\operatorname{tr}(|\Omega \rangle \! \langle \Omega |)=1\).

Let \(E_{k\ell }\) denote the matrix unit with its only nonzero entry in row \(k\) and column \(\ell \). The transfer matrix \(\widehat T\) is the matrix indexed by pairs of bond indices with entries

Equivalently, this is the matrix of \(T\) under the column-stacking identification \(M_D(\mathbb C)\cong \mathbb C^{D^2}\): the matrix-units special case of Definition 3.18.2.

The Naimark isometry \(V:\mathbb {C}^D\to \mathbb {C}^D\otimes \mathbb {C}^n\) of a POVM is the Stinespring-type construction

For a POVM \(\{ E_i\} \), each effect \(E_i\ge 0\) admits a square-root factorisation \(E_i=M_i^\dagger M_i\) with \(M_i\in M_{D}(\mathbb {C})\). The operators \(M_i\) are the Naimark Kraus square roots.

The Naimark projectors on \(\mathbb {C}^D\otimes \mathbb {C}^n\) are

Let \(V:\mathbb {C}^d\to \mathbb {C}^{d'}\otimes \mathbb {C}^r\) be a supplied Stinespring matrix, and let \(|\Omega \rangle =d'^{-1/2}\sum _{a=1}^{d'}|a,a\rangle \) be normalized. Wolf’s auxiliary operator is

or, in coordinates, \(W_{j,(i,a)}=d'^{-1/2}V_{(a,j),i}\).

For a matrix \(V:\mathbb C^K\to \mathbb C^J\) and a complex-linear map \(\Phi :\mathbb C^{I\times I}\to \mathbb C^{J\times J}\), define \(\Psi (X)=V^\dagger \Phi (X)V\).

Let \(\{ \psi _k\} _{k {\lt} m}\) be an ensemble and let \(n\) be a natural number. The padded ensemble \(\{ \psi _k^{\mathrm{pad}}\} _{k {\lt} n}\) is \(\psi _k\) for \(k {\lt} m\) and the zero vector for \(m \le k {\lt} n\). For \(m \le n\) the padded family agrees with \(\psi \) on every index \(k {\lt} m\) and retains the whole ensemble; for \(n {\lt} m\) only the first \(n\) vectors survive. The intended use is \(m \le n\).

Let \(X \in M_{d \cdot d'}(\mathbb {C})\) be a bipartite matrix indexed by \((\{ 0,\ldots ,d-1\} \times \{ 0,\ldots ,d'-1\} )^2\). The left partial trace \(\operatorname{tr}_A(X)\) and right partial trace \(\operatorname{tr}_B(X)\) are the \(d' \times d'\) and \(d \times d\) matrices defined by

Define Pauli time reversal by \(\Theta (X)=\operatorname{tr}(X)\mathbb {1}-X\). Its Pauli transfer matrix is

For a linear map \(T:M_{2}(\mathbb {C})\to M_{2}(\mathbb {C})\), define its Pauli-block diagonal truncation by

This is the truncation in [ Wol12 , Proposition 2.10, Section 2.4 ] ; see also the local source Notes/WolfNoteTexSource/ch02_representations.tex, lines 984–998.

Every Hermitian matrix \(M\in M_2^\dagger (\mathbb {C})\) has unique real Pauli coordinates \(x=(x_0,x_1,x_2,x_3)\) such that

This identifies \(M_2^\dagger (\mathbb {C})\) with \(\mathbb {R}^4\). Write

for the Minkowski quadratic form, its matrix, and the closed future cone. These are the coordinates used immediately before [ Wol12 , Eq. (2.41) ] ; see also the local source Notes/WolfNoteTexSource/ch02_representations.tex, lines 1040–1044.

Let \(\sigma _0=\mathbb {1},\sigma _1,\sigma _2,\sigma _3\) be the Pauli matrices. The Pauli transfer matrix of a linear map \(T:M_{2}(\mathbb {C})\to M_{2}(\mathbb {C})\) is \(\widehat T\in M_4(\mathbb {C})\) with entries

We write

where \(\Delta =(\widehat T_{ij})_{i,j=1}^{3}\). Thus \(r^{\mathsf T}\) and \(v\) are exactly the two off-diagonal Pauli blocks. This is the representation used in [ Wol12 , Section 2.4, Eq. (2.39) ] .

A linear map \(E : M_{n}(\mathbb {C}) \to M_{m}(\mathbb {C})\) is positive if \(E(X) \ge 0\) whenever \(X \ge 0\), where \(X \ge 0\) means that \(X\) is positive semidefinite.

A positive operator-valued measure with \(n\) outcomes on \(\mathbb {C}^D\) is a family \(\{ E_i\} _{i=0}^{n-1}\) of positive semidefinite operators on \(\mathbb {C}^D\) satisfying the resolution of identity

Given an isometry \(V:\mathbb {C}^D\to \mathbb {C}^{d'}\) with \(V^\dagger V=\mathbb {1}_D\) and a family \(\{ P_i\} _{i=0}^{n-1}\) of positive semidefinite operators on \(\mathbb {C}^{d'}\) summing to the identity, the pulled-back operators \(E_i:=V^\dagger P_iV\) form a POVM. In particular, any projective measurement on the dilation (a special case of a PSD resolution of identity) pulls back to a POVM.

A linear map is primitive if its only peripheral eigenvalue is \(\lambda = 1\), i.e. \(\mathrm{peripheral}(E) = \{ 1\} \). For channels this corresponds to [ Wol12 , Theorem 6.7 ] . This definition applies to any linear endomorphism, not only to channels.

Given a finite family \(\{ \psi _i\} _{i \in \iota }\) of (unnormalized) vectors in \(\mathbb {C}^D\), its pure-ensemble density is \(\rho = \sum _i |\psi _i\rangle \! \langle \psi _i|\). Weights \(p_i \geq 0\) with \(\sum p_i = 1\) can be absorbed by replacing \(\psi _i \mapsto \sqrt{p_i} \psi _i\).

A real-eigenvalue block is the ordinary real Jordan block \(J_n(a)\) with eigenvalue \(a\) and ones on its first superdiagonal. If \(a\pm \tau i\) is a non-real conjugate pair, put

Its real Jordan chain of length \(m\) has \(B(a,\tau )\) on the block diagonal and \(I_2\) on the first block superdiagonal. On the product indexing of this \(2m\)-dimensional real block, its unsigned metric reverses both the chain coordinate and the two-dimensional realification coordinate. No sign characteristic is attached to a non-real block.

For a complex-linear map \(\mathcal L:M_{d}(\mathbb {C})\to M_{e}(\mathbb {C})\), its unnormalized rectangular Choi matrix \(J(\mathcal L)\in M_{de}(\mathbb {C})\) is the reshaping

When \(d=e\), this is \(d\) times the normalized Choi matrix convention in [ Wol12 , Proposition 2.1 ] .

A complex-linear map \(\mathcal L:M_{d}(\mathbb {C})\to M_{e}(\mathbb {C})\) is a Hilbert–Schmidt contraction if, for every \(X\in M_{d}(\mathbb {C})\),

Let \(d\ge 1\). A symmetric informationally complete family in dimension \(d\) consists of \(d^2\) rank-one orthogonal projections \(P_i\in M_{d}(\mathbb {C})\) satisfying

A matrix \(P\in M_{D}(\mathbb {C})\) is a rank-one orthogonal projection if \(P=P^\dagger \), \(P^2=P\), and \(\operatorname{rank}P=1\).

The sip matrix of size \(n\) is \(P_n\). A signed sip block is \(\varepsilon P_n\), where \(\varepsilon =1\) or \(\varepsilon =-1\). Each such block is symmetric, invertible, and its own inverse.

An SL-filtering for \(D \times D\) matrices is a completely positive map of the form \(\Phi (X)=SXS^{\dagger }\), where \(S \in M_{D}(\mathbb {C})\) satisfies \(\det S=1\). Such maps are invertible; when \(D\geq 1\), they have Kraus rank 1.

For \(X\in \operatorname{SL}(2,\mathbb {C})\), let \(L(X)\) be the real linear transformation of \(\mathbb {R}^4\) determined by Wolf’s Hermitian congruence

Its Pauli-basis entries are

The trace is real. This is the four-dimensional action in [ Wol12 , Eq. (2.41) ] ; it is not the three-dimensional adjoint action \(M\mapsto UMU^{-1}\) on traceless Pauli matrices. The special orthochronous Lorentz group is described, in the row-action convention of the source, by

Regard \(\mathrm{SO}^+(1,3)\) from (95) as a matrix group. The spinor action defines the homomorphism

Given possibly rectangular Kraus operators \(K_j:\mathbb {C}^{d_{\mathrm{in}}}\to \mathbb {C}^{d_{\mathrm{out}}}\), the Stinespring matrix \(V:\mathbb {C}^{d_{\mathrm{in}}}\to \mathbb {C}^{d_{\mathrm{out}}}\otimes \mathbb {C}^r\) is defined by

Thus \(V=\sum _j K_j\otimes |j\rangle \). It is an isometry precisely when the Kraus family satisfies the trace-preserving normalization.

