Dependencies

For \(s\in [0,1]\), \(t\in [0,1]\), and positive-definite matrices \(A_1,A_2,B_1,B_2\), the map \((A,B)\mapsto \Re \operatorname{tr}(A^s B^{1-s})\) is jointly concave:

Obtained from Theorem 7.7.13 by taking \(K=\mathbb {1}\).

For \(s\in [0,1]\), \(t\in [0,1]\), matrix \(K\), and positive-definite matrices \(A_1,A_2,B\), the map \(A\mapsto \Re \operatorname{tr}(K^\dagger A^s K B^{1-s})\) is concave:

For \(s\in [0,1]\), \(t\in [0,1]\), matrix \(K\), and positive-definite matrices \(A,B_1,B_2\), the map \(B\mapsto \Re \operatorname{tr}(K^\dagger A^s K B^{1-s})\) is concave:

For a leaf of positive terminal weight, the trace adjoint \(\Phi _j^\dagger \), defined by \(\operatorname {Tr}(\Phi _j^\dagger (\rho )Z)=\operatorname {Tr}(\rho \, \Phi _j(Z))\), maps density matrices to density matrices.

The adjoint Kraus map is

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

A weighted binary tree is either a leaf with a label \(j\) or an internal vertex with a parameter \(p\in [0,1]\) and two subtrees. Its root on inputs \((A_j)\) is \(A_j\) at a leaf and \(M_l\# _pM_r\) at an internal vertex with subtree roots \(M_l\), \(M_r\). The terminal weights are \(\delta _j\) at a leaf and \((1-p)w^{(l)}+p\, w^{(r)}\) at an internal vertex.

For a finite family of matrices \(\{ C_i\} _{i\in \iota }\) and a defect block \(S\), one forms an isometric dilation whose rows are the adjoints of the matrices \(C_i\) together with the adjoint of \(S\). One also forms the scalar block-diagonal matrix whose \(\iota \)-blocks carry prescribed weights and whose defect block carries a single scalar.

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

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

For positive definite complex matrices \(A,B\) and \(p\in \mathbb R\), set \(A\# _pB=A^{1/2}(A^{-1/2}BA^{-1/2})^pA^{1/2}\), where the real powers are taken in the functional calculus of positive definite matrices.

Let \(f:\mathbb {R}\to \mathbb {R}\) be convex on \([0,\infty )\), let \(A\) be a positive semidefinite matrix on a finite-dimensional space indexed by \(n\), and let \(v\in \mathbb {C}^n\) satisfy \(\langle v,v\rangle =1\). Then

where \(f(A)\) is defined by the Hermitian continuous functional calculus.

For positive diagonal matrices \(M\) and \(N\) and \(s\in (0,1)\),

For \(a,b{\gt}0\) and \(s\in (0,1)\),

For \(a,b{\gt}0\) and \(s\in (0,1)\), the function \(t\mapsto t^{s-1} \dfrac {ab}{a+tb}\) is integrable on \((0,\infty )\).

For positive-definite matrices \(A\) and \(B\) with \(A\le B\) in the Loewner order, the inverses satisfy \(B^{-1}\le A^{-1}\).

The terminal weights are nonnegative and sum to one. An edge of weight zero may be omitted and the vertex with one remaining child contracted: this changes neither the root nor the terminal weights.

Let \(A,B\) be positive definite, \(0\le p\le 1\) and \(M=A\# _pB\). At a child \(D\) of positive edge weight \(w\), the map \(\Phi _D(Z)=w^{-1}M^{-1/2}\, \partial _DM[D^{1/2}ZD^{1/2}]\, M^{-1/2}\) is unital and completely positive.

For positive-definite matrices \(A_1,A_2,B_1,B_2\), \(s\in (0,1)\), and \(\theta \in [0,1]\), with \(\hat{A}=A\otimes \mathbb {1}\) and \(\hat{B}=\mathbb {1}\otimes B^\top \), setting \(A_\theta =\theta A_1+(1-\theta )A_2\) and \(B_\theta =\theta B_1+(1-\theta )B_2\), the fractional product \(\hat A^s\hat B^{1-s}\) is jointly concave in the Loewner order:

For positive-definite matrices \(A\), \(B\) and \(s\in (0,1)\),

where \(A\otimes \mathbb {1}\) and \(\mathbb {1}\otimes B^\top \) are regarded as operators on the Kronecker model space \(\mathbb {C}^{D\times D}\).

