Dependencies

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 ] .

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 Fitting decomposition of a linear endomorphism \(f:V\to V\) on a finite-dimensional vector space over an algebraically closed field consists of:

1. \(f\) is nilpotent on the generalized \(0\)-eigenspace,

2. \(f\) is invertible on each generalized \(\mu \)-eigenspace for \(\mu \neq 0\),

3. the generalized eigenspaces span \(V\),

4. the generalized eigenspaces are linearly independent.

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.

Let \(K=\{ K^i\} _{i=0}^{d-1}\) be a finite family of \(D\times D\) matrices. For a word \(w=(i_1,\ldots ,i_n)\), its word evaluation is

with the empty word evaluated as the identity matrix.

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.

Given operators \(\{ K_i\} _{i=0}^{d-1}\) with \(K_i \in M_{D}(\mathbb {C})\), the Kraus map is

A linear map is completely positive (Definition 2.1.6) if and only if it can be written in this form, as in (1).

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 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.

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 \(X\).

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

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

At a threshold \(n\), the vector-spread clause says that, for every \(m\geq n\) and every nonzero \(\psi \in \mathbb {C}^D\), \(K_m|\psi \rangle =\mathbb {C}^D\). At a threshold \(q\), the word-span clause says \(K_m=M_{D}(\mathbb {C})\) for every \(m\geq q\), while the Choi clause says \(\tau _m{\gt}0\) for every \(m\geq q\). These are the direct quantified clauses in [ Wol12 , Theorem 6.8(2–4) ] .

For a finite Kraus family on \(M_{d}(\mathbb {C})\), define \(k=\dim K_1\). When the family has eventually full word spans, define

For a primitive quantum channel the set in the second definition is nonempty by Theorem 8.10.3.

Suppose \(K^{i_0}\varphi =\mu \varphi \) with \(\mu \neq 0\). For every \(n\), the span of \(K^w\varphi \) over words with \(|w|\leq n\) equals \(H_n(K,\varphi )\).

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

This is the matrix-factorization input used in  [ PGVWC07 , Lemma 3 ] .

Let \(T:M_{d}(\mathbb {C})\to M_{d}(\mathbb {C})\) be a primitive quantum channel of Kraus rank \(k\). Then there are an integer \(n\) and a word \(K^{(n)}\in K_n\) such that

Let \(T:M_{d}(\mathbb {C})\to M_{d}(\mathbb {C})\) be a primitive quantum channel and suppose that one of its Kraus operators, denoted by \(A\), satisfies \(A|\varphi \rangle =\mu |\varphi \rangle \), where \(\varphi \neq 0\) and \(\mu \neq 0\). Then

1. \(K_{d-1}|\varphi \rangle =\mathbb {C}^d\);

2. if \(A\) is not invertible, then \(|\varphi \rangle \! \langle \psi |\in K_{d^2-d+1}\) for every \(\psi \in \mathbb {C}^d\).

If the exact word spans are eventually full, then the least threshold \(q\) is itself admissible. It is no larger than any other admissible threshold. In particular, for a trace-preserving Kraus family, \(K_N=M_{d}(\mathbb {C})\) implies \(q\leq N\).

Every quantum channel \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\), with \(D\geq 1\), has a nonzero positive semidefinite fixed point: there exists \(\rho \geq 0\), \(\rho \neq 0\), with \(E(\rho )=\rho \).

If \(E\) is a channel, then \(E^p\) is a channel for every \(p\geq 0\).

If \(E\) is completely positive, then \(E^p\) is completely positive for every \(p\geq 0\).

Let \(T_N(K)\) be the span of the products of at most \(N\) matrices from a finite family \(K\). If \(D{\gt}0\) and \(T_N(K)=M_{D}(\mathbb {C})\), then there are a word \(w\), a nonzero scalar \(\mu \), and a nonzero vector \(\varphi \in \mathbb {C}^D\) such that

If \(M\in M_{D}(\mathbb {C})\) has \(\operatorname{tr}(M)\neq 0\), then \(M\) has a nonzero eigenvalue \(\mu \neq 0\) and a corresponding eigenvector \(\varphi \neq 0\) satisfying \(M\varphi =\mu \varphi \).

Let \(K=(K^i)_{i=0}^{d-1}\) be a finite family of \(D\times D\) matrices, where \(D{\gt}0\), and suppose that \(S_n(K)=M_{D}(\mathbb {C})\) for every sufficiently large \(n\). If \(\varphi \neq 0\), \(\mu \neq 0\), and \(K^{i_0}\varphi =\mu \varphi \), then \(H_{D-1}(K,\varphi )=\mathbb {C}^D\).

