1 Deconstructing Quantum Mechanics
Following Wolf’s opening chapter [ Wol12 , Chapter 1 ] , this chapter collects the finite-dimensional groundwork of the volume: states as density operators and observables as Hermitian matrices under the trace pairing; the Schmidt decomposition, purification, and steering; positive maps on commutative ranges, the partial trace, and the extension of completely positive maps from operator systems; and the spectral facts behind these, from trace-power identities to Wigner’s theorem.
1.1 Bipartite systems and the Schmidt decomposition
Every pure state of a bipartite system can be brought into a normal form that exhibits its entanglement directly.
For \(\psi \in \mathbb {C}^{d_A}\otimes \mathbb {C}^{d_B}\) with \(d=\min \{ d_A,d_B\} \), a Schmidt decomposition of \(\psi \) is a choice of orthonormal bases \(\{ e_j\in \mathbb {C}^{d_A}\} \), \(\{ f_j\in \mathbb {C}^{d_B}\} \) and nonnegative reals \(\lambda _j\), \(j=1,\ldots ,d\), satisfying
If \(d_A\le d_B\), every \(\psi \in \mathbb {C}^{d_A}\otimes \mathbb {C}^{d_B}\) admits a Schmidt decomposition.
Identify \(\psi \) with its coefficient matrix \(C\in M_{d_A,d_B}(\mathbb {C})\). The spectral theorem applied to the positive semidefinite matrix \(C\, C^\dagger \) supplies the eigenbasis \(\{ e_j\} \) of \(\mathbb {C}^{d_A}\) and the eigenvalues \(\lambda _j\); the normalized images of the eigenvectors under \(C^\dagger \), conjugated and completed to an orthonormal basis, supply \(\{ f_j\} \).
Every vector \(\psi \in \mathbb {C}^{d_A}\otimes \mathbb {C}^{d_B}\) admits a Schmidt decomposition. This is [ Wol12 , Proposition 1.1 ] .
The case \(d_A\le d_B\) is Lemma 1.1.2. The case \(d_B{\lt}d_A\) reduces to it by applying the lemma to \(\psi \) with its two factors exchanged, then exchanging the two resulting bases back. This matches Wolf’s proof sketch: a change of local bases in the coefficient matrix corresponds to left and right multiplication by two unitaries, so the Schmidt coefficients are its singular values.
For any Schmidt decomposition of \(\psi \) with coefficients \(\{ \lambda _j\} \), the number of nonzero \(\lambda _j\) equals \(\operatorname{SR}(\psi )\). This is [ Wol12 , Proposition 1.1 ] .
The Schmidt decomposition factors the coefficient matrix of \(\psi \) through the isometries with columns \(\{ e_j\} \) and \(\{ f_j\} \), so its rank equals the rank of the diagonal matrix of Schmidt coefficients, which is the number of nonzero entries.
1.2 Quantum steering
A purification of a density operator lets an observer on the purifying system remotely prepare, or “steer,” any ensemble consistent with the reduced state.
A vector \(\psi \in \mathbb {C}^{d_A}\otimes \mathbb {C}^{d_B}\) purifies a density operator \(\rho \in M_{d_A}(\mathbb {C})\) if \(\operatorname{tr}_B[|\psi \rangle \! \langle \psi |]=\rho \).
A convex decomposition of a density operator \(\rho \in M_{d_A}(\mathbb {C})\) is a finite family of nonnegative weights \(\lambda _i\) summing to one together with density operators \(\rho _i\in M_{d_A}(\mathbb {C})\) satisfying \(\rho =\sum _i\lambda _i\rho _i\).
For a bipartite vector \(\psi \in \mathbb {C}^{d_A}\otimes \mathbb {C}^{d_B}\) with coefficient matrix \(C\) and any linear map \(T\) with domain \(M_{d_B}(\mathbb {C})\),
Tracing out the second factor of \((\operatorname{id}\otimes T)(|\psi \rangle \! \langle \psi |)\) applies \(T\) to each rank-one block \(|a\rangle \! \langle b|\mapsto C_{i,a}\, \overline{C_{j,b}}\) of the coefficient matrix and takes its trace; the trace-pairing identity \(\operatorname{tr}[T(M)]=\operatorname{tr}[M\, T^*(\mathbb {1})]\) turns this into the displayed bilinear form in the rows of \(C\). Wolf derives this identity for the canonical purification \(\psi =(\sqrt\rho \otimes \mathbb {1})|\Omega \rangle \); the argument here uses only the coefficient matrix of \(\psi \) and linearity, so it holds for an arbitrary purification.
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) ] .
Write \(C\) for the coefficient matrix of \(\psi \), so \(C\, C^\dagger =\rho \), and let \(C^+:=C^\dagger \rho ^+\) using the generalized inverse of \(\rho \) on its support. Then \(C\, C^+\) is the support projection of \(\rho \), and \(C^+C\) is an orthogonal projection on \(\mathbb {C}^{d_B}\) (the subspace of \(\mathbb {C}^{d_B}\) needed to purify \(\rho \)). For each \(i\), the support projection of \(\rho \) absorbs \(\lambda _i\rho _i\) on both sides (its range lies in the range of \(\rho \)), so \(C\, (C^+(\lambda _i\rho _i)(C^+)^\dagger )\, C^\dagger =\lambda _i\rho _i\). Transposing \(C^+(\lambda _i\rho _i)(C^+)^\dagger \) gives an effect operator \(E_i\) with \(C\, E_i^{\mathsf T}\, C^\dagger =\lambda _i\rho _i\); adding the transpose of the orthogonal complement of \(C^+C\) to one fixed \(E_{i_0}\) keeps this identity (the complement contributes zero under \(C(\cdot )C^\dagger \)) while making \(\sum _i E_i=\mathbb {1}\). Taking \(T_i\) to be the Lüders instrument \(\sigma \mapsto \sqrt{E_i}\, \sigma \, \sqrt{E_i}\) gives \(T_i^*(\mathbb {1})=E_i\), so \(\{ T_i\} \) is an instrument, and Theorem 1.2.3 recovers the displayed identity.
This route proves the proposition directly for an arbitrary purification and an arbitrary convex decomposition, in place of Wolf’s route through the canonical purification \(\psi =(\sqrt\rho \otimes \mathbb {1})|\Omega \rangle \) and the transposition operator \(\theta \) with respect to its Schmidt basis.
1.3 Maximal weight in convex decomposition
Given a density operator \(\rho \) and a candidate term \(\rho _1\), how large a weight can \(\rho _1\) carry in a convex decomposition of \(\rho \)?
For density operators \(\rho ,\rho _1\) on the same space and \(c\in \mathbb {R}\), \(\rho \) has a convex decomposition with \(\rho _1\) carrying weight \(c\) when \(\rho =\sum _i\lambda _i\rho _i\) for some finite index set, with every \(\rho _i\) a density operator, every \(\lambda _i\ge 0\), \(\sum _i\lambda _i=1\), and some index \(i_1\) with \(\lambda _{i_1}=c\), \(\rho _{i_1}=\rho _1\).
Let \(A\), \(B\) be positive semidefinite operators on the same space with \(\ker (A)\subseteq \ker (B)\), and let \(c{\gt}0\). Then
where the inverse is taken on the range of \(A\).
Write \(P\) for the support projection of \(A\). The kernel inclusion compresses \(B\) onto the range of \(A\), so \(A^{-1/2}BA^{-1/2}\) lies entirely inside \(\mathrm{ran}(P)\), and conjugating by \(A^{-1/2}\) and by \(A^{1/2}\) give mutually inverse congruences
so \(A-cB\ge 0\) iff \(P-c\, A^{-1/2}BA^{-1/2}\ge 0\). Since \(A^{-1/2}BA^{-1/2}\) is supported on \(\mathrm{ran}(P)\), this last inequality is in turn equivalent to \(1-c\, A^{-1/2}BA^{-1/2}\ge 0\), which by the \(C^*\)-identity \(\| X\| _\infty \le 1\iff X\le 1\) (for \(X\ge 0\)) holds iff \(c\| A^{-1/2}BA^{-1/2}\| _\infty \le 1\).
If \(\rho \) has a convex decomposition \(\rho =\sum _i\lambda _i\rho _i\) giving \(\rho _1\) a positive weight \(c\), then \(\ker (\rho )\subseteq \ker (\rho _1)\).
If \(\rho v=0\) for some vector \(v\), every term of \(\rho =\sum _i\lambda _i\rho _i\) is positive semidefinite, so each summand’s quadratic form at \(v\) vanishes; in particular \(c\langle v,\rho _1v\rangle =0\), and since \(c{\gt}0\) and \(\rho _1\) is positive semidefinite, \(\rho _1v=0\).
Let \(\rho \) and \(\rho _1\) be two density operators acting on the same space and let \(c{\gt}0\). There is a convex decomposition of the form \(\rho =\sum _i\lambda _i\rho _i\) with \(\rho _1\) carrying weight \(c\) iff \(\ker (\rho )\subseteq \ker (\rho _1)\) and
where the inverse is taken on the range of \(\rho \). Wolf’s Proposition “Maximal weight in convex decomposition” [ Wol12 ] states this with the inverse form \(c\le \| \rho ^{-1/2}\rho _1 \rho ^{-1/2}\| _\infty ^{-1}\); the two are equivalent here because the kernel inclusion together with \(\operatorname{tr}(\rho _1)=1\) (so \(\rho _1\ne 0\)) forces \(\| \rho ^{-1/2}\rho _1\rho ^{-1/2}\| _\infty {\gt}0\), so dividing the displayed multiplicative inequality through by that norm is valid.
Necessity of the kernel inclusion is Lemma 1.3.3.
The existence of a decomposition with weight \(c\) on \(\rho _1\) is equivalent to \(\rho -c\rho _1\ge 0\): given the kernel inclusion, the remaining terms of the decomposition sum to \(\rho -c\rho _1\), itself a positive-semidefinite combination; conversely, when \(\rho -c\rho _1\ge 0\) its trace is \(1-c\ge 0\), so either \(c=1\) and \(\rho =\rho _1\) (a zero-trace positive-semidefinite matrix vanishes), or \(c{\lt}1\) and \(\sigma :=(1-c)^{-1}(\rho -c\rho _1)\) is a density operator with \(\rho =c\rho _1+(1-c)\sigma \).
Finally \(\rho -c\rho _1\ge 0\) is equivalent to \(c\| \rho ^{-1/2}\rho _1\rho ^{-1/2}\| _\infty \le 1\) by Lemma 1.3.2, applied with \(A=\rho \), \(B=\rho _1\) under the kernel inclusion.
1.4 Operator norms in orthonormal coordinates
Let \(E\) and \(F\) be inner-product spaces over \(\mathbb K=\mathbb {R}\) or \(\mathbb {C}\), with finite orthonormal bases \((e_j)_{j\in J}\) and \((f_i)_{i\in I}\). For a continuous linear map \(T:E\to F\), let \(M_{ij}=\langle f_i,Te_j\rangle \) be its matrix in these bases. The Euclidean operator norm of \(M\) equals the operator norm of \(T\): \(\| M\| _{2\to 2}=\| T\| \). The two basis index sets may have different cardinalities, and either may be empty.