Let \(f\) be a function from a measure space to a finite matrix algebra. If \(f(x)\ge 0\) for almost every \(x\), then \(\int f(x)\, d\mu (x)\ge 0\).

Let \(Y\) be positive definite and let \(W\) be an isometry. Then

Let \(T\) be a positive subunital map, let \(A\ge 0\), and let \(t{\gt}0\). Then

Let \(T\) be a positive subunital map, let \(A\ge 0\), let \(p\in (0,1)\), and let \(t{\gt}0\). Then

Let \(T\) be a positive subunital map, let \(A\ge 0\), let \(p\in (1,2)\), and let \(t{\gt}0\). Then

The inequality is reversed relative to the concave integrand, since \(g_{p,t}\) is operator convex.

If the defect block satisfies

then the dilation is an isometry, and compressing the weighted scalar block-diagonal matrix gives \(\sum _{i\in \iota }w_iC_iC_i^\dagger +tSS^\dagger \). The defect relation can also be written as \(\sum _{i\in \iota }C_iC_i^\dagger +SS^\dagger =\mathbb {1}\). Strict positivity of all weights together with \(t{\gt}0\) implies positive definiteness of the scalar block-diagonal matrix. If all weights are nonzero and the defect scalar is nonzero, the inverse diagonal has reciprocal weights, and compressing this inverse gives \(\sum _{i\in \iota }w_i^{-1}C_iC_i^\dagger +t^{-1}SS^\dagger \).

Let \(w_i\ge 0\) and \(t{\gt}0\). If the defect block satisfies \(SS^\dagger =\mathbb {1}-\sum _{i\in \iota }C_iC_i^\dagger \), then

Let \(A\ge 0\) be a finite matrix. There are positive semidefinite matrices \(P_i\) and non-negative real numbers \(\lambda _i\) such that

Let \(P_i\) be Hermitian idempotents with \(P_iP_j=\delta _{ij}P_i\) and \(\sum _iP_i=I\), and let \(g\) be real. Then \(f(\sum _ig_iP_i)=\sum _if(g_i)P_i\) for every real function \(f\). If \(|c_i|\le M\) whenever \(P_i\ne 0\), then \(\| \sum _ic_iP_i\| \le M\) in operator norm. For every vector \(z\), \(\| \sum _ic_iP_iz\| ^2=\sum _i|c_i|^2\| P_iz\| ^2\) and \(\sum _i\| P_iz\| ^2=\| z\| ^2\).

For positive-definite matrices \(X_1,X_2,Y_1,Y_2\) and \(\theta \in [0,1]\), the parallel sum \((X,Y)\mapsto X(X+Y)^{-1}Y\) is jointly concave in the Loewner order:

where \(X_\theta =\theta X_1+(1-\theta )X_2\) and \(Y_\theta =\theta Y_1+(1-\theta )Y_2\).

If \(p\) lies strictly inside a rectangle, the boundary integral of \((u-p)^{-1}\) is \(2\pi i\). If \(f\) is holomorphic on the open rectangle away from finitely many interior points \(p\) and \(f-\sum _pc_p(u-p)^{-1}\) extends continuously to the closed rectangle, the boundary integral of \(f\) is \(2\pi i\sum _pc_p\).

For \(s\in (0,1)\),

Let \(0{\lt}p{\lt}1\) and let \(C\) be positive definite. Along Hermitian matrices, the map \(X\mapsto X^p\) is Fréchet differentiable at \(C\) with derivative

This derivative is completely positive and satisfies \(D(C^p)[C]=pC^p\).

For \(a,b{\gt}0\) and \(s\in (0,1)\),

The function \(g_s\) is continuous, \(m_s\) is positive and integrable with \(\int _{\mathbb R}m_s=2s\), for \(z\ne 0\) one has \((e^{\pm sz}-1)/(2\sinh (z/2))=\pm e^{\pm sz/2}g_{s/2}(z)\), and for \(s=1/4\), \(|m_{s/2}(u\pm is/2)|\le m_s(u)/\sqrt2\) for real \(u\).

For real \(z\), \(g_s(z)=\int _{\mathbb R}e^{iuz}m_s(u)\, du\), \(m_s(u){\gt}0\) and \(\int _{\mathbb R}m_s(u)\, du=2s\). For \(s=1/4\) the functions \(h_\pm (z)=(e^{\pm sz}-1)/(2\sinh (z/2))=\pm e^{\pm sz/2}g_{s/2}(z)\) have Fourier densities \(q_\pm (u)=\pm m_{s/2}(u\pm is/2)\) with \(|q_\pm (u)|\le m_s(u)/\sqrt2\). The quotients are interpreted continuously at zero.