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

Every linear endomorphism on a finite-dimensional vector space over an algebraically closed field admits a Fitting decomposition.

Let \(E\) be a channel, and suppose \(E(X)=\mu X\) with \(X\neq 0\), \(\mu \neq 1\), and \(\mu ^p=1\). Then there is a nonzero Hermitian matrix \(H\) such that \(\operatorname{tr}(H)=0\) and \(E^p(H)=H\).

Let \(K\) be trace-preserving and suppose \(H_q(K,\varphi )=\mathbb {C}^D\) for every nonzero \(\varphi \). If \(p{\gt}0\) and \(H=H^\dagger \) satisfies \(\operatorname{tr}(H)=0\) and \(\mathcal{E}_K^p(H)=H\), then \(H=0\).

Suppose \(H_q(K,\varphi )=\mathbb {C}^D\) for every nonzero \(\varphi \). If \(p{\gt}0\) and \(\rho ,\sigma \) are nonzero positive-semidefinite fixed points of \(\mathcal{E}_K^p\), then \(\sigma =c\rho \) for some \(c\in \mathbb {C}\).

Fix a block length \(L\) and regard the words of length \(L\) in a Kraus family \(K\) as a new family \(K^{[L]}\). Evaluation of a blocked word agrees with evaluation of its flattened word, and consequently

If \(L{\gt}0\) and the exact word spans of \(K\) are eventually full, then the exact word spans of \(K^{[L]}\) are eventually full as well.

Let \(K=\{ K^i\} _{i=0}^{d-1}\) be a finite family of \(D\times D\) matrices, where \(D{\gt}0\), and let \(\varphi \neq 0\). If, for some \(N\), words in \(K\) of length at most \(N\) span \(M_{D}(\mathbb {C})\), then the vectors \(K^w\varphi \) with \(|w|\leq D-1\) span \(\mathbb {C}^D\).

For every choice of \(d\), \(D\), and \(K\) as above and every \(q\ge 0\), suppose \(H_q(K,\varphi )=\mathbb {C}^D\) for every nonzero \(\varphi \). Then every nonzero positive-semidefinite fixed point of \(\mathcal{E}_K\) is positive definite.

Let \(K\) be a finite matrix family with Kraus map \(E\), and let \(Q\) be an orthogonal projection. If

for every \(X\in M_{D}(\mathbb {C})\), then \((\mathbb {1}-Q)K_iQ=0\) for every \(i\).

For every choice of \(d\), \(D\), and \(K\) as above and every \(q\ge 0\), if \(H_q(K,\varphi )=\mathbb {C}^D\) for every nonzero \(\varphi \), then the Kraus map \(\mathcal{E}_K\) is irreducible.

For every choice of \(d\), \(D\), and \(K\) as above and every \(n\ge 0\), the iterate \(\mathcal{E}_K^n\) maps positive-semidefinite matrices to positive-semidefinite matrices.

For every choice of \(d\), \(D\), and \(K\) as above, every \(q\ge 0\), and every \(\varphi \in \mathbb {C}^D\),

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.

Let \(D{\gt}0\), and let \(K=\{ K^i\} _{i=0}^{d-1}\) be a finite family of \(D\times D\) matrices satisfying

Suppose there is a common length \(q\ge 0\) such that \(H_q(K,\varphi )=\mathbb {C}^D\) for every nonzero \(\varphi \in \mathbb {C}^D\). Then there is a positive-definite matrix \(\rho \) such that \(\mathcal{E}_K(\rho )=\rho \).

For every choice of \(d\), \(D\), and \(K\) as above and every \(q\ge 0\), suppose \(H_q(K,\varphi )=\mathbb {C}^D\) for every nonzero \(\varphi \). Then \(\mathcal{E}_K^q(\rho ){\gt}0\) for every nonzero positive-semidefinite matrix \(\rho \).

For every choice of \(d\), \(D\), and \(K\) as above and every \(q\ge 0\), suppose \(H_q(K,\varphi )=\mathbb {C}^D\) for every nonzero \(\varphi \). For every \(p{\gt}0\), each nonzero positive-semidefinite fixed point of \(\mathcal{E}_K^p\) is positive definite.

