26 The Conditional Skew Estimate
This chapter formalizes the conditional skew estimate of [ Ope26b , Lemma 5.3 ] : a local positive expectation conjugated by marginal real powers departs from the nonnegative real axis by at most a fractional power of a conditional mutual information. Throughout, \(\theta \) is a unit vector on finite-dimensional systems \(P,x,U,F\) with \(P=P_0P_1\), \(Y=xU\), \(\rho _D\) denotes a marginal of \(\theta \), powers \(\rho ^{[z]}\) vanish on the kernel, and \(0\le h\le I\) acts on \(P_0x\). Put \(d_h=(\dim P_0)(\dim x)\), \(\ell _h=\log (ed_h)\), \(\eta =I(x:F|U)_\theta \) and
26.1 The scalar function
The function \(f\) is entire, \(f(0)=\langle \theta ,h\theta \rangle \ge 0\) and \(f(-\bar z)=\overline{f(z)}\). On the imaginary axis \(|f|\le 1\), and \(|f(z)|\le d_h^2\) for \(|\operatorname {Re}z|\le \frac12\).
Each matrix entry of a support power is an entire function of the exponent. On the imaginary axis \(f(iy)\) is an expectation of \(h\) in a unit vector. For the strip bound, expand \(h\) in matrix units on \(P_0x\); each of the at most \(d_h^2\) terms is an inner product of two vectors of the form \(\rho ^{[z]}c\, \rho ^{[-z]}\theta \) with \(\lVert c\rVert \le 1\), and \(\lVert \rho ^{[z]}c\, \rho ^{[1/2-z]}\rVert _2\le \lVert c\rVert \) for \(0\le \operatorname {Re}z\le \frac12\). The reflection treats negative real parts.
Let \(g\) be entire with \(g(-\bar z)=\overline{g(z)}\), \(g(0)=p\ge 0\), \(|g|\le 1\) on the imaginary axis and \(|g|\le m\) on \(|\operatorname {Re}z|\le \frac14\), where \(1\le m\), \(1\le \ell \) and \(\log m\le 2\ell \). Suppose that for some \(R\ge 0\) and every real \(u\),
Then for real \(|t|\le 1/(16\ell )\),
Three lines on the two half-strips bounds \(g\) by a universal constant on \(|\operatorname {Re}z|\le 1/(8\ell )\). The phase hypothesis gives \(g'(0)\in i\mathbb R\) and \(|g'(0)|\le 4\pi \sqrt p\, R\). The function \((g(z)-p)e^{z^2}\) is bounded by \(C\min (1,R)\) on the imaginary axis and by a constant on the outer lines, so three lines and a Cauchy estimate bound the second-order Taylor remainder of \(g\) at zero. Finally \(|p+itb|-p\le t^2b^2/p\) for real \(b\) when \(p{\gt}0\), and \(g'(0)=0\) when \(p=0\).
26.2 The imaginary axis
For a vector \(\theta \) on \(A\otimes B\) and any function \(g\) on the real line, \((g(\rho _A)\otimes I)\theta =(I\otimes g(\rho _B))\theta \).
Write \(\theta \) as a coefficient matrix \(C\), so \(\rho _A=CC^*\) and \(\rho _B=(C^*C)^T\). The operators act by \(C\mapsto g(CC^*)C\) and \(C\mapsto C\, g(C^*C)\), which agree since \(CC^*\cdot C=C\cdot C^*C\).
With \(p=\langle \theta ,h\theta \rangle \) and \(\mathcal R_\eta \) the rate of Theorem 25.3.2 for \(d=\dim x\), every real \(u\) satisfies
The kernel parts of the support powers annihilate \(\theta \), and Schmidt mirroring across \(P\) and \(YF\) gives \(f(iu)=\langle w_u,hw_u\rangle \) with \(w_u=\widehat\rho _Y^{\, iu}\widehat\rho _{YF}^{\, -iu}\theta \). The vector \(w_u^0=\widehat\rho _U^{\, iu}\widehat\rho _{UF}^{\, -iu}\theta \) is a unitary on \(UF\) applied to \(\theta \), so \(\langle w_u^0,hw_u^0\rangle =p\). Multiplying \(w_u-w_u^0\) by \(\widehat\rho _P^{\, iu}\) and \(\widehat\rho _U^{\, -iu}\) and mirroring again gives \(\lVert w_u-w_u^0\rVert =\lVert (\widehat\rho _U^{\, -iu}\widehat\rho _Y^{\, iu} -\widehat\rho _{UF}^{\, -iu}\widehat\rho _{YF}^{\, iu})\theta \rVert \), which Theorem 25.3.2 bounds by \(2|\sinh \pi u|\, \mathcal R_\eta \). For \(0\le h\le I\), \(|\langle w,hw\rangle -\langle w^0,hw^0\rangle |\le 2\sqrt{\langle w^0,hw^0\rangle }\, \lVert w-w^0\rVert +\lVert w-w^0\rVert ^2\).
26.3 The estimate
For a density matrix on \(xUF\), \(0\le I(x:F|U)\le 2\log \dim x\). If \(1\le D\), \(\log (eD)\le \ell \) and \(\eta \le 2\ell \), then the rate of Theorem 25.3.2 satisfies \(\mathcal R_\eta \le 3\ell \min (1,\eta )^{1/4}\).
Nonnegativity is strong subadditivity, and the upper bound writes \(I(x:F|U)=S(x|U)-S(x|UF)\) with both conditional entropies bounded in absolute value by \(\log \dim x\). For \(0{\lt}\eta \le 1\) use \(-\eta \log \eta \le 2\sqrt\eta \); for \(\eta {\gt}1\) the rate is at most \(1+\sqrt{2}\, \ell \).
There is a universal constant \(C\) such that for \(0{\lt}a\ell _h\le \frac18\) and \(t=a/2\),
In particular the left side is zero if \(\eta =0\). One may take \(C=18\pi ^2+6144\, e^{\pi ^2+1}\).
Apply Theorem 26.1.2 to \(f\) with \(m=d_h^2\), \(\ell =\ell _h\), \(R=\mathcal R_\eta \) and \(p=\langle \theta ,h\theta \rangle \), using Lemmas 26.1.1 and 26.2.2. Since \(\dim x\le d_h\) and \(\eta \le 2\log \dim x\le 2\ell _h\), Lemma 26.3.1 gives \(\mathcal R_\eta ^2\le 9\ell _h^2\eta ^{1/8}\) and \(\sqrt{\min (1,\mathcal R_\eta )}\le 2\ell _h\eta ^{1/8}\).