Quantum Information and Quantum Computation
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Optimal precision dependence of low-energy Hamiltonian simulation

What is the tight precision dependence of worst-case query complexity for low-energy simulation in the regime (2)? Let \(A\in\mathbb{C}^{N\times N}\), \(\lVert A\rVert\leq1\), \(H=\lambda A^\dagger A\), and \(P_\Delta=\mathbf{1}_{[0,\Delta]}(H)\), where \(\lambda>0\) and \(0<\Delta\leq\lambda\). Assume an exact block encoding \((\langle0^m\rvert\otimes I_N)V_A(\lvert0^m\rangle\otimes I_N)=A\), with controlled and inverse calls. Count these queries; input-state preparation is excluded. For known \(t>0\) and \(0<\epsilon<1/2\), a unitary simulator \(W\) must satisfy (1) uniformly on the promised subspace:

\begin{equation} \sup_{\substack{\lVert\psi\rVert=1\\P_\Delta\lvert\psi\rangle=\lvert\psi\rangle}} \left\lVert W(\lvert0^a\rangle\lvert\psi\rangle) -\lvert0^a\rangle e^{-itH}\lvert\psi\rangle\right\rVert\leq\epsilon. \tag{1} \end{equation}

Here \(a\) counts workspace qubits. Consider asymptotic families with \(\epsilon\to0\) satisfying

\begin{equation} \epsilon=o(t\Delta),\qquad t\Delta=o\bigl(\log(1/\epsilon)\bigr),\qquad \log(1/\epsilon)=o(t\lambda). \tag{2} \end{equation}

Determine whether the known \(O(\sqrt{t\lambda\log(1/\epsilon)})\) upper bound has optimal precision dependence.

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Additivity of the relative entropy of entanglement

Does the relative entropy of entanglement of every bipartite state equal its regularization, or is regularization genuinely necessary? For a finite-dimensional bipartite system \(A{:}B\), write \(\operatorname{Sep}(A{:}B)\) for the set of separable states and \(D(\rho\Vert\sigma)=\operatorname{Tr}[\rho(\log_2\rho-\log_2\sigma)]\) for the Umegaki relative entropy, defined when \(\operatorname{supp}\rho\subseteq\operatorname{supp}\sigma\), with the trace evaluated on \(\operatorname{supp}\rho\) and \(0\log_2 0:=0\). Define the relative entropy of entanglement and its regularization by

\begin{equation} E_R(\rho) :=\min_{\sigma\in\operatorname{Sep}(A{:}B)}D(\rho\Vert\sigma), \qquad E_R^\infty(\rho) :=\lim_{n\to\infty}\frac1nE_R\bigl(\rho^{\otimes n}\bigr). \tag{1} \end{equation}

The limit in Eq. (1) exists and equals \(\inf_{n\geq1}\frac1nE_R(\rho^{\otimes n})\) by Fekete’s lemma: the product of minimizing separable states is separable and \(D\) is additive on tensor products, which gives the subadditivity \(E_R(\rho\otimes\sigma)\leq E_R(\rho)+E_R(\sigma)\), and subadditivity implies convergence of the normalized terms to their infimum, not that each of them is nonincreasing. The archived question is whether single copies already suffice, that is, whether

\begin{equation} E_R^\infty(\rho)=E_R(\rho) \quad\text{for every finite-dimensional bipartite state }\rho. \tag{2} \end{equation}

Since \(E_R^\infty(\rho)\leq E_R(\rho)\) always holds, Eq. (2) can only fail strictly, through a single state \(\rho\) with

\begin{equation} E_R^\infty(\rho)<E_R(\rho). \tag{3} \end{equation}
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