For a Kraus family \((K_j)_{j\in J}\) indexed by an arbitrary finite set \(J\), define \((V_J)_{(i,j),k}=(K_j)_{ik}\).

Let \(\tau \geq 0\). Its support-restricted negative quarter power \(\tau ^{-1/4}_{\operatorname {supp}}\) acts by \(x^{-1/4}\) on every positive eigenspace and vanishes on the kernel. For a complex-linear map \(\Phi \), define

Given a linear map \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\), the tensor extension \(T \otimes \operatorname{id}\) acts on bipartite matrices \(X \in M_{D \times D}(\mathbb {C})\) by applying \(T\) to each “slice”:

where \(X^{(i_2,j_2)}_{ab} = X_{(a,i_2),(b,j_2)}\) is the bipartite slice.

The trace-pairing adjoint \(E^* : M_{D'}(\mathbb {C}) \to M_{D}(\mathbb {C})\) of a linear map \(E : M_{D}(\mathbb {C}) \to M_{D'}(\mathbb {C})\) between matrix algebras of possibly different dimensions is the adjoint for the bilinear pairing \((A,B)\mapsto \operatorname{tr}(AB)\).

For a linear map \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) and a trace-self-dual basis \(\{ \sigma _i\} \), define coefficients \(t_{ij}:=\operatorname{tr}(\sigma _i T(\sigma _j))\).

A basis \(\{ \sigma _i\} _i\) of \(M_{D}(\mathbb {C})\) is trace-self-dual when its coordinate functionals are given by trace pairing:

A linear map \(E : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) is trace-preserving if \(\operatorname{tr}(E(X)) = \operatorname{tr}(X)\) for all \(X\).

For a linear map \(T:M_{d}(\mathbb {C})\to M_{d}(\mathbb {C})\) and a family \((\sigma _\alpha )\) in \(M_{d}(\mathbb {C})\) used on both sides, the transfer matrix of \(T\) in \((\sigma _\alpha )\) is

which is [ Wol12 , Equation (2.20) ] with \(F_\alpha =G_\alpha \).

For a unitary matrix \(U \in \mathcal{U}(D)\), the unitary channel is \(T(\rho )=U\rho U^\dagger \). It is automatically a quantum channel.

The four source-indexed patterns are: four real one-dimensional blocks; one two-dimensional non-real conjugate-pair block and two real one-dimensional blocks; one real Jordan block of size two and two real one-dimensional blocks; or one real Jordan block of size three and one real one-dimensional block.

The fixed-length vector span at length \(n\) is

This is \(S_n(K)|\varphi \rangle \) in the notation of  [ SPGWC10 ] .

For matrices \(\sigma \in \mathbb C^{I\times I}\) and \(\tau \in \mathbb C^{J\times J}\) and a complex-linear map \(\Phi :\mathbb C^{I\times I}\to \mathbb C^{J\times J}\), define

For a non-negative family \(f_{0},\ldots ,f_{D-1}\) of real numbers, or more generally for a non-negative family indexed by a finite set with \(D\) elements,

The block-top matrix satisfies \(CC^\dagger =\mathbb {1}_r\).

The block-top matrix satisfies \(C^\dagger C\le \mathbb {1}_{r+s}\).

Let \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) be a linear map, \(\widehat{T}\) its matrix with respect to the matrix-unit basis and \(\tau \) its Choi–Jamiołkowski operator. Then

Equivalently \(\widehat{T}=D\, \tau ^{\Gamma }\), where the involution \(\tau \mapsto \tau ^{\Gamma }\) is defined by \(\langle m,n|\tau ^{\Gamma }|k,\ell \rangle =\langle m,k|\tau |n,\ell \rangle \). This is [ Wol12 , Eq. (2.22) ] .

Let \(D\geq 1\). Two linear maps \(T,S:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) with the same Choi matrix are equal.

With the notation of Lemma 3.17.2,

For \(D\geq 1\) this is [ Wol12 , Eq. (6.28) ] , which states it in the divided form \(\operatorname{tr}[\tau ^\dagger \tau ]=D^{-2}\operatorname{tr}[\widehat{T}^\dagger \widehat{T}]\).

Let \(A\in M_{D}(\mathbb {C})\) with \(D\geq 1\). If \(|\det A|=1\), then \(D\leq \operatorname{tr}(A^{\dagger }A)\).

For a square complex matrix \(A\) whose index set has \(n\) elements,

Let \(A,B:\mathbb {C}^n\to \mathbb {C}^m\) be linear maps with \(B^\dagger B=A^\dagger A\). Then there is a unitary \(U\) on \(\mathbb {C}^m\) such that \(B=UA\).

The first-environment embedding satisfies \(W_0^\dagger W_0=\mathbb {1}\).

For finite matrices \(A\) and \(B\) of the same shape,

Suppose \(T(X)=\sum _j K_j X K_j^\dagger \) and also \(T(X)=\sum _j K_j^\dagger X K_j\). Then \(T\) has a family of Hermitian Kraus operators, namely \(\tfrac 12(K_j+K_j^\dagger )\) and \(\tfrac {i}{2}(K_j-K_j^\dagger )\).

Let \((\sigma _\alpha )\) be a Hilbert–Schmidt orthonormal basis of \(M_{d}(\mathbb {C})\). The coordinates of \(X\in M_{d}(\mathbb {C})\) in this basis are \(\operatorname{tr}(\sigma _\alpha ^\dagger X)\); consequently \(X=0\) as soon as \(\operatorname{tr}(\sigma _\alpha ^\dagger X)=0\) for every \(\alpha \). If in addition every \(\sigma _\alpha \) is Hermitian, the coordinates are \(\operatorname{tr}(\sigma _\alpha X)\), so the basis is self-dual for the bilinear trace pairing.

For a positive-definite operator \(\tau \) on \(\mathbb {C}^{d_2}\otimes \mathbb {C}^{d_1}\), the infimum of

over \(S_{1}\in M_{d_1}(\mathbb {C})\) and \(S_{2}\in M_{d_2}(\mathbb {C})\) with \(\det S_{1}=\det S_{2}=1\) is attained. The two tensor factors may have different dimensions.

Let \(D\geq 1\) and let \(X\in \mathrm{GL}(D,\mathbb {C})\). There are a nonzero \(c\in \mathbb {C}\) and \(S\in \operatorname{SL}(D,\mathbb {C})\) such that

Moreover, \(\Phi _X\) has Kraus rank exactly one. The matrix scalar \(c\) is complex and need not be positive real; positivity applies only to the map scalar \(|c|^2\).

Suppose \(X_i=c_iS_i\), where \(c_i\neq 0\) and \(S_i\in \operatorname{SL}(d_i,\mathbb {C})\). For every linear map \(T:M_{d_1}(\mathbb {C})\to M_{d_2}(\mathbb {C})\),

For complex matrices \(A\) and \(B\) of compatible rectangular sizes,

Let \(A\) on \(\mathbb {C}^d\otimes \mathbb {C}^d\) be positive semidefinite with \(A\leq |\Omega \rangle \! \langle \Omega |\). Then \(A=a|\Omega \rangle \! \langle \Omega |\) for a non-negative scalar \(a\).

For a POVM \(\{ E_i\} \) with square roots \(E_i=M_i^\dagger M_i\), one has \(\sum _iM_i^\dagger M_i=\mathbb {1}\).

If the supplied dominating dilation is minimal, \(r_2=\operatorname{rank}(\tau _2)\), then \(W_2\) is surjective. Consequently a factorization \(W_1=CW_2\) determines \(C\) uniquely.

Let \(W_i:H\to R_i\) be finite-dimensional complex matrices. If

then there is a contraction \(C:R_2\to R_1\) such that \(W_1=CW_2\).

