Dependencies

For an orthogonal projection \(P\) and linear map \(E\), the corner \(PM_{D}(\mathbb {C})P\) is preserved if \(E(PXP)=PE(PXP)P\) for every \(X\).

The corner subspace associated with \(P\) is \(\{ PXP:X\in M_{D}(\mathbb {C})\} \).

Irreducibility of the corner-restricted map.

The rank of an orthogonal projection \(P\in M_{D}(\mathbb {C})\), equivalently the dimension of its range.

The restriction of \(E\) to an invariant corner \(PM_{D}(\mathbb {C})P\).

Given a linear map \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) and a fixed point \(\rho \) with \(E(\rho )=\rho \) and \(\operatorname{tr}(\rho )\neq 0\), the fixed-point projection is the rank-one map

This projects onto the span of \(\rho \) along the kernel of the trace functional. When the fixed point is unique up to scaling, this span is the full fixed-point space.

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.

The adjoint Kraus map is

When the \(\{ K_i\} \) are the matrices of an MPS tensor \(A\), this is the transfer map of the conjugate-transposed family \(i \mapsto (A^i)^\dagger \).

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 2.1.5) if and only if it can be written in this form, as in (1).

The peripheral eigenvalues of a linear map \(E\) on a finite-dimensional space are the eigenvalues \(\lambda \) on the unit circle: \(|\lambda | = 1\). For a channel, the spectral radius is \(1\) [ Wol12 , Proposition 6.1 ] , so these coincide with the eigenvalues of maximal modulus.

The condition \(|\lambda | = 1\) is appropriate for channels, since the spectral radius of a channel is \(1\); for a general linear map with spectral radius \(r \neq 1\) the condition would generalise to \(|\lambda | = r\).

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 Kraus map is trace-preserving if \(\sum _{i=0}^{d-1} K_i^\dagger K_i = \mathbb {1}\). Equivalently, the adjoint Kraus map is unital. This is the standard MPS normalization condition. In the later gauge language it is the left-canonical condition, so Kadison–Schwarz arguments are often applied to the adjoint map.

A Kraus map is unital if \(\sum _{i=0}^{d-1} K_i K_i^\dagger = \mathbb {1}\). Equivalently, \(E(\mathbb {1}) = \mathbb {1}\).

The channel raised to the period preserves every cyclic corner.

Appropriate powers of the channel preserve each sector corner.

Let \(T\) be irreducible, let \(P_0,\ldots ,P_{m-1}\) be orthogonal projections summing to \(\mathbb {1}\) and cyclically permuted by \(T\), and suppose that every orthogonal projection \(Q\) with \(QP_k=P_kQ=Q\) that is invariant under the corner restriction of \(T^m\) to \(P_kM_{D}(\mathbb {C})P_k\) admits an orthogonal projection \(R\) invariant under \(T\) on the full algebra with \(Q=0\iff R=0\) and \(Q=P_k\iff R=\mathbb {1}\). Then, for every \(k\), the restriction of \(T^m\) to the corner \(P_kM_{D}(\mathbb {C})P_k\) is irreducible.

Each sector restriction is primitive.

A primitive \(m\)-th peripheral root yields \(m\) orthogonal projections summing to \(\mathbb {1}\) and cyclically permuted by the channel.

The fixed-point projection \(P\) has operator trace \(\operatorname{Tr}(P)=1\).

If \(E\) is irreducible and unital, then every fixed point of \(E\) is a scalar multiple of \(\mathbb {1}\).

Let \(E\) be an irreducible unital Kraus map on \(M_{D}(\mathbb {C})\) with a positive-definite adjoint fixed point. If \(X \ne 0\) satisfies \(E(X) = \mu X\) with \(|\mu | = 1\), then \(X\) is invertible.

Peripheral unitary eigenvectors can be normalized compatibly with the faithful fixed-point state.

Let \(E(X)=\sum _iK_iXK_i^\dagger \) be a Kraus map that is unital (\(\sum _iK_iK_i^\dagger =\mathbb {1}\)), and assume:

1. the adjoint Kraus map \(E^*(X)=\sum _iK_i^\dagger XK_i\) has a positive definite fixed point \(\rho {\gt}0\), and

2. \(E\) is irreducible.

