25 Comparing Marginal Phases
This chapter formalizes the resolvent argument behind the marginal-phase comparison of [ Ope26b , Lemma 5.2 ] : a change in products of marginal imaginary powers is controlled by a conditional mutual information, with constants depending only on the dimension of the moved system. The faithful case is proved first and singular states are reached by a faithful approximation.
25.1 The resolvent defect of a compression
Let \(A\) and \(B\) be positive definite operators with orthonormal eigenbases, \(V\) an isometry with \(V^*AV=B\), \(b_0\) a vector and \(a_0=Vb_0\). For \(v{\gt}0\) put \(h(v)=\langle a_0,(A+v)^{-1}a_0\rangle -\langle b_0,(B+v)^{-1}b_0\rangle \) and \(\delta _v=(A+v)^{-1}a_0-V(B+v)^{-1}b_0\).
\(\langle \delta _v,(A+v)\delta _v\rangle =h(v)\). In particular \(h(v)\ge 0\) and \(v\lVert \delta _v\rVert ^2\le h(v)\).
Expand the quadratic form and use \(V^*(A+v)V=B+v\) and \(V^*V=1\).
For \(v{\gt}0\), \(h(v)\le \langle a_0,A^{-1}a_0\rangle \) and \(h(v)\le \langle b_0,Bb_0\rangle /v^2\). Moreover \(h\) is integrable on \((0,\infty )\) and \(\int _0^\infty h(v)\, dv=\langle b_0,\log B\, b_0\rangle -\langle a_0,\log A\, a_0\rangle \).
In eigencoordinates \(h(v)=\sum _kp_k/(\lambda _k+v)-\sum _jq_j/(\mu _j+v)\) with \(\sum _kp_k=\sum _jq_j\), and \(\int _0^\infty \bigl((\lambda +v)^{-1}-(1+v)^{-1}\bigr)dv=-\log \lambda \).
For \(\lambda {\gt}0\) and \(-1{\lt}\operatorname {Re}z{\lt}0\), \(\int _0^\infty v^z(\lambda +v)^{-1}dv=-\frac{\pi }{\sin \pi z}\lambda ^z\). For real \(u\ne 0\), \(\int _0^\infty v^{-iu}\bigl((\lambda +v)^{-1}-(1+v)^{-1}\bigr)dv =-\frac{\pi }{\sin (-i\pi u)}(\lambda ^{-iu}-1)\).
The substitution \(v=\lambda t/(1-t)\) gives the Beta integral \(B(z+1,-z)=\Gamma (z+1)\Gamma (-z)\), and the reflection formula evaluates it. The second identity follows from the first by dominated convergence as \(\operatorname {Re}z\uparrow 0\).
For every real \(u\), \(\lVert A^{-iu}a_0-VB^{-iu}b_0\rVert \le \frac{|\sinh \pi u|}{\pi } \int _0^\infty \sqrt{h(v)/v}\, dv\). If \(\langle a_0,A^{-1}a_0\rangle \le D^2\), \(\langle b_0,Bb_0\rangle \le 1\) and \(D\ge 1\), then \(\lVert A^{-iu}a_0-VB^{-iu}b_0\rVert \le 2|\sinh \pi u|\, \mathcal R_\eta \) with \(\eta =\int _0^\infty h\), \(y=\min (1,\eta )\), \(\mathcal R_\eta =\sqrt y+\sqrt{\eta \log (eD/y)}\) and \(\mathcal R_0=0\).
Pair the phase gap with a vector and use the second resolvent integral. Split \(\int \sqrt{h/v}\) at \(y/D^2\) and \(1/y\); bound the ends by the tails and the middle by the arithmetic–geometric mean inequality.
25.2 The faithful case
For positive semidefinite \(\rho \) on \(X\otimes Y\), \(\rho \le (\dim X)\, (I_X\otimes \operatorname {tr}_X\rho )\).
Write \(\rho =B^*B\) and apply the Cauchy–Schwarz inequality over the \(\dim X\) blocks.
Let \(\rho \) be a positive definite density matrix on \(xUF\), \(d=\dim x\) and \(\eta =I(x:F|U)_\rho \). For every real \(u\),
On Hilbert–Schmidt spaces take \(A=L_{\sigma _f}R_{\rho ^{-1}}\) and \(B=L_{\sigma _s}R_{\rho _{xU}^{-1}}\) with \(\sigma _f=I_x/d\otimes \rho _{UF}\) and \(\sigma _s=I_x/d\otimes \rho _U\), the isometry \(V(Y\sqrt{\rho _{xU}})=(Y\otimes I_F)\sqrt\rho \) and \(b_0=\sqrt{\rho _{xU}}\). Then \(\langle a_0,A^{-1}a_0\rangle =\operatorname {tr}\rho \sigma _f^{-1}\rho \le d^2\) by the domination \(\rho \le d^2\sigma _f\), \(\langle b_0,Bb_0\rangle =1\), and the logarithmic gap is \(I(x:F|U)_\rho \). The phase gap is the displayed vector up to a unimodular scalar.
25.3 The general case
For a Hermitian matrix \(\rho \) write \(\widehat\rho ^{\, iu}\) for the kernel-completed phase, equal to \(\rho ^{iu}\) on the support of \(\rho \) and to the identity on its kernel.
For a positive semidefinite \(\rho \) on \(xUF\), \((I_x\otimes \Pi _{\ker \rho _{UF}})\sqrt\rho =0\), \((\Pi _{\ker \rho _{xU}}\otimes I_F)\sqrt\rho =0\) and \(\bigl((I_x\otimes \Pi _{\ker \rho _U})\widehat\rho _{xU}^{\, iu}\otimes I_F\bigr) \sqrt\rho =0\).
A Hermitian projection \(K\) with \(\operatorname {tr}(\rho K)=0\) annihilates \(\sqrt\rho \). The traces are traces of a kernel projection against the corresponding marginal. For the last identity, \(\widehat\rho _{xU}^{\, iu}\) maps the support of \(\rho _{xU}\) into itself and that support lies in \(x\otimes \operatorname {supp}\rho _U\).
Let \(\theta \) be a unit vector on finite-dimensional systems \(P,x,U,F\), write \(\rho _D\) for its marginals, and put \(d=\dim x\) and \(\eta =I(x:F|U)_\theta \). For every real \(u\),
In particular the two phase products agree on \(\theta \) if \(\eta =0\). The constant does not depend on \(\dim P\), \(\dim U\) or \(\dim F\).
For an operator \(T\) on \(xUF\), \(\lVert T\theta \rVert =\lVert T\sqrt\rho \rVert _2\) with \(\rho =\rho _{xUF}\). Apply the faithful case to \(\rho _\epsilon =(1-\epsilon )\rho +\epsilon I/N\), whose marginals are affine functions of the marginals of \(\rho \). In the eigenbases of \(\rho \) and its marginals, every phase of \(\rho _\epsilon \) converges off the kernel and equals a common unimodular scalar on the kernel; the support inclusions make the kernel parts vanish in the limit. The conditional mutual information of \(\rho _\epsilon \) converges to \(\eta \) and \(\mathcal R\) is continuous on \([0,\infty )\).