Suppose that \(T:M_{d'}(\mathbb {C})\to M_{d}(\mathbb {C})\) has the supplied representation \(T(A)=V^\dagger (A\otimes \mathbb {1}_r)V\). If \(\tau \) is the normalized Choi matrix of \(T\), then

If \(d'{\gt}0\), the assignment \(V\mapsto W\) is one-to-one. Moreover, for every \(C:\mathbb {C}^{r_2}\to \mathbb {C}^{r_1}\),

Let \(D\geq 1\), let \(M\in M_{D}(\mathbb {C})\) be positive-definite, and let \(\lambda _{\min }(M)\) be its smallest Hermitian eigenvalue. For every \(X\in M_{D}(\mathbb {C})\),

Let \(A\in M_{D}(\mathbb {C})\) be positive semidefinite, let \(c\in \mathbb {C}\), and let \(\psi \in \mathbb {C}^D\). If \(A\leq c|\psi \rangle \! \langle \psi |\), then there is a non-negative scalar \(a\) such that \(A=a|\psi \rangle \! \langle \psi |\).

Let \(e \colon \iota _1 \to \iota \) be injective and let \(\{ \Psi _k\} _{k \in \iota }\) satisfy \(\Psi _{e(i)} = \psi _i\) for all \(i \in \iota _1\) and \(\Psi _k = 0\) for every \(k\) outside the image of \(e\). Then \(\sum _{k} |\Psi _k\rangle \! \langle \Psi _k| = \sum _i |\psi _i\rangle \! \langle \psi _i|\).

For \(m \le n\) the padded ensemble of Definition 3.9.3 has the same density operator as the original ensemble.

Let \(\{ \psi _i\} _{i \in \iota _1}\) and \(\{ \phi _j\} _{j \in \iota _2}\) be ensembles. Extend \(\psi \) by the zero vector on the second summand of \(\iota _1 \sqcup \iota _2\), and extend \(\phi \) by the zero vector on the first summand. Both extended families have the same density operator as the family they extend.

Let \(K:\{ 0,\ldots ,r-1\} \to M_{D}(\mathbb {C})\) and \(L:\{ 0,\ldots ,s-1\} \to M_{D}(\mathbb {C})\) be two Kraus families. Write \(K\mathbin {+\! +}L\) for their concatenation as a family indexed by \(\{ 0,\ldots ,r+s-1\} \). Then

Let \(T:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) be a linear map with Choi matrix \(\tau \) and dual \(T^*\). If for some ancilla dimension \(r\) there is a \(V:\mathbb {C}^{d}\to \mathbb {C}^{d'}\otimes \mathbb {C}^{r}\) such that (27) holds for all \(A\in M_{d'}(\mathbb {C})\), then \(\operatorname{rank}(\tau )\le r\). Together with Theorem 3.10.13, the least admissible ancilla dimension is \(\operatorname{rank}(\tau )\), and dilations with \(r=\operatorname{rank}(\tau )\) are minimal. This is the discussion following [ Wol12 , Theorem 2.2 ] .

For output index \(i\), input index \(k\), and \(j\in \{ 0,\ldots ,r-1\} \), \(V_{(i,j),k}=(K_j)_{ik}\).

The finite-index Stinespring matrix satisfies \((V_J)_{(i,j),k}=(K_j)_{ik}\).

If \(T(X)=\sum _j K_j X K_j^\dagger \) then \(T^*(X)=\sum _j K_j^\dagger X K_j\): the Kraus operators of \(T\) and of \(T^*\) differ by Hermitian conjugation, as in [ Wol12 , Section 2.2, after Proposition 2.4 ] . Such a \(T\) satisfies \(T(X^\dagger )=T(X)^\dagger \), and so does every completely positive map.

For a positive-semidefinite \(D \times D\) matrix \(M\),

and equality holds if and only if \(M=(\operatorname{tr}M/D)\mathbb {1}\). The inequality holds for a positive-semidefinite matrix whose rows and columns are indexed by any finite set with \(D\) elements; the equality characterization is stated only for \(D \times D\) matrices.

Let \(T:M_{d}(\mathbb {C})\to M_{d}(\mathbb {C})\) satisfy \(T(X^\dagger )=T(X)^\dagger \). Then in any family \((\sigma _\alpha )\) used on both sides,

and the same identity holds for the transfer matrix taken in the matrix units. This is the Hermitian-map case of the sentence following [ Wol12 , Equation (2.20) ] .

Let \(T:M_{d}(\mathbb {C})\to M_{d}(\mathbb {C})\) satisfy \(T(X^\dagger )=T(X)^\dagger \) and let \((\sigma _\alpha )\) be a Hilbert–Schmidt orthonormal basis of \(M_{d}(\mathbb {C})\) used on both sides. Then

Let \(T:M_{d}(\mathbb {C})\to M_{d}(\mathbb {C})\) be completely positive and let \(\widehat T\) be its transfer matrix in the matrix units. Then

Let \(\rho \geq 0\) act on a finite-dimensional Hilbert space, and let \(C:\mathbb {C}^r\to \mathcal H\) satisfy \(CC^\dagger =\rho \). If \(\rho =\sum _{i=0}^{n-1}\sigma _i\), where \(n\geq 1\) and every \(\sigma _i\geq 0\), then there is a POVM \(\{ P_i\} _{i=0}^{n-1}\) on \(\mathbb {C}^r\) such that

for every \(i\).

Single-Kraus maps satisfy \(\mathcal K_V\circ \mathcal K_W=\mathcal K_{VW}\). Trace adjoints reverse composition, and the trace adjoint of \(X\mapsto VXV^\dagger \) is \(Y\mapsto V^\dagger YV\). Frobenius vectorization transports both linear equivalences and conjugation by such equivalences.

Let \(d\geq 1\) and let \(T:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) be a quantum channel. Suppose a normalized vector \(\varphi \in \mathbb {C}^{d'}\otimes \mathbb {C}^{d'}\) and a unitary identification

are supplied, so both sides have dimension \(d(d')^2\), and suppose that the system-plus-environment representation from Equation (2.14) holds:

If \(n\geq 1\) and \(T=\sum _{i=0}^{n-1}T_i\) is a decomposition into completely positive maps \(T_i:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\), then there is a POVM \(\{ P_i\} _{i=0}^{n-1}\subset M_{dd'}(\mathbb {C})\) such that

for every \(i\) and \(\rho \). Moreover, the Kraus rank \(k_i\) of \(T_i\) satisfies \(k_i\leq \operatorname{rank}(P_i)\). This is [ Wol12 , Proposition (Environment induced instruments), Equation (2.15) ] .

For every sign \(\varepsilon \) and every real Jordan block,

The real Jordan chain for a non-real conjugate pair satisfies the analogous identity with its unsigned \(2m\)-dimensional reversal metric.

For every linear endomorphism \(T\) of \(M_{D}(\mathbb {C})\),

This is the basis-independence of the determinant after transporting \(T\) through column stacking.

If \(T(\rho )=U\rho U^\dagger \) is a unitary channel, then \(\det T=1\) and hence \(|\det T|=1\).

Let \(d\geq 1\) and let \(T:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) be completely positive and trace-preserving. Then \(d'\geq 1\), and there are a normalized vector \(\varphi \in \mathbb {C}^{d'}\otimes \mathbb {C}^{d'}\) and a unitary \(U\) on \(\mathbb {C}^d\otimes \mathbb {C}^{d'}\otimes \mathbb {C}^{d'}\), a space of total dimension \(d(d')^2\), such that, for every \(\rho \in M_{d}(\mathbb {C})\),

Here the partial trace is over the first two tensor factors \(\mathbb {C}^d\otimes \mathbb {C}^{d'}\) and retains the final factor \(\mathbb {C}^{d'}\). This is [ Wol12 , Theorem 2.5, Equation (2.14) ] .

Every quantum channel \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) admits a Stinespring isometry \(V\) on an ancilla space \(\mathbb {C}^r\) such that

Every quantum channel \(T\) admits an isometric Stinespring dilation \(V\) such that \(T(\rho )=\operatorname{tr}_E(V\rho V^\dagger )\).

For \(D\geq 1\), every quantum channel \(T\) on \(\mathbb {C}^D\) admits an environment dimension \(r\geq 1\) and a unitary \(U\) on \(\mathbb {C}^D\otimes \mathbb {C}^r\) such that, for every matrix \(\rho \) on \(\mathbb {C}^D\),

Let \(\Phi \) be completely positive and trace preserving, let \(\sigma {\gt}0\), set \(\tau =\Phi (\sigma )\), and let \(P\) be the support projection of \(\tau \). Then every matrix \(X\) satisfies \(P\Phi (X)=\Phi (X)=\Phi (X)P\). If \(VV^\dagger =P\), then \(\Psi (X)=V^\dagger \Phi (X)V\) is completely positive and trace preserving.

For linear maps \(T_1,T_2:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\),

where \(T_1T_2=T_1\circ T_2\), so \(T_2\) acts first. This is [ Wol12 , Equation (6.22) ] .