Let \(U:E\to \mathbb K^J\) and \(V:F\to \mathbb K^I\) be the coordinate isometries. The linear map represented by \(M\) is \(VTU^{-1}\). Composition on either side with a surjective linear isometry preserves the operator norm, so \(\| M\| _{2\to 2}=\| VTU^{-1}\| =\| T\| \).
1.5 Quadratic forms under compression and relabelling
The following elementary facts pass from operator inequalities to the expectations of fixed vectors, as in the metric comparisons of [ Ope26b , Section 8 ] . For complex matrices, \(A\leq B\) means that \(B-A\) is positive semidefinite. All index sets below are finite and may be empty.
Let \(A,B\) be complex matrices indexed by the same finite set. If \(A\leq B\), then every complex vector \(v\) on that set satisfies \(\operatorname {Re}(v^*Av)\leq \operatorname {Re}(v^*Bv)\).
Positivity of \(B-A\) gives \(0\leq \operatorname {Re}(v^*(B-A)v)\); expanding this expression gives the conclusion.
Let \(P,M\) be complex matrices indexed by the same finite set, with \(P=P^*\). Suppose that \(c\in \mathbb R\), \(PMP\leq cP\), and \(Pv=v\). Then \(\operatorname {Re}(v^*Mv)\leq c\operatorname {Re}(v^*v)\).
Since \(Pv=v\) and \(P=P^*\), one has \(v^*P=v^*\). Consequently, \(v^*PMPv=v^*Mv\) and \(\operatorname {Re}(v^*(cP)v)=c\operatorname {Re}(v^*v)\). Apply Theorem 1.5.1 to \(PMP\leq cP\).
Let \(e\colon I\to J\) be a bijection of finite sets, let \(H\) be a complex matrix indexed by \(I\), and let \(w\in \mathbb C^J\). Define the matrix \(H^e\) by \(H^e_{a,b}=H_{e^{-1}(a),e^{-1}(b)}\). Then \(w^*H^e w=(w\circ e)^*H(w\circ e)\).
In the finite sum defining the left-hand side, substitute \(a=e(i)\) and \(b=e(j)\). This gives the right-hand side term by term.
1.6 Scalar lower bounds for commuting products
For complex matrices, \(A\le B\) means that \(B-A\) is positive semidefinite. The following estimates combine scalar lower bounds for commuting band metrics in the replica construction of [ Ope26b ] . All matrices are indexed by the same finite set \(I\), which may be empty.
Let \(A,B\) be complex matrices indexed by \(I\), and let \(a,b\in \mathbb R\). Suppose that \(a\ge 0\), \(b\ge 0\), \(aI\le A\), \(bI\le B\), and \(AB=BA\). Then \((ab)I\le AB\).
The bound \(B\ge bI\ge 0\) makes \(B\) positive semidefinite. The positive semidefinite matrices \(A-aI\) and \(B\) commute, so their product is positive semidefinite. Since \(a\ge 0\), the identity
gives the conclusion.
Let \((A_0,\ldots ,A_{m-1})\) be any finite list of complex matrices indexed by \(I\), and let \(b\ge 0\). Suppose that \(bI\le A_j\) for every position \(j\) and that the matrices at distinct positions commute. Then \(b^m I\le A_0\cdots A_{m-1}\). Repeated matrices are allowed. For the empty list, the product is \(I\) and \(b^0=1\), including when \(b=0\).
Induct on the length of the list. The empty product gives equality. Write a nonempty list as \(A\) followed by a list of length \(m\) whose ordered product is \(B\). The induction hypothesis gives \(b^m I\le B\). Since \(A\) commutes with every factor of \(B\), it commutes with \(B\). Theorem 1.6.1, applied to the scalar bounds \(bI\le A\) and \(b^m I\le B\), gives \(b^{m+1}I\le AB\).
1.7 Order of tensor products
For complex matrices, \(A\le B\) means that \(B-A\) is positive semidefinite. The index sets in this section are finite and may be empty. These elementary comparisons give the tensor-order step in the sharp upper comparison of [ Ope26b , Section 8 ] .
Let \(A,B\) be complex matrices indexed by \(I\), and let \(C,D\) be complex matrices indexed by \(J\). Suppose that \(A\le B\), \(C\le D\), \(B\ge 0\), and \(C\ge 0\). Then \(A\otimes C\le B\otimes D\). The matrix \(A\) need not be positive semidefinite.
Since \(B\) and \(C\) are positive semidefinite, the linear maps \(X\mapsto X\otimes C\) and \(Y\mapsto B\otimes Y\) are positive and hence preserve order. Consequently,
Let \(N\ge 0\). For each \(0\le i{\lt}N\), let \(I_i\) be a finite set and let \(A_i,B_i\) be complex matrices indexed by \(I_i\). If \(0\le A_i\le B_i\) for every \(i\), then
The matrices act on \(\bigotimes _{i=0}^{N-1}\mathbb C^{I_i}\). When \(N=0\), both products are the identity on the one-dimensional empty tensor product.
Induct on \(N\). The empty product gives equality. For the induction step, write \(A'\) and \(B'\) for the products of all factors after the first. The induction hypothesis gives \(A'\le B'\), and positivity of tensor products gives \(A'\ge 0\). Since \(B_0\ge A_0\ge 0\), Theorem 1.7.1 gives \(A_0\otimes A'\le B_0\otimes B'\). Separating the first coordinate identifies these two matrices with the original tensor products.
1.8 Positive shifts of density matrices
Let \(I\) be a finite index set and write \(\mathcal D_I=\{ x\in \mathbb {C}^{I\times I}:x\ge 0,\ \operatorname{tr}(x)=1\} \). All norms in this section are operator norms for the Euclidean norm on \(\mathbb {C}^I\). For \(b{\gt}0\), real powers of \(x+b\mathbb {1}\) are taken on the full space.
The set \(\mathcal D_I\) is compact. If \(I\) is nonempty, then \(\mathcal D_I\) is nonempty.
Choose a bijection from \(\{ 0,\ldots ,|I|-1\} \) to \(I\). Reindexing rows and columns preserves positive semidefiniteness and trace, and is continuous. Thus \(\mathcal D_I\) is the continuous image of the compact set of density matrices in dimension \(|I|\) from Theorem 8.17.1. If \(I\) is nonempty, then \(|I|{\gt}0\), and reindexing a density matrix supplied by Theorem 8.17.3 proves existence.
If \(x\ge 0\) and \(b{\gt}0\), then \(x+b\mathbb {1}\) is positive definite.
The matrix \(b\mathbb {1}\) is positive definite. Adding the positive semidefinite matrix \(x\) preserves positive definiteness.
For every fixed \(b{\gt}0\) and \(r\in \mathbb {R}\), the function \(x\mapsto (x+b\mathbb {1})^r\) is continuous on the positive semidefinite cone.
The affine map \(x\mapsto x+b\mathbb {1}\) is continuous and, by Theorem 1.8.2, takes the positive semidefinite cone into the strictly positive matrices. Continuous functional calculus makes \(y\mapsto y^r\) continuous on that set. Composing these two maps proves the assertion.
If \(x\in \mathcal D_I\) and \(b{\gt}0\), then \(\operatorname{spec}(x+b\mathbb {1})\subseteq [b,1+b]\).
Positivity and the trace condition give \(\| x\| \le \operatorname{tr}(x)=1\), hence \(0\le x\le \mathbb {1}\). Adding \(b\mathbb {1}\) yields \(b\mathbb {1}\le x+b\mathbb {1}\le (1+b)\mathbb {1}\). The matrix \(x+b\mathbb {1}\) is Hermitian by Theorem 1.8.2, so these order inequalities give the corresponding lower and upper bounds on its spectrum.
Let \(x\in \mathcal D_I\), \(b{\gt}0\), and \(r\in \mathbb {R}\). If \(r\le 0\), then \(\| (x+b\mathbb {1})^r\| \le b^r\). If \(r\ge 0\), then \(\| (x+b\mathbb {1})^r\| \le (1+b)^r\). These bounds include \(r=0\) and singular density matrices \(x\).
The norm estimate for continuous functional calculus bounds \(\| (x+b\mathbb {1})^r\| \) by any uniform bound on \(|t^r|\) over \(t\in \operatorname{spec}(x+b\mathbb {1})\). By Theorem 1.8.4, \(b\le t\le 1+b\). The function \(t\mapsto t^r\) is positive there, decreasing for \(r\le 0\) and increasing for \(r\ge 0\). Its respective bounds are therefore \(b^r\) and \((1+b)^r\).
1.9 Spectral truncation of shifted powers
Let \(I\) be a finite index set. All matrices in this section act on \(\mathbb {C}^I\), and all norms are Euclidean operator norms. The rank is measured on this space, before tensoring with an identity on any other space.
For a Hermitian matrix \(A\) and \(b\in \mathbb {R}\), let \(\Pi =\mathbf1_{[b,\infty )}(A)\) be the spectral projection onto the eigenvalues at least \(b\). An eigenvalue equal to \(b\) is included.
Let \(A\) be Hermitian and \(b\in \mathbb {R}\). The projection \(\Pi \) is the functional calculus of \(\mathbf1_{[b,\infty )}\) at \(A\), and satisfies \(\Pi ^*=\Pi \), \(\Pi ^2=\Pi \), and \(A\Pi =\Pi A\).
The spectrum of \(A\) is finite, so its indicator is continuous on that spectrum. The functional calculus preserves real-valued functions and products. The scalar identities \(\mathbf1_{[b,\infty )}^2= \mathbf1_{[b,\infty )}\) and \(t\mathbf{1}_{\lbrack b,\infty )}(t)=\mathbf{1}_{\lbrack b,\infty )}(t)t\) give the assertions.
Let \(A\ge 0\), \(b{\gt}0\), and \(r\in \mathbb {R}\). Write \(F=(A+b\mathbb {1})^r\) and define
If \(A\ge 0\), \(b{\gt}0\), and \(r\in \mathbb {R}\), then \(F=f(A)\) for \(f(t)=(t+b)^r\), and \(F\Pi =\Pi F\).
The matrix \(A+b\mathbb {1}\) is positive definite by Theorem 1.8.2. Composing the functional calculus for \(t\mapsto t+b\) and \(t\mapsto t^r\) gives \(F=f(A)\). Both \(F\) and \(\Pi \) are functions of \(A\), so they commute.
Distributivity gives \(F\Pi +F(\mathbb {1}-\Pi )=F\).
Let \(A\ge 0\), \(b{\gt}0\), and \(r\in \mathbb {R}\). Then \(H=h(A)\) and \(T=t(A)\), where
Multiplication in the functional calculus gives \(h(\lambda )=(\lambda +b)^r\mathbf1_{[b,\infty )}(\lambda )\). Subtraction gives \(t(\lambda )=(\lambda +b)^r-h(\lambda )\), yielding the two cases.
For \(A\ge 0\), \(b{\gt}0\), and every \(r\in \mathbb {R}\), one has \(H\ge 0\), \(T\ge 0\), and \(HT=TH=0\).
If \(A\ge 0\), \(b{\gt}0\), and \(r\in \mathbb {R}\), then \((A+b\mathbb {1})^r(A+b\mathbb {1})^{-r}=\mathbb {1}\).
Strict positivity of \(A+b\mathbb {1}\) permits multiplication of its real powers. The exponents add to zero.
