Applying \(E\) to (1) gives \(E(\sigma _N)=N^{-1}\sum _{n=1}^{N}E^n(\rho _0)\). Subtracting \(\sigma _N\) cancels all but the first and last terms, yielding (2).

Pigeonhole principle on the integer lattice of side \(q^m\): the \(q^m+1\) points \((\{ nx_1\} ,\dots ,\{ nx_m\} )\) for \(n=0,\dots ,q^m\) lie in \(q^m\) cells; two share a cell, and their difference yields \(n\).

Write each phase as \(\theta _j=e^{2\pi i x_j}\). Apply simultaneous approximation with successively larger denominators and choose the approximating exponents recursively beyond the preceding one.

Every non-peripheral eigenvalue \(\mu \) has modulus strictly below one. On the corresponding maximal generalized eigenspace write \(T=\mu \mathbb {1}+N\) with \(N^{\ell }=0\); the binomial expansion gives

since each summand carries the factor \(\mu ^{n}\to 0\) against a polynomial in \(n\). The non-peripheral subspace is the sum of these eigenspaces, and convergence to zero is preserved by finite sums.

Induct on \(n\). The induction step follows from linearity: \(T(\mu ^n x)=\mu ^nT(x)=\mu ^{n+1}x\).

Write \(x=\sum _{\mu \in S}x_\mu \) with \(T(x_\mu )=\mu x_\mu \). By Theorem 8.1.8, \(T^{n_k}(x_\mu )=\mu ^{n_k}x_\mu \to x_\mu \). The conclusion follows by summing over the finite set \(S\).

Bounded forward orbits imply that every eigenvalue has modulus at most one. On a non-peripheral generalized eigenspace, write the corresponding Jordan block as \(\mu \mathbb {1}+N\), where \(N^\ell =0\). Then

and every summand tends to zero because \(|\mu |{\lt}1\). Theorem 8.8.3 removes nilpotent parts on the peripheral subspace, while the simultaneous recurrence corollary chooses a common subsequence on which all peripheral phases tend to one. The complementary splitting gives pointwise convergence, which is equivalent to operator-norm convergence in finite dimension.

If \(X\) lies in the range of \(T_\phi \), then \(T_\phi (X)=X\), so the recurrent subsequence of Theorem 8.1.12 gives the required positive exponents. Conversely, put \(Y=X-T_\phi (X)\). Commutation of \(T_\phi \) with \(T\) transfers every sufficiently close return of \(X\) to a close return of \(Y\), while Lemma 8.1.7 gives \(T^n(Y)\to 0\). If a positive power fixed \(Y\), then all of its multiples would fix \(Y\), contradicting this limit unless \(Y=0\). The finitely many remaining small powers therefore stay a positive distance from \(Y\), and all sufficiently large powers stay close to zero. Arbitrarily close returns are possible only when \(Y=0\), hence \(X=T_\phi (X)\).

Every power of \(T\) preserves positivity and trace, and every power of a completely positive map is completely positive. These properties are closed under the operator-norm limit. Composition with \(T\) gives the assertion for \(T_\varphi \).

Fix a basis \(e_1,\dots ,e_m\) of \(E\). Evaluation on the basis,

is a linear bijection from the endomorphisms of \(E\) onto \(E^m\), since an endomorphism is determined by, and may be prescribed arbitrarily on, a basis. In finite dimension every linear map is continuous, so both \(\Phi \) and \(\Phi ^{-1}\) are continuous. The hypothesis says \(\Phi (S_N)\to \Phi (P)\) coordinatewise, hence \(S_N=\Phi ^{-1}(\Phi (S_N))\to \Phi ^{-1}(\Phi (P))=P\).

By Theorem 3.1.10 a map is completely positive precisely when its Choi matrix is positive semidefinite. The Choi matrix depends linearly, hence continuously, on the map, and the positive semidefinite matrices are closed. The set of completely positive maps is therefore the preimage of a closed set under a continuous map.

For \(D=0\) the algebra is trivial and every linear map is completely positive, so assume \(D{\gt}0\). Write

for the \(N\)-th Cesàro average. Powers of a completely positive map are completely positive, and so are finite sums and nonnegative multiples of completely positive maps, so every \(S_N\) is completely positive. For every \(X\in M_{D}(\mathbb {C})\), (29) gives

and Lemma 8.1.15 upgrades this to

in the operator norm. Lemma 8.1.16 then makes \(T_\infty \) completely positive. Trace preservation of \(T_\infty \) is Theorem 10.13.2, and the bounded orbits it requires come from Theorem 10.1.8.