Then \(\mathrm{peripheral}(E)\) is closed under powers: if \(\mu \in \mathrm{peripheral}(E)\), then \(\mu ^n\in \mathrm{peripheral}(E)\) for every \(n\in \mathbb {N}\).

The inverse of a peripheral eigenvalue is peripheral.

The product of two peripheral eigenvalues is peripheral.

Let \(E\) be an irreducible unital Kraus map on \(M_{D}(\mathbb {C})\) with a positive-definite adjoint fixed point. If \(\mu , \nu \) are peripheral eigenvalues of \(E\), then so is \(\mu \nu \).

For an irreducible unital Schwarz map with faithful adjoint fixed point, each peripheral eigenvalue admits a unitary eigenvector.

If \(E(U)=\mu U\) with \(|\mu |=1\) and \(U\) unitary, then \(E(U^n)=\mu ^nU^n\) for every \(n\).

Let \(P\in M_{D}(\mathbb {C})\) be a nonzero idempotent and let \(E\) be a linear map on \(M_{D}(\mathbb {C})\) that preserves the corner \(PM_{D}(\mathbb {C})P\), is primitive, and fixes the corner projection, \(E(P)=P\). Then the restriction of \(E\) to the corner \(PM_{D}(\mathbb {C})P\) is again primitive.

If \(A,B\geq 0\), then \(\operatorname{tr}(AB)\geq 0\).

Let \(a\) be an element of a complex Banach algebra with \(\rho _{\operatorname{spec}}(a){\lt}1\). Then \(a^n\to 0\) as \(n\to \infty \).

For any linear endomorphism \(T\) on \(M_{D_1\times D_2}(\mathbb {C})\), the operator trace expands as

where \(E_{pq}\in M_{D_1\times D_2}(\mathbb {C})\) has entry \(1\) in position \((p,q)\) and zero elsewhere.

Let \(K\) be a finite matrix family on \(M_{D}(\mathbb {C})\) with Kraus map \(\mathcal{E}_K\). Assume that \(\mathcal{E}_K\) is irreducible, that \(K\) is unital and trace-preserving, and that the adjoint Kraus map fixes a positive-definite matrix. Let \(\gamma \) be a primitive \(m\)-th root of unity such that the peripheral eigenvalues are exactly \(\{ \gamma ^k:k\in \{ 0,\ldots ,m{-}1\} \} \). Then \(m\mid D\).

Let \(P\in M_{D}(\mathbb {C})\) be an orthogonal projection of rank \(n\). The corner algebra \(PM_{D}(\mathbb {C})P\) is linearly isomorphic to the full matrix algebra \(M_{n}(\mathbb {C})\). The isomorphism is built from the spectral diagonalisation of \(P\) by conjugating the top-left \(n\times n\) block by the unitary diagonalising \(P\).

The rank of an orthogonal projection \(P\) equals its trace: \(n=\operatorname{tr}(P)\).

Let \(K\) be a finite unital matrix family whose Kraus map is irreducible, and suppose the adjoint Kraus map fixes a positive-definite matrix. If the peripheral spectrum is generated by a primitive \(m\)-th root \(\gamma \), then there are a unitary \(U\) and orthogonal projections \(P_0,\ldots ,P_{m-1}\) summing to \(\mathbb {1}\) such that \(E_K(U^k)=\gamma ^kU^k\), \(U^m=\mathbb {1}\), \(U=\sum _{k=0}^{m-1}\gamma ^kP_k\), and \(E_K(P_{k+1})=P_k\).

Let \(E\) be unital and let \(E^*\) be trace-preserving with a positive definite fixed point \(\rho {\gt} 0\). If \(E(X) = \mu X\) with \(|\mu | = 1\), then \(E(X^\dagger X) = E(X)^\dagger E(X)\).

If \(E\) has a nonzero fixed point \(\rho \neq 0\) with \(E(\rho )=\rho \), then \(1\) is a peripheral eigenvalue of \(E\).