Let \(T_1,T_2:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace-preserving. Then

For arbitrary linear maps, the equality \(|\det (T_1T_2)|=|\det T_1|\) holds exactly when

This is the multiplicative and determinant-bound part of the monotonicity corollary following [ Wol12 , Theorem 6.1 ] .

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a linear map and \(\tau \) the corresponding Choi–Jamiołkowski operator. Then

No positivity, trace preservation or unitality is assumed. This is [ Wol12 , Eq. (6.27) ] .

Let \(D{\gt}0\) and let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace-preserving. Then

This is [ Wol12 , Theorem 6.1(2) ] .

For a CPTP map \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\),

This is the CPTP specialization of [ Wol12 , Theorem 6.1(2) ] ; complete positivity excludes the genuinely transpose-type branch.

For any positive trace-preserving map \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\), \(\det T\) is real and belongs to \([-1,1]\); in particular, \(|\det T|\leq 1\). This is [ Wol12 , Theorem 6.1(1) ] .

Ordinary transposition on \(M_{D}(\mathbb {C})\) satisfies

Consequently it has determinant \(-1\) exactly when \(\lfloor D/2\rfloor \) is odd. For a positive trace-preserving \(T\) in positive dimension, \(\det T=-1\) exactly when that parity condition holds and \(T(A)=UA^{\mathsf T}U^\dagger \) for a unitary \(U\). In those odd-parity dimensions, \(\det T=1\) exactly for unitary conjugations. This is [ Wol12 , Theorem 6.1(3) ] .

Let \(D\geq 1\) and let \(T(X)=\sum _{i=0}^{r-1}K_iXK_i^\dagger \). Define the vector \(v_i\) by \((v_i)_{(a,b)}=D^{-1/2}K_i(a,b)\), without transposing the matrix indices. Then

The identity channel has Choi matrix \(|\Omega \rangle \! \langle \Omega |\).

The map \(T\) is completely positive, in the sense that it admits a Kraus representation \(T(X)=\sum _{j}K_jXK_j^\dagger \) with \(K_j\in M_{d'\times d}(\mathbb {C})\), if and only if \(\tau \geq 0\). This is the complete-positivity clause of [ Wol12 , Proposition 2.1 ] .

Let \(\theta (X)=X^T\) be matrix transposition on \(M_{D}(\mathbb {C})\), with \(D \ge 1\). Then

This is [ Wol12 , Equation 3.1 ] .

The Choi matrix \(\tau \) is Hermitian if and only if \(T(B^\dagger ) = T(B)^\dagger \) for all \(B \in M_{D}(\mathbb {C})\). This is the equal-dimension case of [ Wol12 , Proposition 2.1 ] .

For any finite Kraus-like families \(\{ A_i\} , \{ B_i\} \), the map \(T(X) = \sum _i A_i X B_i^\dagger \) satisfies \(4\, T = T_1 - T_2 + i\, T_3 - i\, T_4\) with each \(T_k\) a CP map given explicitly by a single-side Kraus sum of the combined families. The hypothesis is that \(T\) is presented in sandwich-sum form on a single matrix algebra \(M_{D}(\mathbb {C})\); Theorem 2.6.11 decomposes an arbitrary linear map \(M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\).

Let \(T_i:M_{d'}(\mathbb {C})\to M_{d}(\mathbb {C})\) be completely positive maps with \(T_1\leq T_2\). Suppose that, for possibly distinct ancilla dimensions \(r_i\), matrices \(V_i:\mathbb {C}^d\to \mathbb {C}^{d'}\otimes \mathbb {C}^{r_i}\) are supplied and satisfy

Then there is a contraction \(C:\mathbb {C}^{r_2}\to \mathbb {C}^{r_1}\) such that

If \(V_2\) is minimal, in the source sense \(r_2=\operatorname{rank}(\tau _2)\), then \(C\) is unique. This is [ Wol12 , Theorem 2.3 and Equation (2.13) ] .

Let \(T_1,T_2:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be completely positive. There exist an ancilla dimension \(m\), a Kraus family \(K\) with Stinespring matrix \(V=V_K\), and positive semidefinite operators \(P_1,P_2:\mathbb {C}^m\to \mathbb {C}^m\) with \(P_1+P_2=\mathbb {1}_m\) such that, for \(i=1,2\) and every \(A\in M_{D}(\mathbb {C})\),

Let \(D\geq 1\). A linear map \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) is completely positive (in the Kraus sense) if and only if its Choi matrix \(\tau \ge 0\). This is the \(d=d'\) specialization of [ Wol12 , Proposition 2.1 ] .

Let \(I\) be a nonempty finite set, and let \(T_i,T:M_{d'}(\mathbb {C})\to M_{d}(\mathbb {C})\) be completely positive linear maps such that \(\sum _{i\in I}T_i=T\). Suppose that a Stinespring representation of \(T\) is supplied by a linear map \(V:\mathbb {C}^d\to \mathbb {C}^{d'}\otimes \mathbb {C}^r\) satisfying

Then there are positive semidefinite operators \(P_i\in M_{r}(\mathbb {C})\) such that

This is the rectangular Heisenberg-picture statement of [ Wol12 , Theorem 2.4 ] . The nonempty-family condition is made explicit; see [ con26d ] .

Let \(d\geq 1\), let \(T:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) be a quantum channel, and let \(V:\mathbb {C}^d\to \mathbb {C}^{d'}\otimes \mathbb {C}^r\) be a supplied Stinespring matrix such that

Suppose \(n\geq 1\) and \(T=\sum _{i=0}^{n-1}T_i\), where every \(T_i:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) is completely positive. Then there is a POVM \(\{ P_i\} _{i=0}^{n-1}\) on \(\mathbb {C}^r\) such that, for every \(i\) and \(\rho \),

If \(k_i\) is the Kraus rank, equivalently the Choi rank, of \(T_i\), then \(k_i\leq \operatorname{rank}(P_i)\).

If two pure-state ensembles \(\{ \psi _i\} _{i \in \iota _1}\) and \(\{ \phi _j\} _{j \in \iota _2}\) induce the same pure-ensemble density operator and \(|\iota _2| \le |\iota _1|\), then there exists a tall isometric mixing matrix \(V \in \mathbb {C}^{\iota _1 \times \iota _2}\) with \(V^\dagger V = \mathbb {1}\) and \(\psi _i = \sum _j V_{ij} \phi _j\). The cardinality hypothesis is what makes \(V\) a tall isometry; the symmetric case \(|\iota _1| \le |\iota _2|\) follows by swapping the roles of the ensembles.

Every quantum channel \(E : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) admits a finite family \((K_i)_i\) in \(M_{D}(\mathbb {C})\) such that, for every \(X \in M_{D}(\mathbb {C})\),

and

For every POVM \(\{ E_i\} \) on \(\mathbb {C}^D\) there exist a dilation dimension \(r\), an isometry \(V:\mathbb {C}^D\to \mathbb {C}^D\otimes \mathbb {C}^r\), and a projective measurement \(\{ P_i\} \) on the dilation satisfying \(E_i=V^\dagger P_iV\).

Let \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) be a completely positive map whose Choi matrix is positive-definite (equivalently, \(T\) has full Kraus rank). Then there exist SL-filterings \(\Phi _{1},\Phi _{2}\) such that \(\Phi _{2}\circ T\circ \Phi _{1}\) is doubly-stochastic.

This is the equal-dimension (square) case \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) of [ Wol12 , Proposition 2.9 ] ; the rectangular form with independent dimensions is Theorem 3.15.1.24.

Let \(T : M_{d_1}(\mathbb {C}) \to M_{d_2}(\mathbb {C})\) be a completely positive map whose Choi matrix is positive-definite (equivalently, \(T\) has full Kraus rank). Then there exist SL-filterings \(\Phi _{1}\) on \(M_{d_1}(\mathbb {C})\) and \(\Phi _{2}\) on \(M_{d_2}(\mathbb {C})\) such that \(\Phi _{2}\circ T\circ \Phi _{1}\) is doubly-stochastic. This is [ Wol12 , Proposition 2.9 ] at the source’s generality.

Let \(\tau \in \mathcal{B}(\mathbb {C}^{d_2}\otimes \mathbb {C}^{d_1})\) be positive-definite. Then there exist \(S_i\in \mathrm{SL}(d_i,\mathbb {C})\) which attain the infimum

In particular, for

both partial traces are proportional to the respective identity matrices:

for some \(\kappa _1,\kappa _2\in \mathbb {C}\). This is [ Wol12 , Proposition 2.8, lines 894–919 ] , with the two dimensions kept independent.