The range of \(T_\infty \) consists of fixed points of \(T\), hence every power of \(T\) fixes this range. Passing to the recurrent-subsequence limit shows that \(T_\phi \) also fixes it pointwise.

Write \(V_\mu =\ker (T-\mu )\) for the eigenspace of a peripheral eigenvalue \(\mu \), so that \(\mathcal X_T=\sum _{|\mu |=1}V_\mu \) by the definition of \(\mathcal X_T\). The image of the projection \(T_\phi \) is its set of fixed points, and the fixed points of \(T_\phi \) are the linear combinations of peripheral eigenvectors:

The equality (13) holds because \(T_\phi \) is a projection; a matrix fixed by \(T_\phi \) lies in the peripheral subspace, whose Jordan blocks are trivial for a positive trace-preserving map (Theorem 8.8.3), which gives (14); and (15) is the definition of \(\mathcal X_T\). This is the first claim.

The projection \(T_\phi \) is itself positive and trace-preserving, so its fixed-point space is spanned by its stationary density matrices:

Every such \(\rho \) is positive semidefinite, which is the second claim.

On \(V_\mu \) the map \(T\) acts as multiplication by \(\mu \), and \(|\mu |=1\) forces \(\mu \neq 0\), so multiplication by \(\mu \) carries \(V_\mu \) onto itself and \(T(V_\mu )=\mu V_\mu =V_\mu \). Summing over the peripheral eigenvalues,

which is the third claim.

Start with any \(\rho _0\in \mathcal{D}_D\) (Theorem 8.17.3). Lemma 8.17.7 keeps every \(\sigma _N\), \(N\geq 1\), in the compact set \(\mathcal{D}_D\), so a subsequence converges to some \(\rho \in \mathcal{D}_D\). The telescope identity (2) gives

because the iterates \(E^N(\rho _0)\) lie in the compact set \(\mathcal{D}_D\) and are therefore uniformly bounded. Along the convergent subsequence, continuity of \(E\) now gives \(E(\rho )=\rho \). In  [ Wol12 , Theorem 6.11 ] , existence is proved via Brouwer’s fixed-point theorem; we use the Cesàro argument instead.

Every ratio \(\bar\mu \lambda \) again has modulus one, and \(\bar\mu \lambda =1\) forces \(\lambda =\mu \), since \(\mu \bar\mu =1\). The term \(\mu =\lambda \) therefore equals \(1\) for every \(n\) and contributes \(1\) to the mean. For \(\mu \neq \lambda \) set \(\zeta =\bar\mu \lambda \neq 1\); the geometric sum

is bounded by \(2/\lvert 1-\zeta \rvert \) uniformly in \(N\), so dividing by \(N\) sends it to zero.

The Cesàro mean of a convergent sequence converges to the same limit.

For (i) it suffices to treat a \(\lambda \)-eigenvector \(X\) with \(\lambda \in s\), because the span is the sum of the eigenspaces. There \((\bar\mu T)^{n}(X)=(\bar\mu \lambda )^{n}X\), so \(C_{N}(T,s)(X)\) is the scalar \(\frac{1}{N}\sum _{n=1}^{N}\sum _{\mu \in s}(\bar\mu \lambda )^{n}\) times \(X\), which tends to \(X\) by (18). For (ii), multiplication by a unit-modulus scalar does not change norms, so \(\lVert (\bar\mu T)^{n}(X)\rVert =\lVert T^{n}(X)\rVert \to 0\) for every \(\mu \in s\); the inner sum is finite, hence also tends to zero, and its Cesàro means tend to zero.

Split \(X=T_\phi (X)+(X-T_\phi (X))\) along the complementary spectral subspaces. Bounded forward orbits force every eigenvalue to have modulus at most one, so every eigenvalue outside the peripheral spectrum has modulus strictly below one and \(T^{n}(Y)\to 0\) for every \(Y\) in the non-peripheral subspace; part (ii) of Lemma 8.2.4 sends the second summand to zero. Triviality of the peripheral Jordan blocks identifies the peripheral subspace with the span of the peripheral eigenspaces, on which part (i) of the same lemma acts as the identity, so the first summand converges to \(T_\phi (X)\). Pointwise convergence of endomorphisms of a finite-dimensional space is convergence in operator norm.