Fix \(t{\gt}0\) and positive-definite matrices with \(A_1\le A_2\) and \(B_1\le B_2\). The resolvent of the Kronecker model \(A\otimes \mathbb {1}+t\, (\mathbb {1}\otimes B^\top )\) of the commuting left- and right-multiplication superoperators is antitone:

For \(t{\gt}0\), positive-definite \(A_1,A_2,B_1,B_2\), and \(\theta \in [0,1]\), with \(\hat{A}=A\otimes \mathbb {1}\) and \(\hat{B}=\mathbb {1}\otimes B^\top \), setting \(A_\theta =\theta A_1+(1-\theta )A_2\) and \(B_\theta =\theta B_1+(1-\theta )B_2\):

Let a mean tree have positive weights \(\pi _j\), positive definite inputs \(A_j\) and root \(M\), let \(h\ge 0\) commute with \(A_j\) for every leaf \(j\) outside a finite set \(S\), and put \(W=\sum _{j\in S}\pi _j\). Let \(\mathsf D_j\) be the skew square of \(O_j=A_j^{-1/2}hA_j^{1/2}\). For every vector \(v\),

The transport states are density matrices. For \(0{\lt}p{\lt}1\),

The function \(N\) is continuous and positive on \([0,1]\). For \(0\le p_0\le p_1\le 1\) the right side of the exact derivative is integrable on \([p_0,p_1]\), and its integral equals \(\log N(p_0)^2-\log N(p_1)^2\).

If \(Z\) commutes with every input of a mean tree and fixes the vector \(v\), then \(Z\sigma _{j,u}=\sigma _{j,u}\) for every leaf \(j\) and every real \(u\).

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

Let \(A,B\) be positive definite. Then \(A\# _pB\) is positive definite, \(A\# _0B=A\), \(A\# _1B=B\), and:

1. \((S^*AS)\# _p(S^*BS)=S^*(A\# _pB)S\) for every invertible \(S\);

2. \((cA)\# _p(dB)=c^{1-p}d^p\, (A\# _pB)\) for \(c,d{\gt}0\);

3. \(A\# _pB=B\# _{1-p}A\);

4. for \(0\le p\le 1\) the mean is monotone in each argument, and for every vector \(z\), \(\langle z,(A\# _pB)z\rangle \le \langle z,Az\rangle ^{1-p}\langle z,Bz\rangle ^p\);

5. if \(A_1,B_1\) commute with \(A_2,B_2\), then \((A_1A_2)\# _p(B_1B_2)=(A_1\# _pB_1)(A_2\# _pB_2)\), and the two factors commute.

Let \(P\) be an orthogonal projection on a finite-dimensional complex Hilbert space. For arbitrary \(b,d\in \mathbb R\) and an arbitrary function \(f:\mathbb R\to \mathbb R\),

Here functional calculus uses the restriction of \(f\) to the finite spectrum. Neither continuity of \(f\) nor distinctness of \(b,d\) is required. Either projection range, or the entire space, may have dimension zero. This is the complementary-range calculation used in [ Ope26b , Section 7, lines 603–640 ] .

Let \(A_0,\ldots ,A_{G-1}\) be positive definite operators on a finite-dimensional complex Hilbert space. Suppose that \(A_gA_h=A_hA_g\) whenever \(g\ne h\). Then

The empty product is the identity and the empty sum is zero; repeated operator values are allowed. This is the logarithmic identity used to separate commuting bands in [ Ope26b , Sections 7 and 8 ] .

Let \(A,B\) be commuting Hermitian matrices. If \(A\leq B\), then \(e^A\leq e^B\).

Let \(A,B\) be Hermitian, let \(P=P^*=P^2\), and suppose that \(AP=PA\) and \(AB=BA\). If \(Pe^BP\leq cP\) for \(c\in \mathbb R\), then \(Pe^{A+B}P\leq cPe^AP\).

Let \(A\) and \(P\) be Hermitian, let \(B\) be a complex matrix, and assume \(AP=PA\) and \(AB=BA\). If \(Pe^BP\leq cP\) for \(c\in \mathbb R\), then every vector \(v\) with \(Pv=v\) satisfies \(\operatorname {Re}(v^*e^{A+B}v)\leq c\operatorname {Re}(v^*e^Av)\).

Under the matrix hypotheses of Theorem 7.16.5, \(Pe^BP\leq P\) implies \(\operatorname {Re}(v^*e^{A+B}v)\leq \operatorname {Re}(v^*e^Av)\) for every \(Pv=v\).