For \(A\ge 0\), \(b{\gt}0\), and \(r\in \mathbb {R}\), the head satisfies \(\operatorname {ran}H=\operatorname {ran}\Pi \) and \(\operatorname {rank}H=\operatorname {rank}\Pi \).
Commutation gives \(H=\Pi F\), so \(\operatorname {ran}H\subseteq \operatorname {ran}\Pi \). The inverse identity yields \(\Pi =H(A+b\mathbb {1})^{-r}\), proving the reverse inclusion. Taking dimensions gives equality of ranks.
For every \(A\ge 0\) and every real \(b\), \(b\operatorname {rank}\Pi \le \operatorname {Re}\operatorname{tr}A\).
The rank of an orthogonal projection equals its trace. If \(\lambda _i\) are the eigenvalues of \(A\), then
Each scalar inequality follows from \(b\le \lambda _i\) on the selected spectrum and \(0\le \lambda _i\) on its complement.
If \(A\ge 0\), \(\operatorname{tr}A=1\), \(b{\gt}0\), and \(r\in \mathbb {R}\), then \(\operatorname {rank}H\le b^{-1}\).
Substitute \(\operatorname {rank}H=\operatorname {rank}\Pi \) into the trace inequality and divide by \(b{\gt}0\).
If \(A\ge 0\), \(b{\gt}0\), and \(r\ge 0\), then \(\| T\| \le (2b)^r\). No trace normalization is required.
The tail function vanishes for \(\lambda \ge b\). For the remaining spectral values, \(0\le \lambda {\lt}b\), so \(0{\lt}\lambda +b{\lt}2b\) and \((\lambda +b)^r\le (2b)^r\) by monotonicity for \(r\ge 0\). The norm estimate for functional calculus gives the result.
For \(A\ge 0\), \(b{\gt}0\), and \(r\in \mathbb {R}\), consider the orthogonal decomposition \(\mathbb {C}^I=\operatorname {ran}\Pi \oplus \ker \Pi \). Let \(F_{\ge b}\) and \(F_{{\lt}b}\) be the restrictions of \(F\) to the two summands. The split has the form
The restriction \(F_{\ge b}\) is invertible on \(\operatorname {ran}\Pi \), which explains why truncating the power preserves the projection’s rank.
If \(I\) is empty, the matrices, ranges, and ranks are zero, and the stated identities and estimates remain valid. A trace-one matrix cannot occur in that case. The exponent \(r=0\) gives \(F=\mathbb {1}\), \(H=\Pi \), and \(T=\mathbb {1}-\Pi \).
1.10 Derivatives with respect to a scalar shift
Let \(I\) be a finite index set and let \(A\in \mathbb {C}^{I\times I}\) be positive semidefinite. Norms are Euclidean operator norms. The matrix \(A\) may be singular, and \(I\) may be empty.
For every \(b{\gt}0\) and \(r\in \mathbb {R}\), the function \(s\mapsto (A+s\mathbb {1})^r\) has derivative \(r(A+b\mathbb {1})^{r-1}\) at \(b\).
Fix a unitary eigenbasis \(A=U\operatorname {diag}(\lambda _i)U^*\). For \(s{\gt}0\), the shifted power is \(U\operatorname {diag}((\lambda _i+s)^r)U^*\). Each scalar coordinate has derivative \(r(\lambda _i+b)^{r-1}\) because \(\lambda _i+b{\gt}0\). The maps taking diagonal entries to a matrix and multiplying by the fixed matrices \(U\) and \(U^*\) are continuous linear maps. Applying them to the coordinate derivatives gives the claimed derivative.
For every \(R,r\in \mathbb {R}\), the function \(s\mapsto (A+e^{-s}\mathbb {1})^r\) has derivative \(-re^{-R}(A+e^{-R}\mathbb {1})^{r-1}\) at \(R\).
The chain rule and \(\frac{d}{dR}e^{-R}=-e^{-R}\) give the assertion.
For every \(b{\gt}0\), one has \(\| b(A+b\mathbb {1})^{-1}\| \le 1\).
The matrix is the functional calculus of \(t\mapsto b/(t+b)\) at \(A\). Every eigenvalue \(t\) of \(A\) is nonnegative, so \(0\le b/(t+b)\le 1\). The operator norm bound follows from this scalar bound on the spectrum.
1.11 Positive regulators and spectral tails
Let \(I\) be a finite index set. Matrix inequalities refer to the order defined by positive semidefiniteness. The following estimates give the operator and trace consequences used in [ Ope26a , Section 4, regulator estimate ] .
Let \(Q\in \mathbb {C}^{I\times I}\) be an orthogonal projection, \(b\in \mathbb {R}\), and \(T,\rho \in \mathbb {C}^{I\times I}\). If \(T\le b\mathbb {1}\) and \(T\le \rho \), then
No commutation with \(Q\) is required.
Positivity of the two trace pairings gives
Adding these inequalities proves the claim.
Suppose \(x,\rho \in \mathbb {C}^{I\times I}\) are positive semidefinite and commute. Let \(b{\gt}0\) and assume \(\rho \le x+b\mathbb {1}\). Then the matrix \(T=b\rho (x+b\mathbb {1})^{-1}\) satisfies \(0\le T\le b\mathbb {1}\) and \(T\le \rho \).
Set \(K=x+b\mathbb {1}\). The matrices \(K\) and \(K^{-1}\) are positive definite, and \(\rho \) and \(x\) commute with \(K^{-1}\). Thus \(T\ge 0\), and
Each product on the right consists of commuting positive matrices.
Let \(x,\rho \in \mathbb {C}^{I\times I}\) be commuting positive semidefinite matrices, \(Q\) an orthogonal projection, and \(C,u,R\in \mathbb {R}\). Assume \(u\ge 0\), \(R\ge (C+\log 8)(1+u)\), \(\rho \le x+e^{-R}\mathbb {1}\), \(\operatorname{rank}Q\le e^{C(1+u)}\), and \(\operatorname{tr}((\mathbb {1}-Q)\rho )\le 1/8\). Then \(T=e^{-R}\rho (x+e^{-R}\mathbb {1})^{-1}\) satisfies \(\operatorname{tr}T\le 1/4\).
The preceding two theorems give
Since \(\log 8\ge 0\) and \(u\ge 0\), the radius assumption implies \(-R+C(1+u)\le -\log 8\). The exponential term is therefore at most \(1/8\).
Let \(X_1\subseteq \cdots \subseteq X_m\) be regions of a finite system, let \(a_1,\dots ,a_m{\gt}0\) and \(b{\gt}0\), and let \(\Omega \) be a vector. For density matrices \(x_j\) on \(X_j\) put \(L_j=(x_j+b\mathbb {1})^{-a_j/2}\), extended by the identity, and \(M=L_m\cdots L_1\). Suppose \((x_1,\dots ,x_m)\) minimizes \(\lVert M\Omega \rVert \) among all such tuples of density matrices. Let \(\rho _j\) be the state on \(X_j\) of the normalized vector \(\phi =M\Omega /\lVert M\Omega \rVert \), with \(\phi =0\) when \(M\Omega =0\). Then for every \(j\)
Consequently the regulator \(b\rho _j(x_j+b\mathbb {1})^{-1}\) satisfies the bounds of Theorem 1.11.2. This is the descending stationarity step in the proof of [ Ope26a , Proposition 4.1 ] .
Proceed from the largest region inward. When the filters \(L_k\) with \(k{\gt}j\) already commute with \(\rho _k\), conjugating \(x_j\) by a unitary keeps it a density matrix, and Lemma 21.5.8 applied to the minimum shows that \(L_j\), and hence \(x_j\), commutes with \(\rho _j\).
For the order, set \(G=\rho _j(x_j+b\mathbb {1})^{-1}\), which is positive and commutes with \(x_j\). For a density matrix \(y\) commuting with \(x_j\), the segment \(x_j+s(y-x_j)\) stays feasible for \(0\le s\le 1\). Its one-sided derivative at \(s=0\), after the outer filters are removed by the descending trace argument, is \(-\tfrac {a_j}{2}\operatorname {Re}\operatorname{Tr}(G(y-x_j))\), so minimality gives \(\operatorname {Re}\operatorname{Tr}(Gy)\le \lambda :=\operatorname {Re}\operatorname{Tr}(Gx_j)\). Testing \(y\) on rank-one projections of a common eigenbasis of \(G\) and \(x_j\) (Lemma 21.5.9) gives \(G\le \lambda \mathbb {1}\). Since \(\lambda =\operatorname{Tr}\rho _j-b\operatorname{Tr}G\le 1\), we get \(\rho _j\le \lambda (x_j+b\mathbb {1})\le x_j+b\mathbb {1}\).
1.12 Complete positivity from positivity
For a linear map \(T:M_D(\mathbb {C})\to M_D(\mathbb {C})\), write \(T^*\) for its trace-pairing adjoint (Definition 2.5.1). Complete positivity is in general strictly stronger than positivity, but the two coincide once \(T\)’s range, or its adjoint’s range, is commutative.
A linear map \(T:M_D(\mathbb {C})\to M_D(\mathbb {C})\) has commutative range if \(T(X)\) and \(T(Y)\) commute for all \(X,Y\in M_D(\mathbb {C})\). Since a family of pairwise-commuting elements generates a commutative subalgebra, this is the statement that \(T\) maps into a commutative subalgebra of \(M_D(\mathbb {C})\).
Let \(T:M_D(\mathbb {C})\to M_D(\mathbb {C})\) be a positive linear map and let \([a_{ij}]_{i,j=1}^n\) be a positive semidefinite block matrix with entries in \(M_D(\mathbb {C})\) such that the images \(T(a_{ij})\) pairwise commute. Then
for every family of vectors \(\{ \psi _i\} _{i=1}^n\subset \mathbb {C}^D\).
As the \(T(a_{ij})\) mutually commute, choose a basis in which they are simultaneously diagonal, and write \(\psi _{j,\beta }\) for the components of \(|\psi _j\rangle \) in this basis. Then
by positivity of \(T\) and of \([a_{ij}]\): the bracketed matrix is \(B_\beta ^\dagger [a_{ij}]B_\beta \) for the column vector \(B_\beta =(\psi _{j,\beta })_j\), hence positive semidefinite.
Let \(T:M_D(\mathbb {C})\to M_D(\mathbb {C})\) be a positive linear map with commutative range. Then \(T\) is completely positive.
By Theorem 3.1.10 it suffices to show that the Choi matrix \(\tau \) of \(T\) satisfies \(\langle \phi |\tau |\phi \rangle \ge 0\) for every \(\phi \in \mathbb {C}^D\otimes \mathbb {C}^D\). Let \(a_{ij}\in M_D(\mathbb {C})\), for \(i,j\in \{ 1,\dots ,D\} \), be the blocks of \(|\Omega \rangle \! \langle \Omega |\), the Choi matrix of the identity map (Theorem 3.1.6), and let \(\psi _k:=\phi (\cdot ,k)\in \mathbb {C}^D\) be the reshaping of \(\phi \) into a \(D\)-indexed family of vectors. Then
the block quadratic form of Lemma 1.12.2 at \(T\), \([a_{ij}]\), and \(\{ \psi _i\} \). Since \([a_{ij}]\ge 0\) (it is the block family of the rank-one projector \(|\Omega \rangle \! \langle \Omega |\)) and, by commutative range, the images \(T(a_{ij})\) pairwise commute, Lemma 1.12.2 gives \(\langle \phi |\tau |\phi \rangle \ge 0\).