Set \(S(B)=B+E(B)\) for nonzero \(B\geq 0\). Always \(\ker S(B)\subseteq \ker B\): if \(v\in \ker S(B)\), then

and both terms are non-negative, so \(v^\dagger Bv=0\) and hence \(Bv=0\). If \(B\) is not positive definite, irreducibility forces

Indeed, equality of the kernel dimensions would turn the inclusion above into \(\ker B\subseteq \ker E(B)\). Let \(Q\) be the support projection of \(B\). The latter inclusion places \(E(B)\) in the corner \(QM_{D}(\mathbb {C})Q\). Every Hermitian element of this corner lies between two real multiples of \(B\). Positivity and order preservation therefore place its image under \(E\) in the same corner; the Hermitian decomposition gives this for every element of \(QM_{D}(\mathbb {C})Q\). Thus \(Q\) is an invariant projection. Irreducibility gives \(Q=0\) or \(Q=\mathbb {1}\), and \(B\neq 0\) excludes the first case. Hence equality of the kernels would force \(B{\gt}0\).

We prove by induction on \(n\) that every nonzero \(B\geq 0\) with \(\dim \ker B\leq n\) satisfies \(S^n(B){\gt}0\). For \(n=0\), the kernel is trivial, so \(B{\gt}0\). Suppose the claim holds for \(n\) and let \(\dim \ker B\leq n+1\). If \(B{\gt}0\), then \(S(B)=B+E(B){\gt}0\), and every subsequent iterate remains positive definite. Otherwise the strict kernel decrease gives \(\dim \ker S(B)\leq n\); the induction hypothesis applied to \(S(B)\) yields \(S^n(S(B))=S^{n+1}(B){\gt}0\). Since \(A\neq 0\), rank–nullity gives \(\dim \ker A\leq D-1\), which proves (23).

Let \(Q\) be the support projection of \(\rho \). The kernel inclusion and positivity place \(E(\rho )\) in \(QM_{D}(\mathbb {C})Q\). The same two-sided order argument as in Theorem 8.3.1 shows that \(E\) maps the whole corner \(QM_{D}(\mathbb {C})Q\) into itself. Irreducibility forces \(Q\in \{ 0,\mathbb {1}\} \), while \(\rho \neq 0\) rules out \(Q=0\). Thus \(Q=\mathbb {1}\), so \(\rho {\gt}0\).

The eigenvector equation gives \(\ker \rho \subseteq \ker E(\rho )\), including when \(\lambda =0\). Theorem 8.3.2 then forces \(\rho \) to be positive definite.

The matrices \(Q\) and \(\rho \) have the same kernel. For a vector in the orthogonal complement of the range of \(Q\), the fixed-point identity writes its zero quadratic form against \(\rho \) as a sum of nonnegative quadratic forms. Each summand vanishes, so \(QK_i^\dagger (\mathbb {1}-Q)=0\). Taking adjoints gives \((\mathbb {1}-Q)K_iQ=0\).

The lower-corner identity gives \(K_iQ=QK_iQ\); taking adjoints gives \(QK_i^\dagger =QK_i^\dagger Q\). Substituting these two equalities in every summand of the Kraus representation of \(E(QXQ)\) proves the compression identity.

Set \(M_i=(\mathbb {1}-Q)K_iQ\). Multiplying the compression identity by \(\mathbb {1}-Q\) on the left gives the first equality below. Taking \(X=\mathbb {1}\), multiplying on the right by \(\mathbb {1}-Q\), and expanding the Kraus sum gives the second:

Each matrix \(M_iM_i^\dagger \) is positive semidefinite. Hence every term in the vanishing sum is zero, and therefore \(M_i=0\) for every \(i\).

The fixed-point equation makes the complementary Kraus corners vanish, \((\mathbb {1}-Q)K_iQ=0\). Expanding the Kraus sum then gives Equation (25).

A zero support projection would give \(\rho =Q\rho =0\).

The support is the orthogonal complement of the kernel. Thus full support makes the kernel trivial, so \(\rho \) is invertible and positive definite.

Let \(Q\) be the support projection of \(\rho \). Theorem 8.3.8 makes \(Q\) an invariant orthogonal projection, so irreducibility gives \(Q=0\) or \(Q=\mathbb {1}\). Theorem 8.3.9 excludes \(Q=0\), and Theorem 8.3.10 turns \(Q=\mathbb {1}\) into positive definiteness.