Let \(T\) be an operator and let \(c\ge 0\). If \(T^*T\le c\mathbb {1}\), then \(TT^*\le c\mathbb {1}\).

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 \(\operatorname {Re}(v^*Mv)\leq c\operatorname {Re}(v^*v)\).

Let \(A=A^*\) and let \(P=P^*=P^2\) commute with \(A\). Suppose that \(bP\leq PAP\) for some \(b\in \mathbb R\). If \(f\colon \mathbb R\to \mathbb R\) and \(c\in \mathbb R\) satisfy \(f(x)\leq c\) whenever \(x\geq b\), then \(Pf(A)P\leq cP\).

Under the hypotheses on \(A,P,b\) in Theorem 7.16.2, every \(a\geq 0\) satisfies \(Pe^{-aA}P\leq e^{-ab}P\).

Let \(A\) be positive definite, let \(P\) be an orthogonal projection, and let \(c{\gt}0\). Then

No commutation between \(A\) and \(P\) is required. This is the implication from the compressed-inverse estimate to the lower bound in [ Ope26b , Section 8 ] .

Let \(I\) be a finite set, let \(M\) be a positive definite matrix on \(\mathbb {C}^I\), and let \(\varphi \in \mathbb {C}^I\) satisfy \(\| \varphi \| =1\). For every \(s\in \mathbb {R}\),

Let \(I\) be a finite set, let \(M\) be a positive definite matrix on \(\mathbb {C}^I\), and let \(p\in \mathbb {C}^I\) satisfy \(q=\| p\| ^2{\gt}0\). Put \(\varphi =p/\| p\| \). If \(L\in \mathbb {R}\) satisfies \(\operatorname {Re}\langle \varphi ,(\log M)\varphi \rangle \le L\), then, for every \(s\ge 0\),

If \(\sum _i K_i^\dagger K_i = \mathbb {1}\), then the adjoint Kraus map satisfies, for every \(X \in M_{D}(\mathbb {C})\),

Let \(E^*\) be the adjoint of a trace-preserving Kraus map. If \(A\in M_{D}(\mathbb {C})\) admits a positive semidefinite dominant \(B\) commuting with \(A\) and satisfying \(A^\dagger A\le B\), then

Let \(E^*(X)=\sum _i K_i^\dagger X K_i\) be the adjoint Kraus map of a trace-preserving Kraus family, and let \(A\in M_{D}(\mathbb {C})\) be normal. Then the Schwarz gap

is positive semidefinite. Equivalently,

This is the completely positive case of [ Wol12 , Proposition 5.1 ] .

For \(s\in [0,1]\), any matrix \(K\), and positive-definite matrices \(A_1,A_2,B_1,B_2\), the function \((A,B)\mapsto \Re \operatorname{tr}(K^\dagger A^s K B^{1-s})\) is jointly concave, satisfying (8) of Theorem 7.7.10.

For \(s\in [0,1]\), any matrix \(K\), and positive-definite matrices \(A_1,A_2,B_1,B_2\), write \(F_s(A,B):=\Re \operatorname{tr}(K^\dagger A^s K B^{1-s})\). This map is jointly concave. For \(t\in [0,1]\), set \(A_t:=tA_1+(1-t)A_2\) and \(B_t:=tB_1+(1-t)B_2\). Then

See [ Lie73 ; And79 ] . This statement is the positive-definite, boundary-exponent (\(x+y=1\)) case of the Ando–Lieb theorem [ Wol12 , Theorem 5.15 ] , which holds for positive semidefinite \(A,B\) and all exponents \(x,y\ge 0\) with \(x+y\le 1\).

For \(s\in [0,1]\), any matrix \(K\), and positive-semidefinite matrices \(A_1,A_2,B_1,B_2\), the map \((A,B)\mapsto \Re \operatorname{tr}(K^\dagger A^s K B^{1-s})\) is jointly concave, satisfying (8) of Theorem 7.7.10. This is the full Ando–Lieb theorem [ Wol12 , Theorem 5.15 ] on the boundary line \(x+y=1\) (with \(x=s\), \(y=1-s\)): the positive-definiteness restriction of Theorem 7.7.10 is lifted. The general two-exponent region is obtained in Theorem 7.7.12.