If \(T:M_D(\mathbb {C})\to M_D(\mathbb {C})\) is positive, so is \(T^*\).
For \(X\ge 0\), \(T^*(X)\) is Hermitian since \(T\) preserves the adjoint. For every \(B\ge 0\), the trace-pairing identity \(\operatorname{tr}[T^*(X)B]=\operatorname{tr}[X\, T(B)]\) and positivity of \(X\) and of \(T(B)\) (as \(T\) is positive) give \(\operatorname{tr}[T^*(X)B]\ge 0\); a Hermitian matrix with nonnegative trace pairing against every positive semidefinite matrix is itself positive semidefinite, so \(T^*(X)\ge 0\).
A linear map \(T:M_D(\mathbb {C})\to M_D(\mathbb {C})\) is completely positive if and only if \(T^*\) is.
Complete positivity in Kraus form is preserved by the trace-pairing adjoint, and the adjoint is an involution, so the two directions are the same statement applied to \(T\) and to \(T^*\).
Let \(T:M_D(\mathbb {C})\to M_D(\mathbb {C})\) be a positive linear map whose trace-pairing adjoint \(T^*\) has commutative range. Then \(T\) is completely positive.
Let \(T:M_D(\mathbb {C})\to M_D(\mathbb {C})\) be a positive linear map. If \(T\) has commutative range, or \(T^*\) has commutative range, then \(T\) is completely positive. This is [ Wol12 , Proposition 1.6 ] ; the commutative-range hypothesis is the range-only reading of Wolf’s “\(\mathcal A\) or \(\mathcal B\) commutative” hypothesis that his own proof establishes and that his remark immediately after the proof licenses for an arbitrary operator system mapping into a commutative algebra. See [ con26m ] for the precise relationship between the two phrasings.
1.13 Partial trace over subsystems
The following support-projector absorption theorems and the partial trace over the left factor of a bipartite space are generic bipartite-operator facts, with no tensor-network content; they are relocated here from the parent-Hamiltonian marginal-support chapter (the tripartite-lift content stays in TNLean and cites them across the chapter boundary).
The next two adjoint identities and the positivity statement are likewise generic bipartite-operator facts, relocated here from the same chapter for the same reason.
For every operator \(M\) on \(H_L\) and every operator \(\rho \) on \(H_L\otimes H_R\), one has \(\operatorname{tr}((M\otimes \mathbb {1}_R)\rho )=\operatorname{tr}(M\operatorname{tr}_R\rho )\).
Expand both traces in product bases and sum first over the identity entry on \(H_R\).
For finite-dimensional \(H_L\) and \(H_R\), if \(\rho \geq 0\), then \(\operatorname{tr}_L\rho \geq 0\).
If \(\rho ^{(k)}\) is the principal submatrix indexed by \(\{ k\} \times H_R\), then
Each \(\rho ^{(k)}\) is positive semidefinite, and hence so is their finite sum.
For every operator \(M\) on \(H_R\), one has \(\operatorname{tr}((\mathbb {1}_L\otimes M)\rho )=\operatorname{tr}(M\operatorname{tr}_L\rho )\).
Expanding in product bases gives
Let \(\rho \) be positive semidefinite on \(H_L\otimes H_R\), and let \(P_L\) be the orthogonal projector onto the support of \(\operatorname{tr}_R\rho \). Then \((P_L\otimes \mathbb {1}_R)\rho =\rho \).
Set \(Q_L=\mathbb {1}_L-P_L\). Since \(Q_L\operatorname{tr}_R\rho =0\), the adjoint identity gives
Positivity of \(\rho \) then implies \((Q_L\otimes \mathbb {1}_R)\rho =0\). Finally, \(\mathbb {1}_{L\otimes R}-(Q_L\otimes \mathbb {1}_R)=P_L\otimes \mathbb {1}_R\) gives the claim.
Under the same assumptions, \(\rho (P_L\otimes \mathbb {1}_R)=\rho \). For \(H_L=H_A\otimes H_X\) and \(H_R=H_B\), the two absorption identities are precisely the \(P_{AX}\) support-projector identities in [ CPGSV16 , Appendix D.2, lines 2228–2235 ] .
Apply \((\cdot )^\dagger \) to the left absorption identity. Since \(\rho ^\dagger =\rho \) and \((P_L\otimes \mathbb {1}_R)^\dagger =P_L\otimes \mathbb {1}_R\),
For an operator \(\rho \) on \(H_L\otimes H_R\), define \((\operatorname{tr}_L\rho )_{r,s}:=\sum _l\rho _{(l,r),(l,s)}\).
Let \(\rho \) be positive semidefinite. If \(P_R\) is the support projector of \(\operatorname{tr}_L\rho \), then \((\mathbb {1}_L\otimes P_R)\rho =\rho \).
Set \(Q=\mathbb {1}_R-P_R\). The preceding trace identity gives
since \(P_R\) is the support projector of \(\operatorname{tr}_L\rho \). Positivity of \(\rho \) then gives \((\mathbb {1}_L\otimes Q)\rho =0\), and hence \((\mathbb {1}_L\otimes P_R)\rho =\rho \).
Under the same assumptions, \(\rho (\mathbb {1}_L\otimes P_R)=\rho \).
Take adjoints in the left absorption identity.
Let \(\rho \) be a matrix on a finite-dimensional space \(B\). The state-preparation map from matrices on \(A\) to matrices on \(A\otimes B\) is
This is the elementary preparation operation used in the maps \(\mathcal T_1\) and \(\mathcal S_1\) of [ CPGSV16 , Appendix C.2, lines 1527–1533 and 1551–1555 ] . It is relocated here from the MPDO renormalization chapter; its Kraus-action and conditional-expectation consequences for the density-operator setting appear in Section 1.16 below.
1.14 Extending completely positive maps from operator systems
An operator system inside a finite-dimensional \(C^*\)-algebra is a subspace closed under the adjoint and containing the unit. Every positive map defined only on such a subspace, with values in a commutative algebra, is automatically completely positive by the remark following Theorem 1.12.7; extending it to a completely positive map on the whole algebra makes that observation useful for arbitrary target algebras as well.
The source states the extension theorem for an operator system inside any finite-dimensional \(C^*\)-algebra, that is, any direct sum of matrix algebras. What follows treats the one-summand case, where the ambient algebra is a single full matrix algebra \(M_{m}(\mathbb {C})\); see [ con26o ] .
For square matrices \(X\) on \(A\) and \(Y\) on \(B\), \(\operatorname{tr}_{A}(X\otimes Y)=\operatorname{tr}(X)Y\).
For all \(i,j\), \([\operatorname{tr}_{A}(X\otimes Y)]_{ij} =\sum _t X_{tt}Y_{ij}=\operatorname{tr}(X)Y_{ij}\).
For every natural number \(m\), the matrix \(m^{-1}\mathbb 1_m\) is positive semidefinite.
The identity matrix is positive semidefinite, and \(m^{-1}\) is nonnegative.
If \(m\geq 1\), then \(\operatorname{tr}(m^{-1}\mathbb 1_m)=1\).
Since \(\operatorname{tr}(\mathbb 1_m)=m\), scalar linearity of the trace gives the result.
Let \(m\geq 1\). The linear map \(E:M_{m}(\mathbb {C})\otimes M_{d}(\mathbb {C})\to M_{m}(\mathbb {C})\otimes M_{d}(\mathbb {C})\) defined by
is trace-preserving and completely positive. It is a conditional expectation onto the unital \(*\)-subalgebra \(\mathbb 1_m\otimes M_{d}(\mathbb {C})\): it is unital and idempotent, its range lies in that subalgebra, and it fixes the subalgebra pointwise. This is the one-block trace-preserving choice in the family described by [ Wol12 , Proposition 1.5 and Equation (1.40) ] ; it supplies the corresponding one-block codomain retraction for the extension theorem, not the general finite-dimensional codomain construction.
Interchange the two tensor factors, trace out the new right factor, prepare the maximally mixed state \(m^{-1}\mathbb 1_m\), and interchange the factors back. Theorem 1.14.2 and Theorem 1.14.3 show that the prepared matrix is positive semidefinite and has trace one. Theorem 1.16.8, Lemma 1.16.22, Lemma 1.16.27, and Lemma 1.16.9 then show that the resulting map is trace-preserving and completely positive. Expanding the composition gives
The Kronecker-product partial-trace identity gives
so \(E(\mathbb 1_m\otimes X)=\mathbb 1_m\otimes X\). Pointwise fixation gives both idempotence and unitality, while the evaluation identity gives the range inclusion.
Let \(K\) be finite, let \(m_k\geq 1\), and set
For \(A\in \operatorname{End}(\mathcal H)\), write \(A_{kk}\) for its \(k\)-th diagonal block. The map
has zero off-diagonal output blocks. It is trace-preserving and completely positive, and it is a conditional expectation onto the coordinate subalgebra
Thus its range lies in this subalgebra, it fixes the subalgebra pointwise, and it is unital and idempotent. This is the normalized trace-preserving coordinate choice obtained by specializing the block densities in [ Wol12 , Proposition 1.5 and Equation (1.40) ] to maximally mixed densities. Relative to Wolf’s \(M_{d_k}(\mathbb {C})\otimes \mathbb 1_{m_k}\) convention, the displayed \(\mathbb 1_{m_k}\otimes M_{d_k}(\mathbb {C})\) order is obtained by interchanging the tensor factors. This theorem concerns only the displayed direct-sum coordinates; it does not conjugate the subalgebra by an arbitrary unitary or corestrict the codomain. This coordinate restriction is documented in [ con26l ] .
Compress \(A\) to the family \((A_{kk})_{k\in K}\) and apply the normalized partial-trace expectation from Theorem 1.14.4 in each coordinate. The coordinatewise Kraus theorem makes the resulting map on the finite sum completely positive, and its canonical full-matrix extension discards the off-diagonal blocks. Each coordinate preserves trace, so preservation of the total block trace implies preservation of the trace on \(\operatorname{End}(\mathcal H)\). The displayed formula shows that the range lies in the coordinate subalgebra. The one-block fixation identity holds in every summand, hence \(E\) fixes that subalgebra pointwise; idempotence and unitality follow.
Let \(e:H\simeq I\) be an equivalence of finite index sets and let \(U\) be a unitary matrix indexed by \(I\). The mutually inverse coordinate changes
and
both preserve positive semidefiniteness.
Unitary conjugation preserves positive semidefiniteness, as does reindexing a matrix along an equivalence. Apply these facts in the two possible orders.
Let \(e:H\simeq I\) be an equivalence of finite index sets and let \(U\) be a unitary matrix indexed by \(I\). Define
and
Both \(\Phi \) and \(\Phi ^{-1}\) are trace-preserving and completely positive.
The forward map is the composition of conjugation by \(U^\dagger \) with reindexing by \(e^{-1}\), while the inverse is reindexing by \(e\) followed by conjugation by \(U\). Unitarity makes both single-Kraus conjugations trace preserving. Equivalence reindexing is trace preserving and completely positive, and composition preserves these properties.