Every completely positive map admits an ancilla dimension \(r\), Kraus operators \(\{ K_j\} _{j=0}^{r-1}\), and the concrete representation \(\pi (A)=A\otimes \mathbb {1}_r\) such that \(E(A)=V^\dagger \pi (A)V\).

Every quantum channel admits a Stinespring dilation whose matrix \(V\) is an isometry, equivalently \(V^\dagger V=\mathbb {1}\).

Let \(S\) be a finite set and let \((x_i)_{i\in S}\) be real numbers. Then

if and only if \(x_i=x_j\) for every \(i,j\in S\).

Let \((v_i)_{i\in I}\) be a finite family in \(\mathbb {C}^n\). Then

For complex-linear maps \(E:\mathbb C^{I\times I}\to \mathbb C^{J\times J}\) and \(F:\mathbb C^{J\times J}\to \mathbb C^{K\times K}\),

Let \(\tau \geq 0\), and let \(P\) be its support projection. Equal positive-semidefinite matrices have equal support projections, and taking the positive square root leaves \(P\) unchanged. If \(V:\mathbb C^K\to \mathbb C^J\) satisfies \(VV^\dagger =P\), then \(\sqrt{V^\dagger \tau V}=V^\dagger \sqrt\tau V\). Moreover, both \(P\tau ^{-1/2}_{\operatorname {supp}} =\tau ^{-1/2}_{\operatorname {supp}}\) and \(\tau ^{-1/2}_{\operatorname {supp}}P =\tau ^{-1/2}_{\operatorname {supp}}\). If, in addition, \(V^\dagger V=\mathbb {1}\), then

Independently, if \(W:\mathbb C^L\to \mathbb C^J\) satisfies \(W^\dagger W=\mathbb {1}\), then \(\| WZW^\dagger \| _2=\| Z\| _2\) for every \(Z\in \mathbb C^{L\times L}\).

If \(\rho \in M_{D}(\mathbb {C})\) is positive semidefinite, then each outcome probability \(p_i(\rho )=\operatorname{tr}(\Phi _i(\rho ))\) is a non-negative real number.

For every state \(\rho \in M_{D}(\mathbb {C})\), the outcome probabilities \(p_i(\rho ):=\operatorname{tr}(\Phi _i(\rho ))\) sum to \(\operatorname{tr}(\rho )\).

The total map \(\sum _i\Phi _i\) of an instrument is a quantum channel.

If \(\mathcal S:M_{D}(\mathbb {C})\to M_{D'}(\mathbb {C})\) is trace-preserving and completely positive, then its trace-pairing adjoint satisfies

This is the Schrödinger–Heisenberg duality of [ Wol12 , Section 1.2 ] .

The explicit isometry \(V=\sum _iM_i\otimes |i\rangle \) and projectors \(P_i=\mathbb {1}_D\otimes |i\rangle \! \langle i|\) satisfy the defining axioms of a Naimark dilation.

If two Kraus families define the same CPM, then \(\sum _\alpha B_\alpha ^\dagger B_\alpha = \sum _j A_j^\dagger A_j\).

If two Kraus families satisfy \(\sum _\alpha B_\alpha X B_\alpha ^\dagger = \sum _j A_j X A_j^\dagger \) for all \(X\), then \(\sum _\alpha B_\alpha ^\dagger Y B_\alpha = \sum _j A_j^\dagger Y A_j\) for all \(Y\).

If \(\tau _{E^m}{\gt}0\) for every sufficiently large \(m\), then the Kraus family has eventually full vector spread.

If \(\mathcal S_m=M_{D}(\mathbb {C})\) for every sufficiently large \(m\), then the Kraus family has eventually full vector spread.

Let \(\{ B_\alpha \} _{\alpha \in \iota _1}\) and \(\{ A_j\} _{j \in \iota _2}\) be finite Kraus families with \(|\iota _2| \le |\iota _1|\). Then the following are equivalent:

1. \(\sum _\alpha B_\alpha X B_\alpha ^\dagger = \sum _j A_j X A_j^\dagger \) for all \(X \in M_{D}(\mathbb {C})\).

2. There exists a matrix \(V = (V_{\alpha j})\) with \(V^\dagger V = \mathbb {1}\) such that \(B_\alpha = \sum _j V_{\alpha j} A_j\) for every \(\alpha \).

Let \(\tau _{E^m}\) be the normalized Choi matrix of the \(m\)-fold iterate of \(E\). Then

Let \(I\) be finite, let \((K_i)_{i\in I}\) be a family in \(M_{D}(\mathbb {C})\), and set \(E(X)=\sum _i K_iXK_i^\dagger \). For every non-negative integer \(N\),

For \(N=0\), the product is the identity matrix.

Suppose that \(K\) is trace preserving and that its Kraus map \(E\) is irreducible with peripheral spectrum \(\{ 1\} \). This is the implication from item 1 to items 3 and 4 of [ Wol12 , Theorem 6.8(1,3,4) ] . Then, for every sufficiently large \(m\),

Moreover,

Let \(T(X)=\sum _{j=1}^{r}K_jXK_j^\dagger \) with \(K_j\in M_{d'\times d}(\mathbb {C})\). Then \(T\) is trace preserving if and only if \(\sum _j K_j^\dagger K_j=\mathbb {1}_d\), and unital if and only if \(\sum _j K_jK_j^\dagger =\mathbb {1}_{d'}\). This is item 1 of [ Wol12 , Theorem 2.1 ] .

The minimal number of Kraus operators of a completely positive map \(T:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) is \(r=\operatorname {rank}(\tau )\leq dd'\). This is item 2 of [ Wol12 , Theorem 2.1 ] .

If \(\sum _\alpha B_\alpha X B_\alpha ^\dagger = \sum _j A_j X A_j^\dagger \) for all \(X\) and \(r_2 \le r_1\) (where \(\{ B_\alpha \} _{\alpha =0}^{r_1-1}\) is the larger family and \(\{ A_j\} _{j=0}^{r_2-1}\) the smaller), then there exists a rectangular isometry \(V\) (\(r_1 \times r_2\), \(V^\dagger V = \mathbb {1}_{r_2}\)) such that \(B_\alpha = \sum _j V_{\alpha j}\, A_j\).

Variant of Theorem 3.1.12 with general finite index sets \(\iota _1, \iota _2\) in place of standard index sets of cardinalities \(r_1\) and \(r_2\).

Let \(W\) be an isometry between two Kraus index spaces, so \(W^\dagger W = \mathbb {1}\). If \(K_j = \sum _\ell W_{j\ell } \widetilde K_\ell \), then the Kraus families \(\{ K_j\} \) and \(\{ \widetilde K_\ell \} \) define the same completely positive map.

Let \(U\) be a unitary \(r \times r\) matrix (\(U^\dagger U = \mathbb {1}\)) and suppose \(K_j = \sum _\ell U_{j\ell } \tilde{K}_\ell \). Then \(\{ K_j\} \) and \(\{ \tilde{K}_\ell \} \) define the same Kraus map: for every \(X \in M_{D}(\mathbb {C})\),

If \(T(X) = \sum _i K_i X K_i^\dagger \) is trace-preserving, then \(\sum _i K_i^\dagger K_i = \mathbb {1}\).

If \(T(X) = \sum _i K_i X K_i^\dagger \) satisfies \(T(\mathbb {1}) = \mathbb {1}\), then \(\sum _i K_i K_i^\dagger = \mathbb {1}\).

If \(\sum _i K_i^\dagger K_i = \mathbb {1}\), then \(T(X) = \sum _i K_i X K_i^\dagger \) is trace-preserving.

Let \(E:\mathbb C^{I\times I}\to \mathbb C^{J\times J}\) be a completely positive map with a Kraus representation, and let \(E^*\) be its trace-pairing adjoint. Then the Hilbert-space adjoint of \(\widehat E\) is

This is the adjoint identity used in the direct exponent-two specialization of [ Bei13 , Theorem 6, Equation (18) ] .

Let \(\{ K_j\} _{j=0}^{r-1}\) and \(\{ K'_j\} _{j=0}^{r-1}\) be two Hilbert–Schmidt orthonormal Kraus families, and suppose \(K_j = \sum _{\ell =0}^{r-1} U_{j\ell }\, K'_\ell \). Then the transition matrix \(U\) is unitary: \(U^\dagger U = \mathbb {1}_r\).

Let \(\{ B_\alpha \} _{\alpha \in \iota }\) and \(\{ A_j\} _{j \in \iota }\) be two Kraus families with the same finite index set. Then the following are equivalent:

1. \(\sum _\alpha B_\alpha X B_\alpha ^\dagger = \sum _j A_j X A_j^\dagger \) for all \(X \in M_{D}(\mathbb {C})\).