The transfer map is completely positive, so Theorem 8.3.11 applies.

Injectivity implies irreducibility of the transfer map. Apply Theorem 8.3.12.

Theorem 8.3.1 gives \((\mathbb {1}+ E)^{D-1}(A) {\gt} 0\). Since \(B \ge 0\) is nonzero, \(\operatorname{tr}(B(\mathbb {1}+ E)^{D-1}(A)) {\gt} 0\). Expanding by the binomial theorem gives

Each term \(\operatorname{tr}(BE^k(A))\) is non-negative since both factors are positive semidefinite. The \(k = 0\) term vanishes by the hypothesis \(\operatorname{tr}(BA) = 0\). Hence at least one \(k \in \{ 1,\ldots ,D-1\} \) contributes a strictly positive term. This proves the implication (2)\(\Rightarrow \)(4) in [ Wol12 , Theorem 6.2(4) ] by a binomial expansion and positivity.

Write \(\rho =SS^\dagger \) with \(S\) invertible and set \(H=S^{-1}\sigma (S^\dagger )^{-1}\). Take \(c\) to be the smallest eigenvalue of \(H\). Then \(H-c\mathbb {1}\) is positive semidefinite and singular. Congruence by \(S\) gives the corresponding assertions for \(\sigma -c\rho \).

Apply Lemma 8.4.1 to choose \(c{\gt}0\) for which \(\sigma -c\rho \) is positive semidefinite but not positive definite. This difference is fixed by \(E\). If it were nonzero, the hypothesis would make it positive definite, a contradiction. Hence \(\sigma -c\rho =0\).

If \(\sigma =0\), take \(c=0\). Otherwise, Theorem 8.3.11 makes both \(\rho \) and \(\sigma \) positive definite. The same theorem makes every nonzero positive-semidefinite fixed point positive definite. Apply Theorem 8.4.2.

The transfer map is completely positive, so Theorem 8.4.3 applies.

Injectivity implies irreducibility of the transfer map. Apply Theorem 8.4.4.

Extract Kraus operators \(K\) for \(E\). Irreducibility of \(E\) passes to the adjoint transfer map \(E^\dagger (X) = \sum _i K_i^\dagger X K_i\), which is a positive CP map. The Perron–Frobenius existence theorem (Theorem 8.8.2) together with the irreducible positive-definiteness upgrade (Theorem 8.3.3) gives a positive definite eigenvector \(\tau {\gt} 0\) with eigenvalue \(t {\gt} 0\): \(E^\dagger (\tau ) = t\tau \). For any nonzero positive semidefinite \(X\) with \(E(X) = sX\), the trace-pairing identity (20) yields

Since \(\tau {\gt} 0\) and \(X \ge 0\), \(X \neq 0\), the scalar \(\operatorname{tr}(\tau X) {\gt} 0\), so \(s = t\). Applying this to both \((\rho ,r_1)\) and \((\sigma ,r_2)\) gives \(r_1 = t = r_2\). This is the completely positive specialization of [ Wol12 , Theorem 6.3(3) ] : uniqueness of a positive eigenvalue admitting a nonzero positive semidefinite eigenvector. The proof follows Wolf’s dual-map trace argument.

Choose Kraus operators for \(E\) and rescale each by \(r^{-1/2}\). Their transfer map is \(r^{-1}E\), so both \(\rho \) and \(\sigma \) are fixed points. The rescaled map remains irreducible and completely positive, and Theorem 8.4.3 gives the claimed proportionality.

Under the normalization \(\sum _i (A^i)^\dagger A^i = \mathbb {1}\), the transfer map is a quantum channel (Theorem 2.10.5). The Cesàro fixed-point theorem (Theorem 8.1.20) then provides the fixed point; injectivity is not needed for this step.

Combine Theorems 8.5.1, 8.3.13, and 8.4.5.

Existence is the channel fixed-point theorem (Theorem 8.5.1). Irreducibility upgrades the fixed point to positive definite, and Theorem 8.4.4 gives uniqueness up to scalar multiple.

Conjugating each summand gives \((S^{-1}A^iS)(S^{-1}A^iS)^\dagger =S^{-1}A^i\rho (A^i)^\dagger (S^\dagger )^{-1}\). The substitution \(\mathcal{E}_A(\rho )=\rho \) leaves \(S^{-1}\rho (S^\dagger )^{-1}\), and \(\rho =SS^\dagger \) cancels the outer factors. The term-by-term calculation is in Section 8.18.