For every unital \(*\)-subalgebra \(\mathcal B\subseteq M_{n}(\mathbb {C})\), there is a trace-preserving completely positive map \(E:M_{n}(\mathbb {C})\to M_{n}(\mathbb {C})\) that is a conditional expectation onto \(\mathcal B\): it is unital and idempotent, its range lies in \(\mathcal B\), and it fixes every element of \(\mathcal B\). This is the normalized trace-preserving choice in the family described by [ Wol12 , Proposition 1.5 and Equation (1.40) ] .
By the finite-dimensional block form, choose unitary coordinates in which
Let \(E_0\) be the coordinate direct-sum expectation and set
Theorem 1.14.7 and closure under composition show that \(E\) is trace-preserving and completely positive. The identities \(\Phi \Phi ^{-1}=\operatorname{id}\) and \(E_0^2=E_0\) give \(E^2=E\), and all three maps are unital. Finally, the block-membership equivalence identifies \(\Phi (\mathcal B)\) with the coordinate subalgebra. Thus the range of \(E\) lies in \(\mathcal B\), and the pointwise fixation of that coordinate subalgebra by \(E_0\) implies \(E(B)=B\) for every \(B\in \mathcal B\).
A subspace \(S\subseteq M_{m}(\mathbb {C})\) is an operator system if it contains the unit and is closed under the adjoint: \(\mathbb 1\in S\) and \(X\in S\implies X^\dagger \in S\).
Let \(S\subseteq M_{m}(\mathbb {C})\) be a subspace and \(T:S\to M_{n}(\mathbb {C})\) a linear map. For a natural number \(k\), write \(T\otimes \mathrm{id}_k\) for the map sending a \(k\times k\) block matrix with entries in \(S\) to the block matrix obtained by applying \(T\) to every entry. Then \(T\) is completely positive on \(S\) if \(T\otimes \mathrm{id}_k\) sends every positive semidefinite such block matrix to a positive semidefinite matrix, for every \(k\) (the condition is vacuous at \(k=0\)). This makes no reference to \(S\) being an operator system; it is applied below to an operator system \(S\) because that is the situation in which the extension theorem holds.
For the rest of this section fix an operator system \(S\subseteq M_{m}(\mathbb {C})\) (Definition 1.14.9) and a linear map \(T:S\to M_{n}(\mathbb {C})\) completely positive on \(S\) (Definition 1.14.10), and write
for Wolf’s functional, with \(\Omega \) the maximally entangled vector of dimension \(n\).
For every Hermitian \(A\in S\otimes M_{n}(\mathbb {C})\),
The matrix \(\| A\| _\infty \mathbb 1-A\) is positive semidefinite and lies in \(S\otimes M_{n}(\mathbb {C})\), so complete positivity of \(T\) at level \(n\) gives that its image under \(T\otimes \mathrm{id}_n\) is positive semidefinite. Testing against \(\Omega \) and using linearity of \(\tau \),
both sides are real since \(\| A\| _\infty \mathbb 1-A\) is Hermitian, and rearranging gives the bound.
There is a \(\mathbb {C}\)-linear functional \(\tau '\) on \(M_{m}(\mathbb {C})\otimes M_{n}(\mathbb {C})\) agreeing with \(\tau \) on \(S\otimes M_{n}(\mathbb {C})\) and satisfying \(\operatorname{Re}\tau '(A)\le \| A\| _\infty \, \operatorname{Re}\tau '(\mathbb 1)\) for every Hermitian \(A\in M_{m}(\mathbb {C})\otimes M_{n}(\mathbb {C})\).
The functional \(f(A):=\| A\| _\infty \, \tau (\mathbb 1)\) is sublinear, so by Lemma 1.14.11 the Hahn–Banach theorem extends \(\tau \), dominated by \(f\), from the Hermitian matrices of \(S\otimes M_{n}(\mathbb {C})\) to those of \(M_{m}(\mathbb {C})\otimes M_{n}(\mathbb {C})\) as a real-linear functional. Decomposing each matrix of \(M_{m}(\mathbb {C})\otimes M_{n}(\mathbb {C})\) into its Hermitian and skew-Hermitian parts, both of which stay inside \(S\otimes M_{n}(\mathbb {C})\) whenever the original matrix does (since \(S\) is closed under the adjoint), extends this further to a \(\mathbb {C}\)-linear functional \(\tau '\) agreeing with \(\tau \) on all of \(S\otimes M_{n}(\mathbb {C})\) and obeying the same bound throughout.
Let \(\varphi :M_{d}(\mathbb {C})\to \mathbb {C}\) be a \(\mathbb {C}\)-linear functional satisfying \(\operatorname{Re}\varphi (A)\le \| A\| _\infty \, \operatorname{Re}\varphi (\mathbb 1)\) for every Hermitian \(A\in M_{d}(\mathbb {C})\), with \(\operatorname{Re}\varphi (\mathbb 1)\ge 0\). Then \(\operatorname{Re}\varphi (A)\ge 0\) for every positive semidefinite \(A\in M_{d}(\mathbb {C})\).
The matrix \(\| A\| _\infty \mathbb 1-A\) is positive semidefinite (hence Hermitian) of norm at most \(\| A\| _\infty \) whenever \(A\) is positive semidefinite, so the domination hypothesis applied to it gives
since \(\operatorname{Re}\varphi (\mathbb 1)\ge 0\), rearranging gives \(\operatorname{Re}\varphi (A)\ge 0\).
Let \(\tau '\) be a \(\mathbb {C}\)-linear functional on \(M_{m}(\mathbb {C})\otimes M_{n}(\mathbb {C})\) built from a \(\mathbb {R}\)-linear functional by the Hermitian-decomposition construction of Lemma 1.14.12, and suppose \(\operatorname{Re}\tau '(X)\ge 0\) for every positive semidefinite \(X\in M_{m}(\mathbb {C})\otimes M_{n}(\mathbb {C})\). Let \(Y\) be the matrix representing \(\tau '\) through the trace pairing, \(\tau '(X)=\operatorname{tr}(YX)\). Then \(Y\) is positive semidefinite.
Because \(\tau '\) is built by the Hermitian-decomposition construction, it sends the adjoint of a matrix to the conjugate of its value, so \(Y\) is Hermitian. For a vector \(v\), the trace-pairing identity gives \(v^\dagger Yv=\tau '(vv^\dagger )\), and \(vv^\dagger \) is positive semidefinite, so the hypothesis gives \(\operatorname{Re}(v^\dagger Yv)\ge 0\); since \(Y\) is Hermitian, \(v^\dagger Yv\) is already real, hence nonnegative.
Let \(\tau '\) be as in Lemma 1.14.12 and let \(Y=\sum _kv_kv_k^\dagger \) be any rank-one decomposition of its Riesz matrix. Then the map \(T'(B)_{ij}:=n\cdot \tau '(B\otimes |i\rangle \langle j|)\) equals \(\sum _kK_kBK_k^\dagger \) for the reshaped operators \(K_k:\mathbb {C}^m\to \mathbb {C}^n\), \(K_k:=\sqrt n\, v_k^\dagger \) (viewing \(v_k\) as an \(m\times n\) matrix). Complete positivity of \(T'\) is assembled from this identity together with the positive semidefiniteness of \(Y\) at Theorem 1.14.17.
Both sides expand, entry by entry, to the same double sum \(n\sum _k\sum _{p,q}\overline{v_k(p,i)}\, B_{pq}\, v_k(q,j)\).
For every \(B\in S\) and every \(i,j\),
Direct computation from the definition of \(\tau \), expanding the trace against \(\Omega \) and collapsing the resulting sum using the structure of \(B\otimes |i\rangle \langle j|\).
Let \(m,n\ge 1\), let \(S\subseteq M_{m}(\mathbb {C})\) be an operator system, and let \(T:S\to M_{n}(\mathbb {C})\) be a completely positive linear map on \(S\). Then there is a completely positive map \(T':M_{m}(\mathbb {C})\to M_{n}(\mathbb {C})\), in rectangular Kraus form, that agrees with \(T\) on \(S\).
By Lemma 1.14.12, \(\tau \) extends to a \(\mathbb {C}\)-linear functional \(\tau '\) on \(M_{m}(\mathbb {C})\otimes M_{n}(\mathbb {C})\) agreeing with \(\tau \) on \(S\otimes M_{n}(\mathbb {C})\) and obeying the same norm-domination bound throughout. By Lemma 1.14.13, \(\tau '\) is nonnegative on positive semidefinite matrices, so by Lemma 1.14.14 the matrix \(Y\) representing \(\tau '\) through the trace pairing is positive semidefinite, hence a sum of rank-one terms \(Y=\sum _kv_kv_k^\dagger \). Lemma 1.14.15 reshapes these into a rectangular Kraus family for \(T'(B)_{ij}:=n\cdot \tau '(B\otimes |i\rangle \langle j|)\), so \(T'\) is completely positive. Lemma 1.14.16 gives \(\tau (B\otimes |i\rangle \langle j|)=\tfrac 1n\, T(B)_{ij}\) for \(B\in S\), and \(\tau '\) agrees with \(\tau \) on \(S\otimes M_{n}(\mathbb {C})\), so \(T'\) agrees with \(T\) on \(S\).
1.15 Extending completely positive maps from operator systems in a direct sum of matrix algebras
The one-summand case above (Theorem 1.14.17) is generalized here to the source’s full domain generality: the domain ranges over every finite-dimensional \(C^*\)-algebra, that is, every finite direct sum of matrix algebras \(\bigoplus _{k{\lt}r}M_{d_k}(\mathbb {C})\). The generalization transports the one-summand theorem along the block-diagonal embedding of \(\bigoplus _{k{\lt}r}M_{d_k}(\mathbb {C})\) into a single matrix algebra \(M_M(\mathbb {C})\), \(M=\sum _kd_k\).
A subspace \(S\subseteq \bigoplus _{k{\lt}r}M_{d_k}(\mathbb {C})\) is an operator system if it contains the unit and is closed under the entrywise adjoint: \(\mathbb 1\in S\) and \(X\in S\implies X^\dagger \in S\), where \(X^\dagger \) denotes the family \((X_k^\dagger )_{k{\lt}r}\).
Let \(S\subseteq \bigoplus _{k{\lt}r}M_{d_k}(\mathbb {C})\) be a subspace and \(T:S\to M_{p}(\mathbb {C})\) a linear map. The level-\(j\) ampliation of \(T\) is defined by bipartite slicing: for a family \(X=(X_k)_{k{\lt}r}\) with \(X_k\in M_{d_k}(\mathbb {C})\otimes M_{j}(\mathbb {C})\) whose slice families lie in \(S\),
and \(T\) is completely positive on \(S\) if entrywise positivity \((\forall k{\lt}r,\ X_k\ge 0)\) always implies \((T\otimes \operatorname{id}_j)(X)\ge 0\), for every \(j\). This is Definition 1.14.10 applied block by block, one summand at a time.
A linear map \(T:\bigoplus _{k{\lt}r}M_{d_k}(\mathbb {C})\to M_{p}(\mathbb {C})\) is completely positive if, for every natural number \(j\) and every family \(X=(X_k)_{k{\lt}r}\) with \(X_k\in M_{d_k}(\mathbb {C})\otimes M_{j}(\mathbb {C})\), entrywise positivity implies
where the \((a,b)\) slice of \((T\otimes \operatorname{id}_j)(X)\) is \(T\) applied to the family of \((a,b)\) slices of the matrices \(X_k\). This is the direct-sum analogue of rectangular Kraus complete positivity (Definition 2.5.2).
