We give an algorithm which, given $n = O(d^2 \cdot (\log\log(d)/\log(d))^2)$ copies of $\rho$, estimates the eigenvalues of $\rho$ to constant error in total variation distance. Thus, we can learn the eigenvalues of a quantum state with fewer copies than the $\Theta(d^2)$ needed to run full state tomography. This is the first improvement to spectrum estimation over the influential Keyl-Werner algorithm, which uses $n = \Theta(d^2)$ copies, thereby resolving a question raised by Keyl and Werner in 2001 and refuting a 2016 conjecture of Wright. Our main technical tool is a new tomography guarantee, where the error of tomography in a particular direction $|w\rangle$ scales with $\langle w | \rho |w\rangle$ for all directions simultaneously. From this stronger"relative-error"bound, we recover better algorithms for principal component analysis in Bures distance and tomography in $\chi^2$-divergence as corollaries.
An optimal estimator is established under the sole promise that one of the two states is pure, without knowing which one, under the sole promise of which state is pure.
It is well-known that learning a pure $n$-qubit stabilizer state $|\psi\rangle$ both requires, and can be accomplished with, access to a number of copies of $|\psi\rangle$ linear in $n$. However, the precise constant coefficient of this scaling does not appear to have been determined. Here we prove that $L_\delta(n)$,...
Rebecca Chang, Matthias C. Caro, Martín Larocca et al.· 4 citations
In Online Shadow Tomography, we are given copies of an unknown $d$-dimensional quantum state $\rho$, an adversary (adaptively) proposes a sequence of bounded observables $A^{(1)},\ldots,A^{(m)}$, and after each $A^{(t)}$ is given we must estimate $\mathrm{Tr}(A^{(t)}\rho)$ to within $\pm \epsilon$. This is the direct q...
Sitan Chen, R. O'Donnell, Angelos Pelecanos et al.· arXiv.org· 1 citation· ⚡1
We study the best separable state problem (BSS), which asks for the maximum acceptance probability of a quantum measurement over unentangled states. In classical terms, the goal is to maximize $\langle(x \otimes y), M (x \otimes y)\rangle$ over unit vectors $x,y$ where $0 \preceq M \preceq I$; we call this value $\math...
Prashanti Anderson, Sam Hopkins, Amit Rajaraman· 1 citation
A randomized fully non-adaptive protocol is constructed that fixes all queries before observing the data and matches the optimal adaptive sample complexity, giving a negative answer to the COLT 2026 open problem asking whether interaction is necessary for order-optimal one-bit mean estimation.
Shadow Tomography is a fundamental problem in quantum information theory. Given multiple copies of an unknown $d$-dimensional quantum state $\rho$ and a known collection of observables $E_1,\ldots,E_M$, the goal is to estimate all expectation values $\{\text{Tr}(\rho E_i)\}_{i=1}^M$ to additive accuracy $\varepsilon$ w...
F. G. Jeronimo, Qi-Zhao Huang, Le Liu· 2 citations
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