Let \(x,y\ge 0\) with \(x+y\le 1\). For any matrix \(K\) and positive-semidefinite matrices \(A_1,A_2,B_1,B_2\), write \(F_{x,y}(A,B):=\Re \operatorname{tr}(K^\dagger A^x K B^y)\). This map is jointly concave. For \(t\in [0,1]\), set \(A_t:=tA_1+(1-t)A_2\) and \(B_t:=tB_1+(1-t)B_2\). Then

This is Wolf’s Theorem 5.15 in the finite-dimensional matrix setting.

For a positive unital map \(T\) and a positive-definite matrix \(A\), the inequality \(T(\log A)\le \log (T(A))\) holds.

For every \(L\in \mathbb R\) and \(b\ge 0\),

If \(b{\gt}0\), then also

These estimates give explicit constant errors in the logarithmic floor comparison of [ Ope26b , Section 8 ] .

If \(A\succ 0\), then, for every \(s\in \mathbb {R}\),

Let \(A,P\) be complex matrices on a finite-dimensional space and let \(b,c\in \mathbb R\). If \(2bI\le A\) and \(cP\le A\), then

No projection or commutation assumption on \(P\) is needed. This is the averaging step preceding the logarithmic comparison in [ Ope26b , Section 7, lines 603–615 ] .

Let \(T,S\) be complex matrices on a finite-dimensional space and let \(c\in \mathbb R\). If \(S\le I\), \(c\le 1\) and \(cS\le T\), then

In particular, this applies to extension by the identity off the range of an orthogonal projection. The inequality itself does not require \(S\) to be a projection. This auxiliary order inequality extends a lower bound to the whole space.

Under the input hypotheses of Theorem 7.9.3.6, the trace adjoints obey, for every matrix \(\rho \) on \(\mathcal H_2\),

Repeated labels and zero total label weights are permitted. This is an embedding identity for arbitrary \(J\); it does not assert preservation of trace.

Let \(T\) be a fixed weighted binary tree, and let \(A_i,B_i{\gt}0\) satisfy \(A_iJ=JB_i\) for every label \(i\). Denote its roots by \(M_A\) and \(M_B\), its recursive linear maps for a label \(j\) by \(\mathcal L_j^A\) and \(\mathcal L_j^B\). For either input family, the map at a leaf \(i\) is the identity when \(i=j\) and zero otherwise. At an internal vertex it is

where \(M_L,M_R\) are the child roots. If \(w_j\) is the total weight of label \(j\), summing over its occurrences, define

and define \(\Phi _j^B\) analogously. For arbitrary matrices \(H\) and \(X\) on the respective spaces,

Repeated labels are retained in the recursive sums. Zero total label weights are permitted, with reciprocal weight interpreted as zero; the normalized map of such a label is zero.

Let \(A\ge 0\) and \(B\ge 0\) satisfy \(AJ=JB\). For every real exponent \(s\), \(A^sJ=JB^s\). For negative exponents the spectral power is zero on the kernel.

Let \(A{\gt}0\) and \(B{\gt}0\) satisfy \(AJ=JB\), and let \(0\le p\le 1\). Write \(\mathcal D_p(A)\) for the complex-linear extension of the derivative of the real power at \(A\). For arbitrary matrices \(H\) on \(\mathcal H_2\) and \(X\) on \(\mathcal H_1\),

The directions need not be Hermitian.

Let \(A_1,B_1{\gt}0\) on \(\mathcal H_1\) and \(A_2,B_2{\gt}0\) on \(\mathcal H_2\) satisfy \(A_1J=JA_2\) and \(B_1J=JB_2\). For every real \(p\),

Let \(T\) be a fixed weighted binary tree whose vertex weights lie in \([0,1]\). If its positive definite inputs \(A_j,B_j\) satisfy \(A_jJ=JB_j\) for every label \(j\), and its roots are \(M_A=T(A)\) and \(M_B=T(B)\), then \(M_AJ=JM_B\). Repeated labels and zero vertex weights are permitted.

Let \(L_1\) and \(L_2\) be Hermitian and satisfy \(L_1J=JL_2\). Then \(L_2J^*=J^*L_1\), and for every matrix \(H\) on \(\mathcal H_2\),

Suppose \(A_1,B_1{\gt}0\), \(A_2,B_2{\gt}0\), \(A_1J=JA_2\) and \(B_1J=JB_2\). Let \(0\le p\le 1\), and let \(\mathcal G_{p,L}(A,B)\) and \(\mathcal G_{p,R}(A,B)\) be the complex-linear extensions of the derivatives of \(A\# _pB\) in its first and second arguments. For each \(\nu \in \{ L,R\} \) and arbitrary matrices \(H\) and \(X\) on the respective spaces,

The same common intertwiner is used for both positive input pairs. The directions in the two arguments may be chosen independently.