Let \(\iota _j\) be the block-diagonal embedding
Then \(\iota _j(X)\) is positive semidefinite exactly when every \(X_k\) is positive semidefinite. Moreover, for all \(a,b{\lt}j\),
The first assertion follows from the corresponding block-diagonal positivity equivalence and invariance of positivity under simultaneous row and column reindexing. The second follows by evaluating both sides of (21) at a pair of ambient matrix indices.
If \(\Phi (X)=\sum _iA_iXA_i^\dagger \) is a rectangular Kraus completely positive map, then \(\Phi \otimes \operatorname{id}_j\) is a Kraus completely positive map with Kraus operators \(A_i\otimes \mathbb 1_j\). In particular, \((\Phi \otimes \operatorname{id}_j)(X)\) is positive semidefinite whenever \(X\) is.
Expanding matrix entries gives
Each summand preserves positive semidefiniteness, and so does their sum.
Let \(\iota \) be the block-diagonal embedding of \(\bigoplus _{k{\lt}r}M_{d_k}(\mathbb {C})\) into \(M_M(\mathbb {C})\), \(M=\sum _kd_k\). If \(S\subseteq \bigoplus _{k{\lt}r}M_{d_k}(\mathbb {C})\) is an operator system, then \(\iota (S)\) is an operator system in \(M_M(\mathbb {C})\); and if \(T:S\to M_{p}(\mathbb {C})\) is completely positive on \(S\), then the induced map \(T'':\iota (S)\to M_{p}(\mathbb {C})\), \(T''(\iota (X))=T(X)\), is completely positive on \(\iota (S)\).
The image of an operator system under an injective unital \(*\)-homomorphism is an operator system. For complete positivity, a level-\(j\) matrix over the image \(\iota (S)\) compresses blockwise to a family over \(S\): the block-diagonal compression left-inverts the embedding on every slice,
and each block is positive semidefinite by invariance of positivity under simultaneous row and column reindexing. Complete positivity of \(T\) on \(S\) applies to this family, and the linear equivalence of \(S\) and \(\iota (S)\) converts the conclusion back to \(T''\).
Let \(d_0,\dots ,d_{r-1}\) be natural numbers, not all zero, let \(p\ge 1\), let \(S\subseteq \bigoplus _{k{\lt}r}M_{d_k}(\mathbb {C})\) be an operator system (Definition 1.15.1), and let \(T:S\to M_{p}(\mathbb {C})\) be completely positive on \(S\) (Definition 1.15.2). Then there is a completely positive map \(T':\bigoplus _{k{\lt}r}M_{d_k}(\mathbb {C})\to M_{p}(\mathbb {C})\) (Definition 1.15.3) that agrees with \(T\) on \(S\).
Let \(\iota =\iota _1\) be the block-diagonal embedding. By Theorem 1.15.6, \(\iota (S)\) is an operator system in \(M_M(\mathbb {C})\) and the induced map \(T''\) is completely positive at every matrix level. Theorem 1.14.17 gives a rectangular Kraus extension \(\widetilde T:M_M(\mathbb {C})\to M_{p}(\mathbb {C})\) of \(T''\). Define \(T'=\widetilde T\circ \iota \).
To verify (19), fix \(j\) and an entrywise positive family \(X\). The level embedding \(\iota _j(X)\) is positive by Theorem 1.15.4. The Kraus ampliation theorem (Theorem 1.15.5) therefore gives \((\widetilde T\otimes \operatorname{id}_j)(\iota _j(X))\ge 0\). Using (21) identifies this matrix with \((T'\otimes \operatorname{id}_j)(X)\), proving complete positivity of \(T'\). The defining agreement of \(\widetilde T\) with \(T''\) shows that \(T'\) agrees with \(T\) on \(S\).
Let \(F:\bigoplus _{k{\lt}r}M_{d_k}(\mathbb {C})\to M_{p}(\mathbb {C})\) be completely positive and let \(E:M_{p}(\mathbb {C})\to M_{q}(\mathbb {C})\) have a Kraus representation. Then \(E\circ F\) is completely positive.
At each matrix level \(j\), the ampliations satisfy
If every component of \(X\) is positive semidefinite, complete positivity of \(F\) makes the inner matrix positive semidefinite. The Kraus representation of \(E\otimes \operatorname{id}_j\) then preserves its positivity.
Let \(d_0,\dots ,d_{r-1}\) be natural numbers, not all zero, let \(p\geq 1\), let \(S\subseteq \bigoplus _{k{\lt}r}M_{d_k}(\mathbb {C})\) be an operator system, and let \(\mathcal B\subseteq M_{p}(\mathbb {C})\) be a unital \(*\)-subalgebra. Suppose that \(T:S\to M_{p}(\mathbb {C})\) is completely positive and \(T(S)\subseteq \mathcal B\). Then there is a completely positive map \(G:\bigoplus _{k{\lt}r}M_{d_k}(\mathbb {C})\to M_{p}(\mathbb {C})\) such that \(G(\bigoplus _{k{\lt}r}M_{d_k}(\mathbb {C}))\subseteq \mathcal B\) and \(G|_S=T\).
Theorem 1.15.7 gives a completely positive extension \(F\) with values in \(M_{p}(\mathbb {C})\). Choose the conditional expectation \(E:M_{p}(\mathbb {C})\to M_{p}(\mathbb {C})\) from Theorem 1.14.8; its range lies in \(\mathcal B\). Set \(G=E\circ F\). Theorem 1.15.8 shows that \(G\) is completely positive, while \(E(M_{p}(\mathbb {C}))\subseteq \mathcal B\). For \(x\in S\), one has \(F(x)=T(x)\in \mathcal B\), and therefore \(G(x)=E(T(x))=T(x)\).
Let \(d_0,\dots ,d_{r-1}\) be natural numbers, not all zero, let \(p\geq 1\), let \(S\subseteq \bigoplus _{k{\lt}r}M_{d_k}(\mathbb {C})\) be an operator system, let \(\mathcal B\subseteq M_{p}(\mathbb {C})\) be a unital \(*\)-subalgebra, and let \(T:S\to \mathcal B\) be completely positive, where positivity in \(\mathcal B\) is read through its inclusion in \(M_{p}(\mathbb {C})\). Then there is a linear map
whose inclusion into \(M_{p}(\mathbb {C})\) is completely positive and which satisfies \(\widetilde T|_S=T\).
Apply Theorem 1.15.9 to the composite of \(T\) with the inclusion \(\mathcal B\hookrightarrow M_{p}(\mathbb {C})\). Since the resulting ambient map \(G\) takes every value in \(\mathcal B\), it defines a map \(\widetilde T\) with codomain \(\mathcal B\). Its composite with the inclusion is \(G\), hence is completely positive, and equality in \(M_{p}(\mathbb {C})\) gives \(\widetilde T|_S=T\).
1.16 Composition and preparation of completely positive Kraus maps
The single-Kraus, rectangular-Kraus, reindexing, controlled, and partial-trace closure results below are generic statements about completely positive and trace-preserving completely positive maps on finite-dimensional matrix spaces, with no tensor-network content; they are relocated here from the MPDO renormalization chapter, whose Kraus-action and conditional-expectation consequences for the density-operator setting reuse them across the chapter boundary.
The identity map \(\operatorname{id}(X)=X\) is trace-preserving completely positive, with single Kraus operator \(A_0=I\).
Take \(r=1\) and \(A_0=I\); then \(\operatorname{id}(X)=IXI^\dagger =X\) and \(A_0^\dagger A_0=I\).
Let \(H\) and \(K\) be finite-dimensional complex vector spaces, and let \(V:H\to K\) satisfy \(V^\dagger V=I_H\). Then the map
is trace-preserving and completely positive. This is the general one-isometry form of the local basis change used in [ CPGSV16 , Appendix C.2, lines 1439 and 1520 ] .
Take the sole Kraus operator to be \(V\). Its resolution of the identity is precisely \(V^\dagger V=I_H\).
For a rectangular matrix \(V:H\to K\), define the linear map
For every matrix \(X\), one has \(\Phi _V(X)=VXV^\dagger \).
If \(V^\dagger V=I_H\), then \(\Phi _V\) is trace-preserving and completely positive.
For every rectangular operator \(V:H\to K\), the map \(X\mapsto VXV^\dagger \) is completely positive.
Let \(e:I\simeq J\) be an equivalence of index sets. The associated matrix reindexing is the linear map \(R_e:\mathbb {C}^{I\times I}\to \mathbb {C}^{J\times J}\) determined by
For every equivalence \(e:I\simeq J\) between finite index sets, the reindexing map \(R_e\) is trace-preserving and completely positive.
Let \(P_e:\mathbb {C}^I\to \mathbb {C}^J\) be the permutation matrix of \(e\). Then \(R_e(X)=P_eXP_e^\dagger \) and \(P_e^\dagger P_e=I_{\mathbb {C}^I}\). The claim follows from the preceding isometry lemma.
The composition of two trace-preserving completely positive maps is again trace-preserving completely positive. If \(\mathcal{S}\) has Kraus operators \(A_i\) and \(\mathcal{T}\) has Kraus operators \(B_j\), then \(\mathcal{S}\circ \mathcal{T}\) has Kraus operators \(A_iB_j\).
The Kraus form of the composite and its resolution of the identity are
The composition of two completely positive rectangular Kraus maps is completely positive. If the two families are \((A_i)_i\) and \((B_j)_j\), the composite family is \((A_iB_j)_{i,j}\).
Let \(I\) be a finite index set, let \(J\) be an arbitrary index set, and let \((A_a\in \mathbb {C}^{J\times I})_a\) be a finite family of matrices. Its rectangular Kraus map is
Let \((A_a:H\to K)_a\) be a finite family of rectangular operators. If \(\sum _a A_a^\dagger A_a=I_H\), then \(X\mapsto \sum _a A_aXA_a^\dagger \) is trace-preserving and completely positive.
The displayed family is already a Kraus representation, and the assumed identity is precisely its trace-preserving normalization.
Every finite rectangular Kraus family defines a completely positive map, without a resolution-of-identity assumption.
If \(e:I\simeq I'\) and \(A'_b=A_{e^{-1}(b)}\), then \(\Phi _{A'}=\Phi _A\).
Reindexing the finite sum gives \(\sum _{b\in I'}A_{e^{-1}(b)}XA_{e^{-1}(b)}^\dagger =\sum _{a\in I}A_aXA_a^\dagger \).
Let \(H=\bigoplus _k H_k\) and \(K=\bigoplus _k K_k\). For each \(k\), let \((A_{k,a}:H_k\to K_k)_a\) be a finite family of operators, and let \(\widetilde A_{k,a}:H\to K\) agree with \(A_{k,a}\) on \(H_k\) and vanish on every other summand. The orthogonally controlled Kraus map is
In particular, \(\mathcal C(X)_{kl}=0\) for \(k\ne l\). This is the sector control in the definitions of \(\mathcal T_1\) and \(\mathcal S_1\) in [ CPGSV16 , Appendix C.2, lines 1523–1535 and 1548–1555 ] .
Let \(X_{kk}:H_k\to H_k\) denote the \(k\)th diagonal block of \(X\). Then
If \(j\ne k\), then every zero-extended Kraus operator satisfies \([\widetilde A_{j,a}X\widetilde A_{j,a}^\dagger ]_{(k,b),(k,c)}=0\). Hence only the \(j=k\) terms survive, and \([\mathcal C(X)]_{kk}=\sum _a A_{k,a}X_{kk}A_{k,a}^\dagger \).