Let \(K\) be a finite matrix family on \(M_{D}(\mathbb {C})\) with Kraus map \(\mathcal{E}_K\). Assume that \(\mathcal{E}_K\) is irreducible, that \(K\) is unital and trace-preserving, and that the adjoint Kraus map has a positive-definite fixed point. Then there exist \(m\geq 1\) and a primitive \(m\)-th root of unity \(\gamma \) such that \(m\mid D\) and the peripheral eigenvalues of \(\mathcal{E}_K\) are exactly \(\{ 1,\gamma ,\gamma ^2,\ldots ,\gamma ^{m-1}\} \).

Let \(E\) be an irreducible unital Kraus map on \(M_{D}(\mathbb {C})\) with a positive-definite adjoint fixed point (trace-preservation is not required). Then there exist \(m \ge 1\) and a primitive \(m\)-th root of unity \(\gamma \) such that the peripheral eigenvalues of \(E\) are exactly \(\{ 1,\gamma ,\gamma ^2,\ldots ,\gamma ^{m-1}\} \).

On a finite-dimensional space, the set of peripheral eigenvalues is finite.

Let \(K\) be a finite unital matrix family whose Kraus map is irreducible, and suppose that the adjoint Kraus map fixes a positive-definite matrix. For every peripheral eigenvalue \(\gamma \) of the transfer map, the \(\gamma \)-eigenspace is one-dimensional.

Let \(E(X)=\sum _iK_iXK_i^\dagger \) be a Kraus map that is unital, and assume that the adjoint Kraus map \(E^*(X)=\sum _iK_i^\dagger XK_i\) has a positive definite fixed point and that \(E\) is irreducible. Then every peripheral eigenvalue of \(E\) is a root of unity.

Let \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be trace-preserving, and let \(\rho \neq 0\) satisfy \(E(\rho )=\rho \) and \(\operatorname{tr}(\rho )\neq 0\). Let \(P\) be the fixed-point projection associated to \(\rho \), as defined in (15). Assume that:

1. \(E\) is primitive;

2. every eigenvalue \(\mu \) of \(E\) satisfies \(|\mu |\le 1\);

3. whenever \(E(X)=X\) and \(\operatorname{tr}(X)=0\), one has \(X=0\).

Then every eigenvalue \(\nu \) of \(E-P\) satisfies \(|\nu |{\lt}1\).

Let \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be an irreducible primitive channel, and let \(\rho \ne 0\) be a positive semidefinite fixed point. Then \(\operatorname{tr}(\rho )\ne 0\) and, for \(P=P_\rho \), \(\rho _{\operatorname{spec}}(E-P){\lt}1\).

If \(E\) is completely positive and irreducible and \(\rho \ge 0\), \(\rho \neq 0\) is a fixed point of \(E\), then \(\rho \) is positive definite.

Let \(E\) be a completely positive irreducible map on \(M_{D}(\mathbb {C})\). If \(\rho ,\sigma \geq 0\) are fixed points of \(E\) and \(\rho \neq 0\), then \(\sigma =c\rho \) for some \(c\in \mathbb {C}\).

Let \(E\) be a unital Kraus map. If \(E(X^\dagger X) = E(X)^\dagger E(X)\), then, for every \(Y \in M_{D}(\mathbb {C})\),

Let \(V\) be a nonzero finite-dimensional complex normed space. If every eigenvalue of an endomorphism \(F:V\to V\) satisfies \(|\lambda |{\lt}1\), then the spectral radius of \(F\) is strictly less than \(1\).

Let \(\mathcal{E}_K\) be the Kraus map of a finite matrix family \(K\). Assume that \(\mathcal{E}_K\) is irreducible and unital and has a positive-definite adjoint fixed point. Then there exist \(m\geq 1\) and a primitive \(m\)-th root of unity \(\gamma \) such that the peripheral eigenvalues of \(\mathcal{E}_K\) are exactly \(\{ 1,\gamma ,\gamma ^2,\ldots ,\gamma ^{m-1}\} \).