Let \(J\) be finite and let \(g\) range over \(0,\ldots ,G-1\). Let \(T_g\) be weighted binary trees with terminal weights \(w_{j,g}\) and positive definite inputs \(A_{j,g}\) on one finite-dimensional complex Hilbert space. Suppose their roots \(M_g=T_g(A_{\cdot ,g})\) commute pairwise, set \(M=M_0\cdots M_{G-1}\), and let \(P_g\) be orthogonal projections. Let \(b_g{\gt}0\), \(\kappa _g\ge 0\), and let \(L_{j,g},U_g,r_g\in \mathbb R\) satisfy

For \(s\ge 0\) and \(0{\lt}\| p\| \le 1\), put \(u=M^{-s}p\) and \(v=u/\| u\| \). Suppose \(\delta _g\ge 0\) and \(1-\delta _g\le \langle v,P_gv\rangle \). Define \(R_g=2|r_g|+|\log b_g|+\log 2+\log (3/2)\). Then

The remainders are independent of the deficits and terminal weights; no upper bound on \(\delta _g\) is imposed. This gives explicit errors in the analytic lower comparison of [ Ope26b , Section 8 ] .

Let a weighted binary tree have distinct leaves with positive definite inputs \(A_j\), terminal weights \(w_j\) and root \(M\). The root is Fréchet differentiable in each input along Hermitian matrices. Composing the normalized vertex maps from a leaf \(j\) to the root gives a completely positive map \(\Phi _j\), the identity along an empty path, such that

and \(\Phi _j\) is unital when \(w_j{\gt}0\).

If all inputs of a weighted binary tree with distinct leaves are block diagonal for a fixed orthogonal decomposition, the root is block diagonal and the normalized leaf maps preserve each matrix block. On the first diagonal block they are the normalized leaf maps of the tree of first diagonal blocks, and they do not depend on the other diagonal blocks.

Let \(T\) be a weighted binary tree with labels in a finite set \(J\) and terminal weights \(w_j\). Let \(b{\gt}0\), \(\kappa \ge 0\), and \(r,U\in \mathbb R\), and suppose \(L_j\le U\) for every \(j\in J\). For \(0\le \delta \le 1\), set

Then

The remainder \(R\) is independent of \(\delta \), the tree, and its terminal weights. This makes explicit the scalar error preceding the logarithmic lower comparison in [ Ope26b , Section 8 ] .

Let \(J\) be finite and let \(g\) range over \(0,\ldots ,G-1\). For each \(g\), let \(T_g\) be a weighted binary tree with terminal weights \(w_{j,g}\) and positive definite inputs \(A_{j,g}\) on one finite-dimensional complex Hilbert space. Set \(M_g=T_g(A_{\cdot ,g})\) and \(M=M_0\cdots M_{G-1}\), and suppose the roots \(M_g\) commute pairwise. Let \(P_g\) be orthogonal projections and let \(b_g,c_{j,g}{\gt}0\) satisfy

For \(s\ge 0\) and a vector \(p\) with \(0{\lt}\| p\| \le 1\), put \(u=M^{-s}p\), \(v=u/\| u\| \), and \(\ell _g=\sum _{j\in J}w_{j,g}\log (b_g+c_{j,g}/2)\). If real numbers \(\delta _g\) satisfy \(1-\delta _g\le \langle v,P_gv\rangle \), then

The projections need not commute with the inputs, the roots, or one another. The empty product and zero terminal weights are included. This is an analytic implication of the floor, projection bounds, and mass estimates used in [ Ope26b , Section 8 ] .

Let \(I\) and \(J\) be finite sets, let \(P\) be an orthogonal projection on \(\mathbb {C}^I\), and let \(T\) be a weighted binary tree with labels in \(J\) and terminal weights \(w_j\). Suppose \(b{\gt}0\), \(d_j{\gt}0\), and \(b(\mathbb {1}-P)+d_jP\le A_j\) for every \(j\in J\). Then

No commutation condition is imposed on the matrices \(A_j\). Repeated leaf labels and zero terminal weights are permitted. This is the operator comparison used before the projection-mass estimate in [ Ope26b , lines 603–640 of the norm-comparison argument ] .