2. There exists a unitary matrix \(U = (U_{\alpha j})\) such that \(B_\alpha = \sum _j U_{\alpha j} A_j\) for every \(\alpha \).

If \(\mathcal S_m=M_{D}(\mathbb {C})\), then \(H_m(K,\psi )=\mathbb {C}^D\) for every nonzero \(\psi \in \mathbb {C}^D\).

Let \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) be linear. If \(T(|v\rangle \! \langle v|) = |v\rangle \! \langle v|\) for every \(v \in \mathbb {C}^D\), then \(T = \operatorname{id}\). In particular, a quantum channel preserving every pure state equals the identity.

Let \(E\) be a complex vector space and let \(T,S:M_{D}(\mathbb {C})\to E\) be complex-linear maps. If

for every \(v\in \mathbb {C}^D\), then \(T=S\).

If \(\{ \sigma _i\} \) is trace-self-dual, then every linear map \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) admits the expansion

If \(V^\dagger V=\mathbb {1}\), then \((V\otimes \mathbb {1}_R)^\dagger (V\otimes \mathbb {1}_R)=\mathbb {1}\).

Put \(q_i=p_i^2\) for the Pauli coefficients in [ VV02 , Theorem 8, Eq. (18) ] , specialized to \(A=B=\mathbb {1}\). Direct conjugation of the Pauli basis gives

Thus, in Wolf’s Pauli-transfer convention, the four candidate Bell weights are

Verstraete–Verschelde instead print \(1-s_1-s_2-s_3\geq 0\) in Theorem 8. Their preceding Equation (16) uses \(R_\Phi \) as the Bloch transfer matrix, so the identity channel has \(R_\Phi =\operatorname {diag}(1,1,1,1)\) and violates the printed inequality. The all-minus sign is therefore inconsistent with their own convention; it is not used in this construction and no silent change of the parameter \(s_3\) is made. They sum to one. If every \(q_i\) is nonnegative, put \(p_i=\sqrt{q_i}\). Then the source family \(\{ p_0\sigma _0,p_1\sigma _1,p_2\sigma _2,p_3\sigma _3\} \) from [ VV02 , Theorem 8, Eq. (18) ] , specialized to \(A=B=\mathbb {1}\), defines a bistochastic channel whose Pauli transfer matrix is \(\operatorname {diag}(1,s_1,s_2,s_3)\). Its Choi rank is exactly \(\# \{ i\mid q_i\ne 0\} \), so all diagonal rank-drop boundaries are retained. Simultaneous nonnegativity of the four weights is equivalent to nonnegativity of their four displayed numerators. This construction shows that those inequalities suffice for the displayed Pauli family to be a channel; it makes no converse assertion for a separately specified diagonal map. Under the ordered convention \(1\geq s_1\geq s_2\geq \lvert s_3\rvert \), Wolf’s \(s_1+s_2\leq 1+s_3\) from Equation (2.40) implies all four.

For every \(x\in [0,1]\), the three Kraus operators displayed in [ VV02 , Theorem 8, Eq. (19) ] define the non-diagonal representative

Its Choi rank, equivalently its minimal Kraus rank, is \(3\) when \(x{\lt}1\) and \(2\) when \(x=1\). The third normal form in [ VV02 , Theorem 8, Eq. (17) ] is the singular trace-to-state channel

it has \(\Delta =0\), \(v=(0,0,1)\), and Choi/Kraus rank \(2\).

These are constructions and rank calculations for the representatives in [ VV02 , Theorem 8, Eqs. (17)–(19) ] . The diagonal rank formula is the direct Bell-family calculation from Equation (18). The non-diagonal and singular rank statements agree with cases 2 and 3 of [ WC08 , Theorem 18 ] . This does not prove the Lorentz-orbit classification or derive that the non-diagonal parameter must lie in \([0,1]\). The normalized Choi convention is \(\tau =\frac14\sum _{ij}\widehat T_{ij}\sigma _i\otimes \sigma _j^{\mathsf T}\); hence its raw Pauli correlation matrix satisfies \(R_{\mathrm{raw}}(\tau )=\widehat T\operatorname {diag}(1,1,-1,1)\) because \(\sigma _2^{\mathsf T}=-\sigma _2\). Verstraete–Verschelde define \(R_\Phi \) only after taking the first-factor partial transpose of their dual state and then use it as the Bloch transfer matrix. Thus \(R_\Phi \) corresponds to \(\widehat T\), not to \(R_{\mathrm{raw}}(\tau )\); the displayed sign matrix is the explicit bridge. In particular, the partial-transpose sign has already been absorbed before their Theorem 8 parameters are introduced and cannot account for its printed all-minus constraint.

For every qubit channel \(T : M_{2}(\mathbb {C}) \to M_{2}(\mathbb {C})\), there exist invertible completely positive maps \(\Phi _{1},\Phi _{2}\), both of Kraus rank one, such that the filtered channel \(T'=\Phi _{2}\circ T\circ \Phi _{1}\) is in one of the three Lorentz normal forms: diagonal, non-diagonal, or singular. These general filters include scalar freedom and are not restricted to determinant-one \(\operatorname{SL}(2,\mathbb {C})\) filterings. Indeed, writing \(X_i=c_iS_i\) with \(S_i\in \operatorname{SL}(2,\mathbb {C})\) separates the Lorentz action from the positive map scalar \(|c_1c_2|^2\); only the latter can normalize the resulting representative to a channel. A proof still requires this scalar normalization and the classification of Lorentz orbits. The displayed diagonal, non-diagonal, and singular representatives and their ranks are already supplied by Theorem 3.15.1.28.

For every \(\rho \in M_D(\mathbb C)\),

Let \(V:\mathbb {C}^D\to \mathbb {C}^D\otimes \mathbb {C}^n\) satisfy \(V^\dagger P_iV=E_i\) for the canonical projectors \(P_i=\mathbb {1}_D\otimes |i\rangle \! \langle i|\). Then there exists an isometry \(W\) on the dilated space such that \(V=WV_0\), where \(V_0\) is the canonical Naimark isometry of \(\{ E_i\} \).

The Naimark isometry satisfies \(V^\dagger V=\mathbb {1}_D\).

The Naimark projectors satisfy

Every POVM \(\{ E_i\} _{i=0}^{n-1}\) arises as a projective measurement on a dilation: for the isometry \(V\) and projectors \(P_i\) above, \(E_i=V^\dagger P_iV\).

Under the hypotheses of Theorem 3.3.2 there is, for every \(\alpha \), a constant \(c_\alpha \geq 0\) with \(\operatorname{tr}[T_\alpha (\rho )]=c_\alpha \) for every \(\rho \) of unit trace. The probability of the outcome \(\alpha \) therefore does not depend on the input, so no information is gained.

Under the hypotheses of Theorem 3.3.2 the constants satisfy \(c_\alpha \geq 0\) and \(\sum _\alpha c_\alpha =1\).

Let \(d\geq 1\) and let \(\{ T_\alpha :M_{d}(\mathbb {C})\to M_{d}(\mathbb {C})\} \) be a finite family of completely positive maps with \(\sum _\alpha T_\alpha =\operatorname{id}\). Then for every \(\alpha \) there is a constant \(c_\alpha \geq 0\) with \(T_\alpha =c_\alpha \operatorname{id}\).

Let \(T:M_{2}(\mathbb {C})\to M_{2}(\mathbb {C})\) be Hermiticity preserving, and let \(T'\) be its Pauli-block diagonal truncation. If \(\widehat T\) is written as in 62, then

Equivalently, entrywise,

If \(T\) is positive, then \(T'\) is positive. If \(T\) is completely positive, then \(T'\) is completely positive. These are precisely the forward implications proved in [ Wol12 , Proposition 2.10, Section 2.4 ] , following Eq. (2.39); the local proof is at Notes/WolfNoteTexSource/ch02_representations.tex, lines 992–998.

For every \(x\in \mathbb {R}^4\),

Let \(T:M_2(\mathbb {C})\to M_2(\mathbb {C})\) be complex linear and let \(X_1,X_2\in \operatorname{SL}(2,\mathbb {C})\). If \(L_i=L(X_i)\) and \(L_{i,\mathbb {C}}\) denotes its complex scalar extension, then

If \(T\) preserves Hermiticity, then \(\widehat T\) has real entries, and Equation 101 is the real identity

in the exact order of [ Wol12 , Eq. (2.43) ] .