Conjugating each summand gives \((SA^iS^{-1})^\dagger (SA^iS^{-1}) =(S^\dagger )^{-1}(A^i)^\dagger \sigma A^iS^{-1}\). The adjoint fixed-point substitution leaves \((S^\dagger )^{-1}\sigma S^{-1}\), and \(\sigma =S^\dagger S\) cancels the outer factors. See Section 8.18 for the full calculation.

Suppose \(Q\neq 0,\mathbb {1}\) is an invariant projection for \(E'\). Thus \((\mathbb {1}-Q)E'(QXQ)Q=0\) for every \(X\). Setting \(R = CQC^\dagger \), the support projection \(P\) of \(R\) is an invariant projection for \(E\). Irreducibility of \(E\) forces \(P = 0\) or \(P = \mathbb {1}\). The invertibility of \(C\) then forces \(Q = 0\) or \(Q = \mathbb {1}\), a contradiction. This is the completely positive specialization of the similarity result in  [ Wol12 , Proposition 6.6 ] ; the scalar case \(C=\mathbb {1}\) gives the scaling result stated next.

An invariant projection for \(cE\) is an invariant projection for \(E\), since the nonzero constant may be removed from the invariance relation.

If \(E\) annihilates a nonzero positive semidefinite matrix \(\rho \), take \(r=0\). Otherwise \(E(\sigma )\neq 0\) for every nonzero \(\sigma \geq 0\), and Theorem 8.8.2 gives (26) with \(r{\gt}0\).

Normalize \(E\) on the density matrices by

Positivity and the nonvanishing hypothesis make this a continuous self-map. Theorem 8.17.6 gives a fixed point \(\rho \). Clearing the positive denominator yields \(E(\rho )=r\rho \) with \(r=\operatorname{tr}(E(\rho )){\gt}0\). Thus this theorem is the positive-eigenvalue specialization of Theorem 8.8.1.

If a rank-\(2\) generalized eigenvector existed, the binomial expansion \(T^n X=\lambda ^n X+n\lambda ^{n-1}(T-\lambda )X\) (when \((T-\lambda )^2X=0\)) would make \(\| T^nX\| \) grow linearly with \(n\). This contradicts the bounded-orbit theorem, which holds under either hypothesis: trace preservation or unitality each give \(\operatorname{tr}[1\, T(1)]=D\), and positivity turns this into a uniform bound on \(\operatorname{tr}[A\, T^n(B)]\) over all powers of \(T\). The rank-\(k\) case reduces to rank \(2\) by induction.

At \(X=0\), both sets in (27) equal \(\mathbb {R}\), whose supremum and infimum in \(\overline{\mathbb {R}}\) are respectively \(+\infty \) and \(-\infty \). The infimum of the empty subset of \(\overline{\mathbb {R}}\) is \(+\infty \).

If \(a\in L_T(X)\) and \(b\in U_T(X)\), adding the two positive semidefinite residuals gives \((b-a)X\geq 0\). Since \(X\geq 0\) is nonzero, \(\operatorname{tr}X{\gt}0\); taking traces therefore gives \(a\leq b\). Taking the supremum over \(a\) and the infimum over \(b\) proves (30), including the cases in which either set has an infinite extremum.

The eigenvalue \(a\) belongs to both sets in (27). Hence \(a\leq r_T(X)\leq \widetilde r_T(X)\leq a\), where the middle inequality is Theorem 8.9.3.

Suppose \(P\) is an invariant projection for \(T^*\). Its complementary leakage vanishes, and trace duality gives

Positivity turns the final scalar equality into the assertion that \(T(\mathbb {1}-P)\) is supported on \(\mathbb {1}-P\). Two-sided order domination then shows that \(T\) preserves the whole corner \((\mathbb {1}-P)M_{D}(\mathbb {C})(\mathbb {1}-P)\). Irreducibility of \(T\) makes \(\mathbb {1}-P\), and hence \(P\), trivial. Applying the same implication to \(T^*\) and using \((T^*)^*=T\) proves the converse.