Let \(P\) be an orthogonal projection on a finite-dimensional complex Hilbert space, and let \(T\) be a weighted binary tree with labels in a finite set \(J\) and terminal weights \(w_j\). Suppose \(b{\gt}0\) and, for each \(j\in J\), the real number \(d_j\) and matrix \(A_j\) satisfy

Let \(v\) be a unit vector and let \(\delta \in \mathbb R\) satisfy \(1-\delta \le \langle v,Pv\rangle \). Put \(\ell =\sum _{j\in J}w_j\log d_j\). Then

No upper bound on \(\delta \) or commutation of \(P\) with the inputs is required. This is the expectation step in the lower comparison of [ Ope26b , Section 8 ] .

With the same \(P\), \(b{\gt}0\), positive family \((d_j)_{j\in J}\), and tree \(T\), suppose matrices \(A_j\) satisfy \(b(\mathbb {1}-P)+d_jP\le A_j\) for every \(j\in J\). Then

The hypotheses impose no commutation condition on the matrices \(A_j\).

Let \(T\) be a finite weighted binary tree with labels in a finite set \(J\) and terminal weights \(w_j\). For \(b{\gt}0\) and \(d_j{\gt}0\) for every \(j\in J\),

Repeated leaf labels and zero terminal weights are permitted.

Let a weighted binary tree have terminal weights \(w_j\) and positive definite inputs \(A_j\), and let \(M\) be its root. Then \(M\) is positive definite, commutes with invertible congruence, is monotone in the inputs, and satisfies \(\mathcal T(c_jA_j:j)=\bigl(\prod _jc_j^{w_j}\bigr)M\) for \(c_j{\gt}0\). A tree with constant positive definite input \(X\) has root \(X\). For every vector \(z\),

If \(P\) is positive semidefinite (in particular, an orthogonal projection) and \(A_j\ge c_jP\) with \(c_j{\gt}0\), then \(M\ge \bigl(\prod _jc_j^{w_j}\bigr)P\). Finally, if \(A_j=\prod _{g=1}^KA_{j,g}\) with positive definite factors and every factor from band \(g\) commutes with every factor from band \(g'\ne g\), then \(M=\prod _{g=1}^KM_g\), where \(M_g\) is the root of the same tree with inputs \(A_{j,g}\), and the factors \(M_g\) commute.

Fix real numbers \(s,G,t,c,C_f\) and a real sequence \((\varepsilon _k)\) such that \(\varepsilon _k/k\to 0\) as \(k\to \infty \). Put

Then \(E_k/k\to 0\). No sign condition on the five fixed coefficients or on \(\varepsilon _k\) is required.

This is the scalar limit for the explicit remainder in the rough lower norm estimate of [ Ope26b , Section 8 ] .

Let \(T\) be a weighted binary tree with labels in a finite set \(J\) and terminal weights \(w_j\). Let \(b{\gt}0\), let \(U\in \mathbb R\), and suppose \(L_j\le U\) for every \(j\in J\). Set

Then

The constants are independent of the tree and its terminal probabilities. Zero weights and repeated leaf labels are allowed. These are the uniform scalar estimates preceding the logarithmic lower comparison in [ Ope26b , Section 8 ] .

Let \(I\) be a finite set, let \(M\) be a positive definite matrix on \(\mathbb {C}^I\), let \(s\in \mathbb {R}\), and let \(p\in \mathbb {C}^I\) satisfy \(p\ne 0\) and \(\| p\| \le 1\). Put \(u=M^{-s}p\) and \(v=u/\| u\| \). Then

For elements \(a,b\) of a unital C\({}^\ast \)-algebra with \(0\le a,b\), an exponent \(p\in [1,2]\), and \(t\in [0,1]\), the map \(a\mapsto a^p\) is convex for the Loewner order on the positive cone:

This is the convex counterpart of the operator concavity of \(x\mapsto x^p\) for exponents \(p\in [0,1]\).

For \(b,d{\gt}0\), the matrix \(b(\mathbb {1}-P)+dP\) is positive definite.

For a positive unital map \(T\) and positive-definite \(A\), one has \(T(\log A)\le \log (T(A))\). This is the corrected logarithmic specialization associated with [ Wol12 , Corollary 5.2(3) ] . The printed common hypothesis \(T(\mathbb {1})\le \mathbb {1}\) is insufficient: in dimension one, for \(0{\lt}c{\lt}1\), the positive subunital map \(T(x)=cx\) and \(A=1\) would give \(0=T(\log A)\le \log (T(A))=\log c\). The corrected theorem instead assumes \(T(\mathbb {1})=\mathbb {1}\); see [ con26k ] .

