Direct measurement of density-matrix elements using a phase-shifting technique Tianfeng

TL;DR

This paper introduces a phase-shifting based direct measurement protocol for quantum density matrices that eliminates the need for ancillary pointers, simplifying the process of extracting specific quantum state elements.

Contribution

The authors propose a novel direct measurement scheme using phase-shifting operations, applicable to both multiqubit and continuous-variable quantum states, without ancillary pointers.

Findings

Requires at most six expectation values to determine density matrix elements

Uses a quantum circuit with single-qubit and controlled-phase gates

Applicable to partial quantum state information extraction

Abstract

A direct measurement protocol allows reconstructing specific elements of the density matrix of a quantum state without using quantum state tomography. However, the direct measurement protocols to date are primarily based on weak or strong measurements with an ancillary pointer, which interacts with the investigated system to extract information about the specified elements. Here, we present a direct measurement scheme based on phase-shifting operations which do not need ancillary pointers. In this method, estimates of at most six expectation values of projective observables suffice to determine any specific element of an unknown quantum density matrix. A concrete quantum circuit to implement this direct measurement protocol for multiqubit states is provided, which is composed of just single-qubit gates and two multiqubit controlled-phase gates. This scheme is also extended for the direct measurement of the density matrix of continuous-variable quantum states. Our method can be used in quantum information applications where only partial information about the quantum state needs to be extracted, for example, problems such as entanglement witnessing, fidelity estimation of quantum systems, and quantum coherence estimation.