First maximize the lower functional, in the order used by Wolf. On density matrices, lower feasibility implies \(a\leq \operatorname {Re}\operatorname{tr}(T(X))\). The latter continuous function has a maximum. Since \(0\) is feasible and every negative candidate is below it, the search for a maximum may be restricted to a compact interval \(0\leq a\leq M\). Hence there is a lower maximizer \((X,r)\) with \(r\geq 0\). Upper feasible values are automatically nonnegative: positivity gives \(\operatorname{tr}T(X)\geq 0\), while tracing \(aX-T(X)\geq 0\) gives \(a\geq \operatorname{tr}T(X)\).

Put \(S=(\operatorname{id}+T)^{D-1}\) and \(R=T(X)-rX\geq 0\). The polynomial identity

is Wolf’s Equation (6.31). Theorem 8.3.1 gives \(S(X){\gt}0\). If \(R\neq 0\), it also gives \(S(R){\gt}0\); subtracting a sufficiently small positive multiple of \(S(X)\) from \(S(R)\) would make a value larger than \(r\) lower feasible at the normalized matrix \(S(X)\). This contradicts maximality. Thus \(R=0\), so \(T(X)=rX\) and \(X{\gt}0\).

For geometric non-degeneracy, split a complex \(r\)-eigenvector into its Hermitian and skew-Hermitian parts. If a Hermitian part \(H\) is not a scalar multiple of \(X\), choose a real boundary perturbation \(W=X+cH\geq 0\) which is nonzero and singular. It is still an \(r\)-eigenvector, whence

This is impossible because a positive multiple of a singular matrix is singular. Both Hermitian parts are therefore proportional to \(X\), proving that the ordinary eigenspace is \(\mathbb {C}X\).

If \(r=0\), positivity and \(X{\gt}0\) imply \(T=0\), so the remaining conclusions follow directly. Hence suppose \(r{\gt}0\). By Theorem 8.9.5, the trace adjoint \(T^*\) is again irreducible and positive. The preceding construction gives \(X_0{\gt}0\) with \(T^*(X_0)=rX_0\); the trace pairing first identifies its Perron value with \(r\). For any nonzero \(Y\geq 0\) satisfying \(T(Y)=\lambda Y\), one then has Wolf’s Equation (6.33):

Since \(\operatorname{tr}(X_0Y){\gt}0\), this gives \(\lambda =r\). Pairing \(X_0\) instead with the upper residual \(aY-T(Y)\geq 0\) from (27) shows that every upper feasible value is at least \(r\). Together with feasibility of the Perron pair, this proves the corrected global maximum–minimum equality.

Finally set

When \(r{\gt}0\), this map is positive and unital. Its spectral radius is one by Theorem 2.11.7; similarity and scalar rescaling give \(\varrho (T)=r\).

The local source at lines 617–619 prints the upper global quantity as a second supremum and then states the reversed inequality \(r\geq \widetilde r\). The valid statement proved here is the minimum of the upper feasible values, equal to the lower maximum by the trace-adjoint argument above. No arbitrary pointwise equality \(r(Y)=\widetilde r(Y)\) is used.

Apply the construction (36) to the supplied pair \((X,r)\). The resulting map is positive and unital, so Theorem 2.11.7 gives spectral radius one. Similarity invariance and the scalar rule give \(\varrho (T)=r\).

The Perron–Frobenius theorem (Theorem 8.8.2) gives a PSD eigenvector \(T(\rho )=r\rho \) with \(r{\gt}0\). Trace preservation forces \(r=1\) because \(\operatorname{tr}(T(\rho ))=\operatorname{tr}(\rho )\).

In each case, “for every sufficiently large \(m\)” means exactly that there is a lower threshold beyond which the property holds.

Primitivity gives full \(K_m\) for all sufficiently large \(m\) by the established positive-definite-fixed-point and nonperipheral-decay argument. At every fixed length, vectorization identifies \(K_m\) with the span generating \(\tau _m\), so items 3 and 4 are equivalent with the same \(q\). A full matrix word span sends every nonzero vector onto all of \(\mathbb {C}^D\), proving item 2 from item 3 at the same threshold; conversely, evaluating item 2 at \(m=n\) gives irreducibility and trivial peripheral spectrum. Thus all four clauses are equivalent. The pointwise word-span–Choi equivalence makes their admissible \(q\)-sets equal, and a word threshold is also a vector threshold, so minimality gives \(n\leq q\).

Choose a nearest point in \(S\) by minimizing squared distance. The first-order variational inequality for this minimizer shows that the metric projection is \(1\)-Lipschitz and is the identity on \(S\). It is thus a continuous retraction of the ambient Euclidean space onto \(S\), so Theorem 8.17.5 applies. Wolf states the theorem without supplying a proof.