For a positive subunital map \(T\), a positive semidefinite matrix \(A\), and \(p\in [0,1]\), one has \(T(A^p)\le (T(A))^p\). This is [ Wol12 , Theorem 5.13 ] for concave \(f(x)=x^p\).

For a positive subunital map \(T\), a positive semidefinite matrix \(A\), and \(p\in [1,2]\), one has \((T(A))^p\le T(A^p)\). This is [ Wol12 , Theorem 5.11 ] for the subunital operator-convex case \(f(x)=x^p\).

For a positive subunital map \(T\), a positive semidefinite matrix \(A\), and \(p\in [0,1]\), the inequality \(T(A^p)\le (T(A))^p\) holds.

For a positive subunital map \(T\), a positive semidefinite matrix \(A\), and \(p\in [1,2]\), the inequality \((T(A))^p\le T(A^p)\) holds.

Let \(M\) be a positive definite complex matrix on a finite-dimensional space, which may have dimension zero. Suppose that \(b{\gt}0\), \(s\le 0\), and \(bI\le M\). Then \(M^s\le b^sI\), with powers defined by spectral functional calculus.

Let \(I\) be a finite set, let \(M\) be a positive definite matrix on \(\mathbb {C}^I\), let \(s\in \mathbb {R}\), and let \(v\in \mathbb {C}^I\) be a unit vector. If \(N{\gt}0\) and \(N\| M^s v\| ^2\le 1\), then

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a positive linear map with \(T(\mathbb {1})\le \mathbb {1}\). If \(A\in M_{D}(\mathbb {C})\) admits a positive semidefinite dominant \(B\ge 0\) with \(A^\dagger A\le B\) and \(B\) commuting with \(A\) (\(BA=AB\)), then

This is [ Wol12 , Theorem 5.6 ] .

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a positive linear map with \(T(\mathbb {1})\le \mathbb {1}\). If \(A\in M_{D}(\mathbb {C})\) is normal, then \(T(A^\dagger A)-T(A^\dagger )T(A)\ge 0\). This is [ Wol12 , Proposition 5.1 ] .

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a positive linear map with \(T(\mathbb {1})\le \mathbb {1}\). If \(A\in M_{D}(\mathbb {C})\) is subnormal (i.e. there exists a normal operator on a larger space whose compression to \(\mathbb {C}^D\) is \(A\)), then \(T(A^\dagger A)-T(A^\dagger )T(A)\ge 0\). This is [ Wol12 , Theorem 5.5 ] .

Let \(A\in M_{n}(\mathbb {C})\) be positive semidefinite and let \(v\in \mathbb {C}^n\) satisfy \(\langle v,v\rangle =1\) and \(P_{\operatorname{supp}(A)}v=v\). Then

Here \(\log A\) is defined by the continuous functional calculus with \(\log 0=0\). Compare the finite-dimensional vector-state Jensen argument in [ Bha97 , Chapter V ] .

Under the hypotheses of Theorem 7.16.7, let \(\rho \geq 0\) satisfy \(P\rho =\rho \). Then \(\operatorname {Re}\operatorname {tr}(\rho e^{A+B}) \leq c\operatorname {Re}\operatorname {tr}(\rho e^A)\). No trace normalization is required.

For \(p\in [0,1]\), PSD matrices \(A_1,A_2\), and \(t\in [0,1]\),

This follows from operator concavity of \(x\mapsto x^p\) composed with trace monotonicity on the Loewner order.

For \(p\in [1,2]\), PSD matrices \(A_1,A_2\), and \(t\in [0,1]\),

This follows from operator convexity of \(x\mapsto x^p\) composed with trace monotonicity on the Loewner order.

Let \(M\ge 0\) and \(N\ge 0\) satisfy \(MJ=JN\). For every \(x\in \mathcal H_2\), the unnormalized filtered vector satisfies \(M^{-1/4}Jx=JN^{-1/4}x\). If in addition \(J^*J=I\), then

Negative powers vanish on the kernels. Each quotient in (25) is defined to be zero when its denominator is zero, so the identities include \(x=0\).

Let \(M{\gt}0\) and \(N{\gt}0\) satisfy \(MJ=JN\). For every real \(u\), \(M^{-iu}J=JN^{-iu}\). For a fixed weighted tree with positive inputs \(A_i,B_i\) satisfying \(A_iJ=JB_i\), roots \(M_A,M_B\), and any vector \(y\) in the corresponding Hilbert space, define, for \(q\in \{ A,B\} \),

The vector need not be normalized; repeated labels and zero total label weights are permitted. No isometry condition on \(J\) is required.