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace-preserving, with \(D{\gt}0\). There is a strictly increasing sequence of positive integers \(n_i\) such that \(T^{n_i}\to T_\phi \) pointwise and in operator norm. This is [ Wol12 , Proposition 6.3(i) ] .

For any \(A, B, X \in M_{D}(\mathbb {C})\),

Each of the four summands on the right is a single-Kraus CP map, so every sesquilinear sandwich decomposes into a signed complex linear combination of CP maps.

The inverse of a bijective trace-preserving linear map is trace preserving. If \(T\) is positive, then \(T^{-1}\) is positive exactly when \(T\) maps the positive-semidefinite cone onto itself. If both maps are positive and trace preserving, then \(|\det T|=1\).

For a unitary \(U\), the inverse of \(A\mapsto UAU^\dagger \) is conjugation by \(U^{-1}\). The inverse of \(A\mapsto UA^{\mathsf T}U^\dagger \) is ordinary transposition after conjugation by \(U^{-1}\). Both inverse maps are positive; the reversed composition order in the transpose branch is essential.

Let \(D{\gt}0\), and let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive and trace-preserving. If \(T(X)=\mu X\) for some nonzero \(X\), then \(|\mu |\leq 1\). This is the unit-disk conclusion of [ Wol12 , Proposition 6.1 ] .

Under the cardinality hypothesis \(|\iota _2| \le |\iota _1|\), two pure-state ensembles induce the same density operator iff they are related by a tall isometric mixing matrix.

Two ensembles \(\{ \psi _j\} \) and \(\{ \widetilde\psi _\ell \} \) of not necessarily normalized vectors satisfy

iff there is a unitary \(U\) with \(\psi _j = \sum _\ell U_{j\ell }\widetilde\psi _\ell \), where both families are first padded with zero vectors onto one common index set. No relation between the two cardinalities is assumed. Two paddings are recorded: onto the disjoint union \(\iota _1 \sqcup \iota _2\) of the two index sets, and, for ensembles of \(m\) and \(n\) vectors, onto \(\{ k : k {\lt} \max (m,n)\} \).

Let \(\{ \Psi _k\} _{k \in \iota }\) and \(\{ \Phi _k\} _{k \in \iota }\) be ensembles on one common finite index set whose density operators agree with those of \(\{ \psi _i\} _{i \in \iota _1}\) and \(\{ \phi _j\} _{j \in \iota _2}\) respectively. Then

iff there is a unitary \(U \in \mathbb {C}^{\iota \times \iota }\) with \(\Psi _k = \sum _\ell U_{k\ell }\Phi _\ell \).

If two pure-state ensembles \(\{ \psi _i\} _{i \in \iota _1}\) and \(\{ \phi _j\} _{j \in \iota _2}\) are related by an isometric mixing matrix \(V \in \mathbb {C}^{\iota _1 \times \iota _2}\) with \(V^\dagger V = \mathbb {1}\) and \(\psi _i = \sum _j V_{ij} \phi _j\), then they induce the same pure-ensemble density operator. This is the sufficient direction of the Hughston–Jozsa–Wootters theorem; the converse is Theorem 3.6.7.

Let \(\rho \in M_{d_A}(\mathbb {C})\) be a density operator with purification \(\psi \in \mathbb {C}^{d_A}\otimes \mathbb {C}^{d_B}\). For every convex decomposition \(\rho =\sum _i\lambda _i\rho _i\) there is an instrument \(\{ T_i:M_{d_B}(\mathbb {C})\to M_{d_B}(\mathbb {C})\} \) acting on Bob’s system such that

for every \(i\). This is [ Wol12 , Proposition (Quantum steering) ] .

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be complex-linear. Suppose that for every \(v\in \mathbb {C}^D\) there is a scalar \(c_v\in \mathbb {C}\) such that \(T(|v\rangle \! \langle v|)=c_v|v\rangle \! \langle v|\). Then there is a scalar \(c\in \mathbb {C}\) such that \(T=c\, \operatorname{id}\).

For every complex-linear map \(\mathcal L:M_{d}(\mathbb {C})\to M_{e}(\mathbb {C})\),

For every complex-linear map \(\mathcal L:M_{d}(\mathbb {C})\to M_{e}(\mathbb {C})\),

If \(\mathcal L:M_{d}(\mathbb {C})\to M_{e}(\mathbb {C})\) is a Hilbert–Schmidt contraction, then

Let \(T:M_{d}(\mathbb {C})\to M_{d}(\mathbb {C})\) be completely positive and let \((\sigma _\alpha )\) be a Hilbert–Schmidt orthonormal basis of \(M_{d}(\mathbb {C})\). The following are equivalent.

1. \(T=T^*\).

2. \(\widehat T=\widehat T^\dagger \), the transfer matrix being taken in the family \((\sigma _\alpha )\) on both sides.

3. \(T\) admits a family of Hermitian Kraus operators.

This is [ Wol12 , Proposition 2.6 ] .

Let \(\{ |\psi _i\rangle \! \langle \psi _i|\} _{i=0}^{n-1}\) be a rank-one POVM on \(\mathbb {C}^d\) with \(n\) outcomes, i.e. \(\sum _i|\psi _i\rangle \! \langle \psi _i|=\mathbb {1}_d\). Then necessarily \(d\le n\), and there exists an orthonormal basis \(\{ \phi _i\} _{i=0}^{n-1}\) of \(\mathbb {C}^n\) such that each \(\psi _i\) is the restriction of \(\phi _i\) to the first \(d\) coordinates. Concretely, the rows of a unitary \(U\in U(n)\) give the \(\phi _i\), and \(\psi _i\) is recovered as \(\psi _i(j)=U_{i,\, j}\) for \(j=0,\dots ,d-1\).

Unlike 3.12.12, whose ambient Hilbert space is \(\mathbb {C}^D\otimes \mathbb {C}^n\) of dimension \(D\cdot n\), this theorem yields the sharp \(n\)-dimensional ambient space asserted by [ Wol12 , Theorem “Neumark’s theorem” ] .

The corresponding corollary starts from a positive operator-valued measure with an explicit rank-one decomposition.

If \(Q\in M_{D}(\mathbb {C})\) is Hermitian, then

A Hermitian matrix \(Q\in M_{D}(\mathbb {C})\) satisfies equality in (149) if and only if \(Q=r\mathbb {1}\) for some \(r\in \mathbb {R}\).

Let \((P_i)_i\) be a symmetric informationally complete family. Every \(\rho \in M_{d}(\mathbb {C})\) satisfies

Suppose, in addition, that \(d\ge 2\). Any family attaining equality in (153) is linearly independent.

Local fix (one-dimensional equality families): The corresponding assertion printed in [ Wol12 , Chapter 2, Proposition “SIC POVMs”, lines 816–823 ] is false when \(d=1{\lt}n\): every \(P_i\) then equals \([1]\). This deviation is recorded in the QICLean paper-gap note [ con26g ] .

For every \(\rho \in M_{d}(\mathbb {C})\),

Equivalently, the left-hand side is the quantum channel with Kraus operators \(K_i=P_i/\sqrt d\).

For a symmetric informationally complete family \((P_i)_i\), the operators

are, respectively, the effects of a POVM and a trace-preserving Kraus family.

The \(d^2\) projectors in a symmetric informationally complete family form a basis of \(M_{d}(\mathbb {C})\).

Suppose that \(2\le n\), \(1\le d\le n\), and that \(P_1,\ldots ,P_n\in M_{d}(\mathbb {C})\) are positive semidefinite matrices satisfying \(\operatorname{tr}(P_i^2)=1\). Then

Under the hypotheses of Theorem 3.22.1, equality holds in (153) if and only if

A singleton family whose sole matrix \(P\) satisfies \(\operatorname{tr}(P^2)=1\) is linearly independent.

Suppose that \(P\in M_{D}(\mathbb {C})\) is positive semidefinite and \(\operatorname{tr}(P^2)=1\). Then \(\operatorname{tr}(P)=1\) if and only if \(P\) is a rank-one orthogonal projection.

If \(P\in M_{D}(\mathbb {C})\) is positive semidefinite and \(\operatorname{tr}(P^2)=1\), then \(\operatorname{tr}(P)\ge 1\).

Let \(n\in \mathbb {R}^3\) be a unit vector. Write \(n\cdot \sigma \) for its Pauli contraction, \(n\cdot B\) for the boost generator with time–space blocks \(n\) and \(n^{\mathsf T}\), and \(n\cdot R\) for Wolf’s rotation generator \(R_i=\sum _{j,k}\varepsilon _{ijk}\lvert k\rangle \langle j\rvert \). For every \(t\in \mathbb {R}\),

If \(u(n,t)=(\cosh (t),\sinh (t)n)\), then \(B_{u(n,t)}=\exp (t\, n\cdot B)\), and the spinor map satisfies

The positive sign in the second Lorentz exponential is the correction to the sign printed in [ Wol12 , Eq. (2.44) ] ; with the displayed Pauli matrices and column-vector convention, the printed negative sign rotates in the opposite direction.