If \(k\ne l\), then \(\mathcal C(X)_{kl}=0\).
For every \((j,a)\) and \(k\ne l\), the zero-extended operator satisfies \([\widetilde A_{j,a}X\widetilde A_{j,a}^\dagger ]_{(k,b),(l,c)}=0\). Therefore \([\mathcal C(X)]_{kl}=\sum _{j,a}0=0\).
Suppose that for every \(k\) the sectorwise Kraus family resolves the identity, \(\sum _a A_{k,a}^\dagger A_{k,a}=I_{H_k}\). Then the orthogonally controlled map \(\mathcal C\) is trace-preserving and completely positive.
Each embedded operator has support in one summand, and therefore
Apply Theorem 1.16.12 to the combined rectangular Kraus family \((\widetilde A_{k,a})_{k,a}\).
Let \((P_s)_{s\in I}\) be a finite family of orthogonal projections on \(H\) such that \(\sum _sP_s=I_H\). For each \(s\), let \(\Phi _s:\operatorname{End}_{\mathbb {C}}(H)\to \operatorname{End}_{\mathbb {C}}(K)\) be trace-preserving and completely positive. Then
is trace-preserving and completely positive.
Choose Kraus operators \((A_{s,a})_a\) for \(\Phi _s\), and set \(B_{s,a}=A_{s,a}P_s\). These operators give the displayed map, and
Apply Theorem 1.16.12.
For a matrix \(X\) on \(A\otimes B\), the right partial trace is the linear map from matrices on \(A\otimes B\) to matrices on \(A\) given by
After the retained and discarded subspins have been regrouped as \(A\otimes B\), this is the partial-trace ingredient of the maps \(\mathcal T_0\) and \(\mathcal S_0\) in [ CPGSV16 , Appendix C.2, lines 1521–1522 and 1547 ] .
For square matrices \(X\) on \(A\) and \(Y\) on \(B\), \(\operatorname{tr}_B(X\otimes Y)=\operatorname{tr}(Y)X\).
For all \(i,j\), \([\operatorname{tr}_B(X\otimes Y)]_{ij} =\sum _t X_{ij}Y_{tt}=\operatorname{tr}(Y)X_{ij}\).
The map \(\operatorname{tr}_B\) is trace-preserving and completely positive for arbitrary finite-dimensional spaces \(A\) and \(B\).
For each basis vector \(e_k\) of \(B\), define \(V_k:A\otimes B\to A\) by \(V_k(e_i\otimes e_\ell )=\delta _{k\ell }e_i\). Then
Theorem 1.16.12 applies.
The state-preparation map \(\mathcal P_\rho (X)=X\otimes \rho \) from matrices on \(A\) to matrices on \(A\otimes B\), the elementary preparation operation used in the maps \(\mathcal T_1\) and \(\mathcal S_1\) of [ CPGSV16 , Appendix C.2, lines 1527–1533 and 1551–1555 ] , is a fully generic construction with no MPDO-specific content, stated as Definition 1.13.9 above.
Let \(R=\sqrt\rho \). For an orthonormal basis \((e_j)_j\) of \(B\), define rectangular operators \(A_j:A\to A\otimes B\) by \(A_j(e_a)=e_a\otimes Re_j\).
If \(\rho \succeq 0\) and \(A_j(e_a)=e_a\otimes \sqrt\rho \, e_j\), then
Put \(R=\sqrt\rho \). Since \(R\) is Hermitian and \(R^2=\rho \), the \(((a,s),(b,t))\) entry of the left-hand side is
If \(\rho \geq 0\) and \(\operatorname{tr}(\rho )=1\), then \(\sum _j A_j^\dagger A_j=I_A\).
Since \(R\) is Hermitian and \(R^2=\rho \), the diagonal entries of the sum are \(\sum _{j,t}\overline{R_{tj}}R_{tj} =\operatorname{tr}(R^2)=\operatorname{tr}(\rho )=1\), while its off-diagonal entries vanish.
If \(\rho \geq 0\), then the map \(X\mapsto X\otimes \rho \) is completely positive. No trace normalization is required.
If \(\rho \geq 0\) and \(\operatorname{tr}(\rho )=1\), then \(\mathcal P_\rho :X\mapsto X\otimes \rho \) is trace-preserving completely positive.
1.17 Set spectra and spectral multiplicities
For a matrix \(A\in M_d(\mathbb {C})\), the spectrum \(\sigma (A)\) is a set: it does not record how often a root of the characteristic polynomial occurs. Wolf’s spectrum-preserver hypothesis is precisely equality of these sets on Hermitian inputs. The next results justify the continuity argument in [ Wol12 , Chapter 1, Spectrum preserving maps ] without replacing that hypothesis by multiplicity preservation.
If \(A\) is Hermitian with eigenvalues \(\lambda _0,\ldots ,\lambda _{d-1}\), listed with multiplicity, then for every \(k\in \mathbb {N}\),
This includes \(d=0\), when both sides are empty sums.
Diagonalize \(A=UDU^\dagger \), where \(D=\operatorname {diag}(\lambda _0,\ldots ,\lambda _{d-1})\). Unitary conjugation commutes with powers and leaves the trace unchanged, while \(D^k=\operatorname {diag}(\lambda _0^k,\ldots ,\lambda _{d-1}^k)\).
Let \(A=U\operatorname {diag}(\lambda _0,\ldots ,\lambda _{d-1})U^\dagger \) be Hermitian with \(\lambda _0\ge \cdots \ge \lambda _{d-1}\). For \(\varepsilon \in \mathbb {R}\), define
Define the corresponding Hermitian matrix by
If \(\varepsilon {\gt}0\), then \(\lambda _0(\varepsilon ){\gt}\cdots {\gt}\lambda _{d-1}(\varepsilon )\), so the roots of \(\chi _{A_\varepsilon }\) have no repetitions. Moreover, \(A_\varepsilon \to A\) as \(\varepsilon \to 0\). In particular, the positive sequence \(\varepsilon _n=1/(n+1)\) gives simple-spectrum matrices \(A_{\varepsilon _n}\to A\). All assertions remain valid for \(d=0\).
For \(i{\lt}j\), monotonicity of the ordered eigenvalues and positivity of \(\varepsilon \) give
Thus the diagonal entries in (28) are distinct. The same formula is continuous in \(\varepsilon \), and at \(\varepsilon =0\) it is the spectral decomposition of \(A\).
Let \(A,B\in M_d(\mathbb {C})\). Suppose \(\sigma (B)=\sigma (A)\) and the roots of \(\chi _A\) have no repetitions. Then the root multisets of \(\chi _A\) and \(\chi _B\) are equal.
The common spectrum is the common finite set underlying the two root multisets. The roots of \(\chi _A\) are distinct, so this set has \(d\) elements. Both characteristic polynomials split over \(\mathbb {C}\) and have degree \(d\), so the root multiset of \(\chi _B\) also has cardinality \(d\). Hence it cannot repeat any root, and the two root multisets agree.
Let \(A,B\in M_d(\mathbb {C})\) be Hermitian. Suppose \(\sigma (B)=\sigma (A)\) and the roots of \(\chi _A\) have no repetitions. Then \(\operatorname{tr}(B^k)=\operatorname{tr}(A^k)\) for every \(k\in \mathbb {N}\).
For each \(k\in \mathbb {N}\), the function \(A\mapsto \operatorname{tr}(A^k)\) on \(M_d(\mathbb {C})\) is continuous. Thus \(A_n\to A\) implies \(\operatorname{tr}(A_n^k)\to \operatorname{tr}(A^k)\).
Finite matrix multiplication is continuous, hence so is \(A\mapsto A^k\); the trace is a finite sum of matrix entries.
If \(A, B \in M_{n}(\mathbb {C})\) satisfy \(\operatorname{tr}(A^k) = \operatorname{tr}(B^k)\) for all \(k \ge 1\), then \(A\) and \(B\) have the same characteristic polynomial. It is enough to assume this equality for \(1 \le k \le n\).
Let \(e_m\) denote the \(m\)th elementary symmetric polynomial in the eigenvalues, with \(e_0 = 1\). The Newton–Girard recursion
expresses the coefficients of the characteristic polynomial in terms of the power-sum traces \(\operatorname{tr}(A^i)\). Equal power sums for \(A\) and \(B\) therefore give equal coefficients, hence equal characteristic polynomials.
Let \(T:M_d(\mathbb {C})\to M_d(\mathbb {C})\) be complex linear. Suppose that \(T(A)\) is Hermitian whenever \(A\) is Hermitian and that
for every Hermitian \(A\). Then every Hermitian \(A\) satisfies
Fix Hermitian \(A\) and use the positive simple-spectrum perturbations \(A_{\varepsilon _n}\) from Theorem 1.17.3. The hypothesis (31) and Theorem 1.17.5 give
for every \(n\) and \(k\). Linearity makes \(T\) continuous in finite dimension. Since \(A_{\varepsilon _n}\to A\), continuity from Theorem 1.17.6 yields \(\operatorname{tr}(T(A)^k)=\operatorname{tr}(A^k)\). For \(1\le k\le d\), Newton–Girard identities determine all coefficients of the degree-\(d\) characteristic polynomial, proving \(\chi _{T(A)}=\chi _A\). When \(d=0\), the same argument consists of empty sums and degree-zero characteristic polynomials.
For a finite Hermitian matrix, the characteristic polynomial records the ordered eigenvalues with their multiplicities. This determines the matrix up to unitary conjugation.
Let \(A,B\in M_d(\mathbb {C})\) be Hermitian. If their characteristic polynomials agree, then there is a unitary matrix \(U\in M_d(\mathbb {C})\) such that
Equality of the characteristic polynomials gives equality of the ordered eigenvalue lists. Write the two spectral decompositions as
Likewise,
Set
This matrix is unitary, and cancellation of \(U_A^\dagger U_A\) gives
1.18 Pure states, projective rays, and Wigner’s theorem
A nonzero vector \(v\in \mathbb {C}^d\) determines a ray \([v]\) and the normalized rank-one matrix
The quotient in (39) is unchanged when \(v\) is multiplied by a nonzero scalar.
For a ray \(p=[v]\) in \(\mathbb {C}^d\), define
For each ray \(p\), fix a nonzero representative \(p^{\mathrm{rep}}\). Define
The matrix \(P_p\) is an orthogonal projection with trace one, and
The map \(p\mapsto P_p\) is injective. Moreover, a unitary \(U\) and coordinatewise conjugation act by
These are the pure-state matrix identities used in [ Wol12 , Chapter 1, Wigner’s theorem ] .
The rank-one identity \(|v\rangle \! \langle v|^2=\langle v,v\rangle |v\rangle \! \langle v|\) gives idempotence, while taking the adjoint gives Hermiticity. The trace of \(|v\rangle \! \langle v|\) is \(\langle v,v\rangle \), and
Equality \(P_p=P_q\) implies that a representative of one ray is a nonzero scalar multiple of a representative of the other. Finally, matrix multiplication sends \(|v\rangle \! \langle v|\) to \(|Uv\rangle \! \langle Uv|\), and coordinatewise conjugation exchanges the two factors of the outer product.