Positivity and trace preservation give \(T(\rho )\in \mathcal D_D\) for every \(\rho \in \mathcal D_D\); continuity of \(T\) gives continuity on \(\mathcal D_D\). Theorem 8.17.6 applied to the restriction \(T|_{\mathcal D_D}\) yields \(\rho \in \mathcal D_D\) with \(T(\rho )=\rho \).

The proof follows Wolf: first handle Hermitian fixed points by the orthogonality argument using the support projection (equation (6.46)–(6.47) in the source). This gives the two-part decomposition for a Hermitian channel fixed point. Decompose an arbitrary fixed point into Hermitian and skew-Hermitian parts and apply the Hermitian case to each.

By Theorem 8.12.1, every fixed point is a \(\mathbb C\)-linear combination of four positive-semidefinite fixed points. Each nonzero positive-semidefinite fixed point can be scaled to a stationary density matrix, so the stationary density matrices span \(\mathcal F_E\). Since \(\mathcal F_E\) is finite-dimensional, the spanning set contains a basis; extracting one yields \(r\) linearly independent stationary density matrices.

By Theorem 8.21.1, the first \(D\) terms in (37) already form a positive definite matrix. Every remaining term is positive semidefinite, so the full exponential series is the sum of a positive definite matrix and a positive semidefinite tail.

The forward implication is Theorem 8.13.1. Conversely, let \(P\) be an invariant orthogonal projection. Invariance gives \(E(PM_{D}(\mathbb {C})P)\subseteq PM_{D}(\mathbb {C})P\), so induction yields \(E^k(P)\in PM_{D}(\mathbb {C})P\) for every \(k\geq 0\). Therefore every term, and hence the convergent series

lies in the same corner. If \(P\neq 0\), then at \(t=1\) the assumed positive definiteness is impossible unless \(P=\mathbb {1}\), because a positive definite matrix cannot be supported on a proper corner. Thus every invariant projection is \(0\) or \(\mathbb {1}\), so \(E\) is irreducible.

Every Cesàro mean (1) of a density matrix stays inside the compact convex set of density matrices, so a subsequence converges to a density matrix \(\sigma \). The telescope identity (2) gives \(E(\sigma )=\sigma \). Theorem 8.3.3 makes \(\sigma \) positive definite. If \(\tau \) is another density-matrix fixed point, Theorem 8.4.7 gives \(\tau =c\sigma \). Taking traces yields \(1=c\), hence \(\tau =\sigma \).

Every subsequential limit of the Cesàro means is again a density-matrix fixed point. By Theorem 8.14.1, there is only one such fixed point, so every convergent subsequence has the same limit. Compactness then forces the whole sequence to converge to that limit.

Combine Perron–Frobenius existence for the map and its adjoint with the uniqueness theorem for positive semidefinite Perron eigenvectors and the spectral-radius identity of Theorem 8.9.7.

Gauge by the positive definite left eigenvector in (38) and rescale by the Perron eigenvalue to obtain a trace-preserving map with a positive definite fixed point. A nontrivial invariant projection would then produce, by Cesàro averaging in its corner algebra, a second positive semidefinite fixed point not proportional to the Perron vector, contradicting the uniqueness clause in Definition 8.15.1.

Combine Theorems 8.15.2 and 8.15.3.

The positive semidefinite cone is closed, being the preimage of the closed non-negative cone under continuous quadratic forms. If \(\rho \ge 0\) with \(\operatorname{tr}(\rho ) = 1\), then each entry satisfies \(|\rho _{ij}|^2 \le \rho _{ii}\rho _{jj} \le 1\): the diagonal entries are non-negative and sum to \(1\), and the off-diagonal bound is the positive semidefinite Cauchy–Schwarz inequality. Hence the matrix entries are uniformly bounded, and Heine–Borel gives compactness.

A convex combination of positive semidefinite matrices is positive semidefinite, and the trace is linear.

\(\frac{1}{D}\mathbb {1}_D\) is a density matrix.

Complete positivity implies positivity by Theorem 2.2.3, so \(E(\rho ) \ge 0\); trace preservation gives \(\operatorname{tr}(E(\rho )) = \operatorname{tr}(\rho ) = 1\).

