Dependencies

For complex square matrices \(H\) and \(A\), set \(\mathcal F_H(A)=\int _{\mathbb {R}} f(t)\, e^{itH}Ae^{-itH}\, dt\), a Bochner integral for the operator norm.

Define \(\chi (\omega )=e\exp [-(1-(\omega /\delta )^2)^{-p}]\) for \(|\omega |{\lt}\delta \) and \(\chi (\omega )=0\) for \(|\omega |\ge \delta \).

Define \(f(t)=(2\pi )^{-1}\int _{\mathbb {R}}\chi (\omega )\cos (t\omega )\, d\omega \).

If \(0{\lt}\delta \le \Delta \), then \(\mathcal F_H(A)\Omega =\langle \Omega ,A\Omega \rangle \Omega \) for every matrix \(A\).

For unit vectors \(\psi ,\phi \) the trace distance of \(|\psi \rangle \langle \psi |\) and \(|\phi \rangle \langle \phi |\) is \((1-|\langle \psi ,\phi \rangle |^2)^{1/2}\).

If \(a\in \mathbb {C}\) satisfies \(|a-1|\le 1/(4p)\), then \(\operatorname {Re}a^{-p}\ge 1/2\). Consequently, for real \(u\in (-1,1)\), \(w=1-u^2\) and \(|z-u|\le w/(12p)\), one has \(1-z^2\ne 0\) and \(\operatorname {Re}(1-z^2)^{-p}\ge w^{-p}/2\).

The kernel is real and even, \(f(t)=(2\pi )^{-1}\int \chi (\omega ) e^{-it\omega }\, d\omega \), and for \(n\le pk\),

There is \(K_0{\gt}0\) with \(|f(t)|\le K_0(1+|t|)^{-2}e^{-(c/2)T^\alpha }\) whenever \(0\le T\le |t|\), where \(c\) is the decay rate of the kernel.

In a unital C*-algebra, \(\| a^{1/2}-b^{1/2}\| \le \| a-b\| ^{1/2}\) for \(a,b\ge 0\). For self-adjoint \(a,b\) one has \(a\le |a|\) and \(\| |a|-|b|\| \le ((\| a\| +\| b\| )\| a-b\| )^{1/2}\).

Let \(0{\lt}\delta \le \Delta \) and \(M_i=\mathcal F_H(h_i)-\langle \Omega ,h_i\Omega \rangle I\). Then \(M_i\) is Hermitian, \(M_i\Omega =0\), \(\sum _iM_i=H-E_0I\), and \(\| M_i\| \le \| h_i\| (\| f\| _1+1)\).

Let \(H\) be Hermitian. The integral defining \(\mathcal F_H(A)\) converges in norm, \(\mathcal F_H\) is linear, \(\| \mathcal F_H(A)\| \le \| f\| _1\| A\| \), \(\mathcal F_H(A)^\dagger =\mathcal F_H(A^\dagger )\), and \(\mathcal F_H(H)=H\).

Let \(H\) be Hermitian with \(Hv=Ev\) and \(Hw=E'w\). Then \(\langle v,\mathcal F_H(A)w\rangle =\chi (E-E')\langle v,Aw\rangle \). In an orthonormal eigenbasis \((u_k)\) of \(H\) with eigenvalues \(\lambda _k\), the entries of \(\mathcal F_H(A)\) are those of \(A\) multiplied by \(\chi (\lambda _k-\lambda _l)\). If moreover every eigenvalue \(\lambda _k\) equals \(E'\) or satisfies \(|\lambda _k-E'|\ge \delta \), then \(\mathcal F_H(A)w=\sum _{\lambda _k=E'}\langle u_k,Aw\rangle u_k\), the projection of \(Aw\) onto the \(E'\)-eigenspace.

Let \(C\ge 0\) and \(v,c{\gt}0\). There are \(C',c'{\gt}0\), depending only on \(p,\delta ,C,v,c\), such that \(\| \tau _t(A)-L\tau _t(A)\| \le \| A\| \min \{ 2,Ce^{v|t|-cl}\} \) for all \(t\) implies \(\| \mathcal F_H(A)-L\mathcal F_H(A)\| \le C'e^{-c'l^\alpha }\| A\| \).

