Sample-optimal tomography of quantum states

TL;DR

This paper introduces a near-optimal measurement scheme for quantum state tomography that significantly reduces the number of copies needed to accurately determine an unknown quantum state, matching lower bounds up to logarithmic factors.

Contribution

It provides a theoretically optimal POVM measurement scheme for quantum state tomography with near-matching lower bounds, improving efficiency over previous methods.

Findings

The measurement scheme requires O(dr/δ) log(d/δ) copies for infidelity error δ.

Matching lower bounds show the scheme's near-optimality.

Implementation is feasible on a quantum computer in polynomial time.

Abstract

It is a fundamental problem to decide how many copies of an unknown mixed quantum state are necessary and sufficient to determine the state. Previously, it was known only that estimating states to error $\epsilon$ in trace distance required $O(dr^2/\epsilon^2)$ copies for a $d$-dimensional density matrix of rank $r$. Here, we give a theoretical measurement scheme (POVM) that requires $O (dr/ \delta ) \ln (d/\delta) $ copies of $\rho$ to error $\delta$ in infidelity, and a matching lower bound up to logarithmic factors. This implies $O( (dr / \epsilon^2) \ln (d/\epsilon) )$ copies suffice to achieve error $\epsilon$ in trace distance. We also prove that for independent (product) measurements, $\Omega(dr^2/\delta^2) / \ln(1/\delta)$ copies are necessary in order to achieve error $\delta$ in infidelity. For fixed $d$, our measurement can be implemented on a quantum computer in time polynomial in $n$.