Let \(D{\gt}0\), and let \(K=\{ K^i\} _{i=0}^{d-1}\) be a finite family of \(D\times D\) matrices satisfying

Suppose there is a common length \(q\ge 0\) such that \(H_q(K,\varphi )=\mathbb {C}^D\) for every nonzero \(\varphi \in \mathbb {C}^D\). Then the Kraus map \(\mathcal{E}_K\) is primitive: it has a nonzero fixed point and \(1\) is its only peripheral eigenvalue.

For every choice of \(d\), \(D\), and \(K\) as above and every \(q\ge 0\), suppose \(H_q(K,\varphi )=\mathbb {C}^D\) for every nonzero \(\varphi \in \mathbb {C}^D\). Then \(\mathcal{E}_K^q(|\varphi \rangle \! \langle \varphi |){\gt}0\) for every \(\varphi \neq 0\).

Let \(K\) be trace preserving. If \(S_N(K)=M_{D}(\mathbb {C})\) and \(m\geq N\), then \(S_m(K)=M_{D}(\mathbb {C})\). Consequently, eventual fullness is equivalent to the existence of a positive \(N\) such that \(S_N(K)=M_{D}(\mathbb {C})\).

If \(f\) is nilpotent on a space of dimension \(n\), then \(f^n=0\).

Let \(D\geq 1\), and let \(E\colon M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be an irreducible quantum channel. Then every peripheral eigenvalue of \(E\) is a root of unity.

Let \(E\colon M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be linear, and let \(\rho \) and \(\sigma \) be positive-definite fixed points of \(E\). Suppose every nonzero positive-semidefinite fixed point of \(E\) is positive definite. Then \(\sigma =c\rho \) for some \(c\in \mathbb {C}\).

Let \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive. If \(E(X)=\mu X\), then \(E(X^\dagger )=\overline\mu X^\dagger \).

Let \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be positive. If \(E(X)=\mu X\) and \(\mu ^p=1\), then \(E^p(X^\dagger )=X^\dagger \).

Let \(E\colon M_D(\mathbb C)\to M_D(\mathbb C)\) be a positive, trace-preserving linear map and \(X=E(X)\) a fixed point. Decompose \(X\) into Hermitian and anti-Hermitian parts and then each into orthogonal positive and negative parts, yielding four positive semidefinite operators \(P_1,\dots ,P_4\). Then \(E(P_j)=P_j\) for \(j=1,\dots ,4\); in other words, every fixed point is a \(\mathbb C\)-linear combination of four positive-semidefinite fixed points. In particular, if \(E\) is a channel and \(H=H^\dagger \) is fixed by \(E\), then there are positive-semidefinite fixed points \(Q_1,Q_2\) such that \(H=Q_1-Q_2\).

This is Wolf Proposition 6.8.

If \(T:M_{n}(\mathbb {C})\to M_{m}(\mathbb {C})\) is positive, then \(T(A^\dagger )=T(A)^\dagger \) for all \(A\in M_{n}(\mathbb {C})\).

Let \(T\) be an endomorphism of a vector space over a field. If \(T(x)=\mu x\), then

for every \(n\geq 0\).

Let \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be linear. If \(E(X)=\mu X\) and \(\mu ^p=1\), then \(E^p(X)=X\).

Let \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) preserve the trace. If \(E(X)=\mu X\) and \(\mu \neq 1\), then \(\operatorname{tr}(X)=0\).

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be completely positive and trace preserving, with the Kraus decomposition above. Then the following are equivalent:

1. \(T\) is primitive: it is irreducible and its peripheral spectrum consists only of \(1\);

2. there is an \(n\in \mathbb N\) such that, for every \(m\geq n\) and every nonzero \(\psi \in \mathbb {C}^D\), \(K_m|\psi \rangle =\mathbb {C}^D\);

3. there is a \(q\in \mathbb N\) such that \(K_m=M_{D}(\mathbb {C})\) for every \(m\geq q\);

4. there is a \(q\in \mathbb N\) such that \(\tau _m{\gt}0\) for every \(m\geq q\).

The thresholds in items 3 and 4 can be chosen equal. If \(n\) and \(q\) are chosen minimal, then \(n\leq q\). This is [ Wol12 , Theorem 6.8 ] .

Let \(T:M_{d}(\mathbb {C})\to M_{d}(\mathbb {C})\) be a primitive quantum channel of Kraus rank \(k\). Then the minimal threshold \(q\) satisfies

1. in general, \(q\leq (d^2-k+1)d^2\);

2. if \(K_1\) contains an invertible element, then \(q\leq d^2-k+1\);

3. if \(K_1\) contains an element with a nonzero eigenvalue, then \(q\leq d^2\).