Let \(A\in M_d(\mathbb {C})\) be Hermitian, and let \(p\) be a projective ray. If
then there is a projective ray \(q\) such that
The matrix \(P_p\) is Hermitian. Equality of the characteristic polynomials therefore gives a unitary \(U\) such that
Taking \(q=U\cdot p\) and using \(P_{U\cdot p}=UP_pU^\dagger \) proves the claim.
For every nonzero \(v\in \mathbb {C}^d\),
Substitute the definition \(P_{[v]}=\langle v,v\rangle ^{-1}|v\rangle \! \langle v|\) and use \(\langle v,v\rangle \ne 0\).
Let \(E\) be a complex module and let \(T,S:M_{d}(\mathbb {C})\to E\) be complex-linear maps. If
for every ray \(p\) in \(\mathbb {C}^d\), then \(T=S\).
For \(v\ne 0\), multiply the equality at \(p=[v]\) by \(\langle v,v\rangle \) to obtain \(T(|v\rangle \! \langle v|)=S(|v\rangle \! \langle v|)\). The same equality is immediate for \(v=0\). Rank-one extensionality now gives \(T=S\).
For arbitrary nonzero representatives \(v,w\in \mathbb {C}^d\),
By the trace-product identity above, \(\operatorname {tp}([v],[w])=\operatorname{tr}(P_{[v]}P_{[w]})\). Moreover,
which gives the displayed quotient.
There are no projective rays when \(d=0\). When \(d=1\), every pure-state matrix is the identity and
If \(d\geq 2\), then
For \(d\geq 2\), choose nonzero representatives \(v,w\) of \(p,q\), and define \(A\in M_{d,2}(\mathbb {C})\) and \(B\in M_{2,d}(\mathbb {C})\) by
Define the second factor by
Then \(AB=P_p+P_q\). The characteristic-polynomial identity for rectangular products gives
Direct calculation yields
It also gives
Substitution into the quadratic characteristic polynomial of \(BA\) proves (53). In dimension one, trace one forces each pure-state matrix to equal the identity. The zero-dimensional claim follows because a projective ray would require a nonzero vector in \(\mathbb {C}^0\).
For projective rays \(p,q,r,s\) in \(\mathbb {C}^d\), if
then
For \(d\geq 2\), apply (53) to both sums and cancel the nonzero polynomial \(X^{d-2}\). Equality of the remaining constant terms gives the result. In dimension one both traces equal one, while in dimension zero there are no rays.
Let \(f\) be a map on the rays of \(\mathbb {C}^d\) such that
for all rays \(p,q\). Then there is a unitary \(U\) such that either \(f(p)=U\cdot p\) for every \(p\), or \(f(p)=U\cdot \overline p\) for every \(p\).
The canonical linear isometry between \(\mathbb {C}^d\) and its Euclidean-space realization induces inverse maps on rays. This identification preserves transition probabilities and commutes with both the unitary action and coordinatewise conjugation. Apply projective Wigner rigidity on the Euclidean-space realization and return the two alternatives through the inverse identification.
Let \(F\) be a bijection of the normalized pure states \(P_p\) such that \(\operatorname{tr}(F(P)F(Q))=\operatorname{tr}(PQ)\) for all normalized pure states \(P,Q\). Then there is a unitary \(U\) such that either \(F(P)=UPU^\dagger \) for every \(P\), or \(F(P)=UP^{\mathsf T}U^\dagger \) for every \(P\). This is a disjunction, not an exclusive alternative. For \(d=0\) the set of pure states is empty, while for \(d=1\) it is a singleton and the two alternatives coincide.
Use the bijection between rays and the range of \(p\mapsto P_p\) to regard \(F\) as a map of projective rays. The trace identity \(\operatorname{tr}(P_pP_q)=\operatorname {tp}(p,q)\) shows that this map preserves transition probabilities. Projective Wigner rigidity gives a unitary \(U\) and either \(p\mapsto U\cdot p\) or \(p\mapsto U\cdot \overline p\). The identities \(P_{U\cdot p}=UP_pU^\dagger \) and \(P_{\overline p}=P_p^{\mathsf T}\) give the two stated formulas.
Let \(T:M_{d}(\mathbb {C})\to M_{d}(\mathbb {C})\) be complex linear. Suppose that
for every \(X\in M_{d}(\mathbb {C})\), and that
for every Hermitian \(A\in M_{d}(\mathbb {C})\). Then there is a unitary \(U\) such that either
for every \(X\in M_{d}(\mathbb {C})\), or
for every \(X\in M_{d}(\mathbb {C})\). This is the spectrum-preserver classification in [ Wol12 , Chapter 1, Spectrum preserving maps ] .
The adjoint identity shows that \(T(A)\) is Hermitian whenever \(A\) is Hermitian. Hence Theorem 1.17.8 gives
for every Hermitian \(A\). For each projective ray \(p\), recognize \(T(P_p)\) by its characteristic polynomial and choose a ray \(f(p)\) such that
For rays \(p,q\), linearity and characteristic-polynomial preservation give
Theorem 1.18.9 therefore shows that \(f\) preserves transition probabilities. Projective Wigner rigidity gives a unitary \(U\) and either
for every \(p\), or
for every \(p\). Thus \(T\) agrees on every \(P_p\) with, respectively, \(X\mapsto UXU^\dagger \) or \(X\mapsto UX^{\mathsf T}U^\dagger \). Pure-state extensionality gives the corresponding equality on all matrices.
1.19 A cyclic trace identity for rectangular powers
For rectangular matrices \(L\) and \(Q\) over a commutative semiring and every integer \(N\geq 1\),
Write \(N=M+1\). Induction on \(M\) gives \((LQ)^{M+1}=L(QL)^M Q\). Moving the first factor cyclically through the trace and reassociating the product gives \(\operatorname{tr}((QL)^{M+1})\).
1.20 Coherent frames and external references
All spaces in this section are finite-dimensional complex Euclidean spaces. The norm on matrices is the induced operator norm. The cross matrix of two column matrices retains their relative phases; its entries are not replaced by absolute overlaps.
If \(W:\mathbb {C}^n\to \mathbb {C}^m\) satisfies \(W^\dagger W=I_n\), then \(\| W\| \le 1\).
The identity \(\| W\| ^2=\| W^\dagger W\| =\| I_n\| \le 1\) gives the conclusion, including a zero-dimensional domain.
Let \(V,W:\mathbb {C}^n\to \mathbb {C}^m\) and suppose \(W^\dagger W=I_n\). Set \(G=V^\dagger V\) and \(Z=W^\dagger V\). Then
Expanding the Gram matrix of the difference gives
Apply the triangle inequality to (72) and use \(\| X^\dagger X\| =\| X\| ^2\) and \(\| X^\dagger \| =\| X\| \).
For every \(A:\mathbb {C}^n\to \mathbb {C}^m\), every finite-dimensional reference space \(\mathbb {C}^k\), and every \(\xi \in \mathbb {C}^n\otimes \mathbb {C}^k\),
In an orthonormal reference basis, write \(\xi =\sum _{r=1}^k\xi _r\otimes e_r\). The coordinate formula for the Kronecker product and orthogonality give
Taking nonnegative square roots proves (73).
Let \(F,G:\mathbb {C}^n\to \mathbb {C}^m\) and \(U:\mathbb {C}^m\to \mathbb {C}^m\) satisfy \(\| UF-G\| \le \varepsilon \). For every finite-dimensional reference \(\mathbb {C}^k\) and every \(\xi \in \mathbb {C}^n\otimes \mathbb {C}^k\),
Neither normalization of \(\xi \) nor isometricity of \(F\) and \(G\) is required.
The difference is \(((UF-G)\otimes I_k)\xi \). Apply Theorem 1.20.3 and the assumed bound.
1.21 A common positive compression on a fixed subspace
Let \(\mathcal H,\mathcal K\) be finite-dimensional complex Hilbert spaces, let \(Z:\mathcal K\to \mathcal H\), and let \(M,N,P\) be operators on \(\mathcal H\). Suppose that \(N\) is positive definite and
Then the actual compression \(B=Z^*MZ\) satisfies
No Hermitian assumption on \(M\) is required, and the spaces may have dimension zero. This is the finite-dimensional compression used for the restricted replica metrics in [ Ope26b , Section 8, lines 80–110, 299–308 and 370–381, equation (comparator:pinned-metric) ] .
The isometry equations give \(PZ=ZZ^*Z=Z\). Hence
Multiplication by \(Z^*\) gives \(B=Z^*NZ\). The map \(Z\) is injective, so \(\langle x,Bx\rangle =\langle Zx,NZx\rangle {\gt}0\) for every nonzero \(x\in \mathcal K\); also \(B\) is Hermitian because \(N\) is Hermitian. Finally, \(NZ=PNZ=ZZ^*NZ=ZB\). Combining this with (83) proves (82).
1.22 A norm comparison through an orthogonal projection
Let \(\mathcal H\) and \(\mathcal K\) be finite-dimensional complex Hilbert spaces, let \(Z:\mathcal K\to \mathcal H\) satisfy \(Z^*Z=I\), and let \(P\) be an orthogonal projection on \(\mathcal H\). Suppose that \(A\) on \(\mathcal K\) and \(N\) on \(\mathcal H\) are self-adjoint and
Then every \(u\in \mathcal H\) satisfies
Neither matrix is required to be positive, and either Hilbert space may have dimension zero.
Taking adjoints gives \(NP=PZAZ^*\). Contractivity of \(P\) and the isometry property of \(Z\) therefore yield
This is the norm comparison used for the restricted metric powers in [ Ope26b , 07-comparators.tex, lines 283–308 and 370–381 ] .
1.23 Isometric coordinates for an orthogonal projection
The restriction of an operator to a projection range can be expressed in orthonormal coordinates without imposing an ordering on the ambient basis. This elementary construction is used for the symmetric restrictions in [ Ope26b , Section 8, lines 80–110 and 370–381 ] .
Let \(I\) be a finite set and let \(P\in \mathbb C^{I\times I}\) satisfy \(P^*=P\) and \(P^2=P\). Then there exist an integer \(n\ge 0\) and \(Z\in \mathbb C^{I\times \{ 0,\ldots ,n-1\} }\) such that
The assertion includes the empty ambient set and the zero projection.
Enumerate the ambient coordinates, choose an orthonormal basis of the projection range in these coordinates, and transport the resulting isometry back to the original index set. The two matrix identities are preserved by the coordinate equivalence.
1.24 Compression of a contained projection
Let \(P\) be an orthogonal projection and let \(Z\) be a rectangular matrix such that \(PZZ^*=P\). Then \(Z^*PZ\) is an orthogonal projection.
The matrix is Hermitian, and \((Z^*PZ)^2=Z^*(PZZ^*)PZ=Z^*P^2Z=Z^*PZ\).
1.25 Commutation in isometric coordinates
Let \(Z\) be a rectangular complex matrix with \(Z^*Z=I\). Suppose \(MZ=ZB\), \(NZ=ZC\), and \(MN=NM\). Then \(BC=CB\). No Hermitian hypothesis is required. This is the elementary restriction argument used in [ Ope26b , Section 7, lines 268–276 ] .
The intertwining identities give \(ZBC=MNZ=NMZ=ZCB\). Multiplication on the left by \(Z^*\) gives the assertion.