24 The Conditional Movement Estimate
This chapter states the first conditional operator estimate of [ Ope26b , Lemma 5.1 ] . A small subsystem \(x\) of a pure state changes sides between two systems \(U\) and \(P\); the norm of a vector obtained by conjugating the state with marginal powers then decays at the rate of the conditional information cost of the move. The constants are universal, and the admissible exponents depend only on the dimension of \(x\).
For a positive semidefinite matrix \(\rho \) write \(\Pi _{\ker \rho }\) for the spectral projection onto its kernel and \(\widehat\rho =\rho +\Pi _{\ker \rho }\) for its kernel completion.
For a square complex matrix \(\rho \), the kernel projection is \(\Pi _{\ker \rho }=\mathbf1_{\{ 0\} }(\rho )\), computed through the continuous functional calculus, and the kernel completion is \(\widehat\rho =\rho +\Pi _{\ker \rho }\).
Let \(\rho \) be Hermitian. Then \(\Pi _{\ker \rho }\rho =0\), \(\Pi _{\ker \rho }^2=\Pi _{\ker \rho }\), and for every real \(s\ne 0\), \(\widehat\rho ^{\, s}=\rho ^s+\Pi _{\ker \rho }\), where \(\rho ^s\) vanishes on the kernel.
All three identities are identities of functions on the spectrum of \(\rho \); for the last one use \(0^s=0\) when \(s\ne 0\).
Let \(\theta \) be a unit vector on four finite-dimensional systems \(P,x,U,F\), and let \(\rho _D\) denote its marginal on \(D\). Set \(d=\dim x\), \(\ell =\log (ed)\) and \(\eta =S(x|P)_\theta +S(x|U)_\theta \). There are universal constants \(c,C{\gt}0\) such that, for \(0{\lt}a\ell \le c\) and arbitrary positive semidefinite \(\sigma \) on \(xP\) and \(\tau \) on \(xU\) of trace at most one,
One may take \(C=200000\) and \(c=1/8\).
Identify \(\theta \) with its coefficient matrix from \(xU\) to \(PF\). The kernel projection of \(\rho _U\) annihilates \(\theta \), since its expectation in \(\theta \) is \(\operatorname {tr}(\Pi _{\ker \rho _U}\rho _U)=0\). The kernel projection of \(\rho _P\) annihilates every vector obtained from \(\theta \) by an operator on \(xU\), because the two operators act on different factors. By the preceding lemma both kernel completions may therefore be replaced by the powers vanishing on the kernels. For these powers the estimate is the interpolation argument of [ Ope26b , proof of Lemma 5.1 ] : the analytic family \(G(z)=\sigma ^{az}\rho _P^{-az}\tau ^{a(1-z)}\rho _U^{-a(1-z)}\theta \) on the strip \(0\le \operatorname {Re}z\le 1\) is bounded on the left boundary by the conditional Schatten estimate with the slack \(\lambda (u)\) retained, the slack controls the movement of the state arriving at the right boundary, continuity of the conditional entropy with controlled system \(x\) bounds the right boundary, and the three-lines inequality with harmonic weights \(du/(2\cosh \pi u)\) combines the two. Faithful \(\sigma ,\tau \) are removed by regularization of their eigenvalues.