Brouwer’s theorem for finite products of simplices transfers to closed cubes. After choosing linear coordinates on \(E\), apply the fixed-point theorem on a cube containing \(K\) to the composite of the self-map with the retraction. The zero-dimensional case is immediate.

The density-matrix set is a compact retract of its finite-dimensional ambient real normed space, using the explicit density retraction. Apply Theorem 8.17.5.

Every iterate \(E^n(\rho )\) belongs to \(\mathcal{D}_D\) by Theorem 8.17.4. Equation (39) is their convex average: positivity is preserved by sums and multiplication by \((N+1)^{-1}\), while its trace is \((N+1)^{-1}\sum _{n=0}^{N}1=1\).

Lemma 8.17.7 and closedness of the compact set \(\mathcal{D}_D\) give \(\sigma \in \mathcal{D}_D\). The telescope identity yields

Both \(E^{\psi (k)+1}(\rho )\) and \(\rho \) lie in the compact set \(\mathcal{D}_D\), so their difference is uniformly bounded. The right-hand side tends to zero. Continuity of \(E\) and convergence of the subsequence therefore give \(E(\sigma )-\sigma =0\).

Self-adjointness of \(\rho ^{1/2}\) gives

Summing, substituting the adjoint fixed-point equation, and cancelling the outer square-root factors yields

Direct computation: the inner conjugation \(C(C^{-1}X(C^\dagger )^{-1})C^\dagger \) collapses to \(X\) using \(CC^{-1} = \mathbb {1}\) and \((C^\dagger )^{-1}C^\dagger = \mathbb {1}\), and the outer conjugation by \(C(\, \cdot \, )C^\dagger \) then cancels the inner \(C^{-1}(\, \cdot \, )(C^\dagger )^{-1}\) on \(E(X)\) for the same reason.

The forward direction follows from Lemma 8.7.1 applied with the inverse \(C^{-1}\), whose determinant is also nonzero: if the similarity transform of \(E\) by \(C\) is irreducible, then so is its similarity transform by \(C^{-1}\), which equals \(E\) by (40). The reverse direction is Lemma 8.7.1 itself with \(c = 1\).

If the map vanishes, evaluating at \(\mathbb {1}\) gives \(\sum _i K_i K_i^\dagger = 0\). Since each summand \(K_i K_i^\dagger \) is positive semidefinite, each \(K_i K_i^\dagger = 0\) and hence each \(K_i = 0\). Applying this to \(K_i = (A^i)^\dagger \) and using \(A^i \neq 0 \iff (A^i)^\dagger \neq 0\) shows that the adjoint transfer map \(\mathcal{E}_A^\dagger (X) = \sum _i (A^i)^\dagger X A^i\) of a tensor with some \(A^i \neq 0\) is nonzero.

The family \(\{ K_i\} \) itself is a Kraus representation of the map.

If \(P\) is invariant for the Kraus map of \(\{ K_i^\dagger \} \), so that \((\mathbb {1}-P)K_i^\dagger P=0\) for all \(i\), then taking adjoints gives \(PK_i(\mathbb {1}-P)=0\); by Theorem 8.3.6, \(\mathbb {1}-P\) is invariant for \(\mathcal K_K\). Irreducibility of \(\mathcal K_K\) forces \(\mathbb {1}-P \in \{ 0,\mathbb {1}\} \), hence \(P \in \{ 0,\mathbb {1}\} \), so the Kraus map of \(\{ K_i^\dagger \} \) is irreducible.

The conjugate-transposed family \(\{ K_i^\dagger \} \) has the same Kraus map as the adjoint of \(\mathcal K_K\), so it is completely positive (Theorem 8.20.2), and by Theorem 8.20.1 it is nonzero. By Theorem 8.20.3, its Kraus map is irreducible, so it does not annihilate any nonzero positive semidefinite matrix, and Theorem 8.8.2 applies, giving \(\sigma \ge 0\), \(\sigma \neq 0\), and \(r {\gt} 0\) with \(\sum _i K_i^\dagger \sigma K_i = r\sigma \). Theorem 8.3.3 then gives that \(\sigma \) is positive definite.

This is the real-valued reformulation of Theorem 8.9.7.

If a nonzero vector \(v\) annihilated the quadratic form in (41), positivity of every summand would make \(E^k(A)v=0\) for \(0\leq k\leq D-1\). Expanding \((\mathbb {1}+E)^{D-1}(A)\) by the binomial theorem would then make it annihilate \(v\), contrary to Theorem 8.3.1.