Let \(H_F\ge g(I-P_\Omega )\) with \(H_F\Omega =0\), \(\| \Omega \| =1\), \(g{\gt}0\), and let \(H'\ge 0\) with \(\| H'-H_F\| \le \epsilon \le g/4\). Then \(H'\) has a unit ground vector \(\Omega _0\) with energy \(e_0\in [0,\epsilon ]\) and \(H'-e_0I\ge (g/2)(I-P_{\Omega _0})\); after a choice of phase \(\| \Omega _0-\Omega \| \le 2(\epsilon /g)^{1/2}\), and the trace distance of the two pure states is at most \((2\epsilon /g)^{1/2}\).

If \(L\) is a contraction, \(M\) and \(LM\) are Hermitian, and \(L\) fixes \(|LM|\), then \(\| |M|-L|M|\| \le 2(2\| M\| \| M-LM\| )^{1/2}\). Consequently, if \(L\) is moreover unital and preserves Hermitian matrices, and \(h\) satisfies the hypothesis of Theorem 20.7.2 with \(\| h\| \le J\), then the positive constraint \(k=|M|/c_*\) built from \(h\) satisfies \(\| k-Lk\| \le C_3e^{-c_3l^\alpha }\), with constants depending only on \(p,\Delta ,J,C,v,c\).

Take \(\delta =\Delta /2\), \(c_*=\max \{ 1,J(\| f\| _1+1)\} \), and \(k_i=|M_i|/c_*\). Then \(0\le k_i\le I\), \(k_i\Omega =0\), and

[ Ope26b , Lemma 4.2 ] . The cutoff \(\chi \) is smooth and compactly supported with \(\chi (0)=1\); the real even function \(f(t)=(2\pi )^{-1}\int \chi (\omega )e^{-it\omega }\, d\omega \) satisfies

with \(C,c{\gt}0\) depending only on \(p\) and \(\delta \). In particular \(\int _{\mathbb {R}} f=1\).

Let \(0\le k,k'\le I\), \(G=(I-k)^{1/2}\), \(K=k^{1/2}\), and define \(G',K'\) from \(k'\). Then \(G^2+K^2=I\), and \(\| G-G'\| ,\| K-K'\| \le \| k-k'\| ^{1/2}\); if \(k\Omega =0\), then \(K\Omega =0\) and \(G\Omega =\Omega \). The map \(\mathcal E(B)=GBG+KBK\) is unital and completely positive, and for every finite auxiliary system the extension \(\mathcal E\otimes \mathrm{id}\) is a contraction in operator norm. Moreover, uniformly in the auxiliary dimension,

and \(\mathcal E\) fixes every matrix commuting with \(k\).

Let \((k_i)_{i\in I}\) be positive contractions on a finite-dimensional complex Hilbert space, and suppose that \(k_i\Omega =0\) for every \(i\). Put \(G_i=(I-k_i)^{1/2}\), \(K_i=k_i^{1/2}\) and \(\mathcal E_i(B)=G_iBG_i+K_iBK_i\). Then, for every matrix \(B\),

More generally, for every finite word \(w=(i_1,\ldots ,i_m)\),

No normalization of \(\Omega \) is required. Repeated letters and the empty word are allowed, with both sides equal to \(B\Omega \) for the empty word.

The cutoff \(\chi \) is infinitely differentiable on \(\mathbb {R}\), even, supported in \([-\delta ,\delta ]\), takes values in \([0,1]\), and \(\chi (0)=1\). For all integers \(n,k\ge 0\) with \(n\le pk\) and all \(\omega \),

Let \(g(z)=e\exp [-(1-z^2)^{-p}]\), analytic away from \(z=\pm 1\). For \(u\in (-1,1)\), \(w=1-u^2\) and every \(n\ge 0\),

and for every integer \(k\) with \(n\le pk\),

For all real \(t\), \(|f(t)|\le C e^{-c|t|^\alpha }\) with \(C=2\delta e/\pi \) and \(c=\log 2\cdot \kappa _p\, \delta ^\alpha \), where \(\kappa _p=(4(12p^2)^p)^{-1/(p+1)}\). Every polynomial moment \(t\mapsto t^nf(t)\) is integrable.