The homomorphism \(\Lambda \) in (97) is surjective. Its fibres contain exactly two points: for all \(X,Y\in \operatorname{SL}(2,\mathbb {C})\),

Thus \(\operatorname{SL}(2,\mathbb {C})\) is a double cover of \(\mathrm{SO}^+(1,3)\), as stated after [ Wol12 , Eq. (2.42) ] .

The assignment \(X\mapsto L(X)\) is multiplicative, and for every \(X\in \operatorname{SL}(2,\mathbb {C})\),

Hence \(L(X)\in \mathrm{SO}^+(1,3)\). It preserves both the Minkowski form and the closed future cone, and \(L(-X)=L(X)\).

Let \(K_j:\mathbb {C}^{d}\to \mathbb {C}^{d'}\), \(j=0,\dots ,r-1\), be an arbitrary family of matrices and let \(V=\sum _j K_j\otimes |j\rangle \). Then, for every observable \(A\in M_{d'}(\mathbb {C})\) on the output space,

an identity in \(M_{d}(\mathbb {C})\) on the input space, valid for arbitrary \(K\) independently of any channel. When the \(K_j\) are Kraus operators of a map \(T:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\), the right-hand side is the dual (Heisenberg-picture) map \(T^*(A)\) of 2.5.1; the square case is \(d=d'\).

Let \(T:M_{d}(\mathbb {C})\to M_{d'}(\mathbb {C})\) be completely positive with Choi matrix \(\tau \) and dual \(T^*\). Then for every \(r\geq \operatorname{rank}(\tau )\) there is a \(V:\mathbb {C}^{d}\to \mathbb {C}^{d'}\otimes \mathbb {C}^{r}\) such that, for all \(A\in M_{d'}(\mathbb {C})\),

and \(V\) is an isometry, \(V^\dagger V=\mathbb {1}_d\), if and only if \(T\) is trace preserving. In the Schrödinger picture one may choose the same type of dilation so that

for every \(\rho \in M_{d}(\mathbb {C})\). In particular \(r=\operatorname{rank}(\tau )\) is an admissible ancilla dimension; by Lemma 3.10.14 no smaller ancilla dimension is admissible, and a dilation with \(r=\operatorname{rank}(\tau )\) is called minimal. This is [ Wol12 , Theorem 2.2 ] .

The Kraus map \(T(\rho )=\sum _jK_j\rho K_j^\dagger \) equals the partial trace over the dilation space:

This is the Schrödinger-picture Stinespring representation.

The Stinespring matrix satisfies

In the square specialization \(d_{\mathrm{in}}=d_{\mathrm{out}}=D\), one has \(V^\dagger V=\mathbb {1}_D\) if and only if \(\sum _j K_j^\dagger K_j=\mathbb {1}_D\). In particular, \(V\) is an isometry precisely when the Kraus map is trace-preserving.

Every invertible complex square matrix \(M \in M_{D}(\mathbb {C})\) admits a singular value decomposition \(M=U\operatorname{diag}(\sigma )V^{\dagger }\), with \(U,V \in \mathcal{U}(D)\) unitary and \(\sigma _i{\gt}0\) for all \(i\).

Every positive semidefinite matrix \(M \in M_{D}(\mathbb {C})\) admits a decomposition \(M=U\operatorname{diag}(\sigma )U^{\dagger }\), where \(U \in \mathcal{U}(D)\) is unitary and \(\sigma : \{ 0,\ldots ,D-1\} \to \mathbb {R}_{\ge 0}\) has non-negative entries.

If \(D\geq 1\) and \(T\) is trace-preserving, then \(\operatorname{tr}(\tau ) = 1\). This is the equal-dimension case of [ Wol12 , Proposition 2.1 ] .

For \(M \in M_{D}(\mathbb {C})\), \(\operatorname{tr}(M N) = 0\) for all \(N \in M_{D}(\mathbb {C})\) if and only if \(M = 0\).

For every linear map \(E : M_{D}(\mathbb {C}) \to M_{D'}(\mathbb {C})\), every \(\rho \in M_{D'}(\mathbb {C})\), and every \(X \in M_{D}(\mathbb {C})\), one has

If \(T\) is trace-preserving, then \(\operatorname{tr}_A(\tau ) = \frac{1}{D} \mathbb {1}_D\). This is the equal-dimension implication in [ Wol12 , Proposition 2.1 ] .

For linear maps \(S,T:M_D(\mathbb C)\to M_D(\mathbb C)\),

For a linear map \(S:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\), let \(\lVert S\rVert _{2\to 2}\) be its operator norm for the Hilbert–Schmidt norm. Then

where the norm on the transfer matrix is its largest-singular-value norm. The transfer representation also preserves subtraction and powers.

If \(T(X)=\sum _j K_jXK_j^\dagger \), then the column-stacking convention gives

Wolf’s Equation (2.21) prints the two Kronecker factors in the opposite order; the difference is the simultaneous product-index permutation induced by the vectorization convention.

For every nonnegative integer \(N\),

where the second trace is the operator trace of the endomorphism of \(M_D(\mathbb C)\).

If \(T:M_D(\mathbb C)\to M_D(\mathbb C)\) preserves trace and \((\Phi \rvert =\operatorname {vec}(\mathbb {1})^{\mathsf T}\), then

Every invertible transfer matrix \(\widehat{T}\) of a linear super-operator \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) admits a singular value decomposition \(\widehat{T}=U\operatorname{diag}(\sigma )V^{\dagger }\) with \(U,V\) unitary on \(\mathbb {C}^{D}\otimes \mathbb {C}^{D}\) and \(\sigma _i{\gt}0\).

The matrix transposition map \(\theta (X)=X^T\) on \(M_{D}(\mathbb {C})\) is positive and trace-preserving.

Let \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a Kraus map satisfying \(E(\mathbb {1})=\mathbb {1}\). Every eigenvalue \(\mu \) of \(E\) satisfies

This is the unital Kraus specialization of [ Wol12 , Proposition 6.1 ] .

Let \(E:\mathbb C^{I\times I}\to \mathbb C^{I\times I}\) be a completely positive map with \(E(\mathbb {1})=\mathbb {1}\). Then the spectral radius of its Frobenius transport is at most one.

Let \(I\) and \(J\) be finite index sets, let \(\Phi :\mathbb C^{I\times I}\to \mathbb C^{J\times J}\) be completely positive and trace preserving, and let \(\sigma \) and \(\tau \) be positive definite matrices satisfying \(\Phi (\sigma )=\tau \). Then, for every \(X\),

Equivalently, \(L\) is a Hilbert–Schmidt contraction. This is the full-support exponent-two specialization of [ Bei13 , Theorem 6, Equation (18) ] ; it does not assert the support-compressed form for a singular output weight.

Let \(\Phi :\mathbb C^{I\times I}\to \mathbb C^{J\times J}\) be completely positive and trace preserving, let \(\sigma {\gt}0\), and set \(\tau =\Phi (\sigma )\). Then, for every \(X\),

Equivalently, \(L_{\operatorname {supp}}\) is a Hilbert–Schmidt contraction. This is the exponent-two specialization of [ Bei13 , Theorem 6, Equation (18) ] , including singular output weights.

Let \(T:M_{d}(\mathbb {C})\to M_{d}(\mathbb {C})\) be a completely positive linear map with Kraus rank at most two. Then

No trace-preservation hypothesis is assumed. This is the proposition “Positive determinant for small Kraus rank” in [ Wol12 , Chapter 6, Equation (6.26) ] .

Let \(D{\gt}0\), and let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive, trace preserving, and bijective. Then \(T^{-1}\) is positive if and only if there is a unitary \(U\) for which

This is the corollary “Positive invertible maps” following [ Wol12 , Theorem 6.1 ] .

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a complex-linear map. The following are equivalent:

1. \(T\) maps the cone of positive semidefinite matrices onto itself;

2. \(T\) is positive and preserves the rank of Hermitian matrices;

3. there is an invertible \(Y\in M_{D}(\mathbb {C})\) such that either \(T(X)=YXY^\dagger \) for every \(X\), or \(T(X)=YX^{\mathsf T}Y^\dagger \) for every \(X\).

This is [ Wol12 , Proposition 3.6 ] .