Complete Quantum-State Tomography with a Local Random Field

Abstract

Single-qubit measurements are typically insufficient for inferring arbitrary quantum states of a multi-qubit system. We show that if the system can be fully controlled by driving a single qubit, then utilizing a local random pulse is almost always sufficient for complete quantum-state tomography. Experimental demonstrations of this principle are presented using a nitrogen-vacancy (NV) center in diamond coupled to a nuclear spin, which is not directly accessible. We report the reconstruction of a highly entangled state between the electron and nuclear spin with fidelity above 95%, by randomly driving and measuring the NV-center electron spin only. Beyond quantum-state tomography, we outline how this principle can be leveraged to characterize and control quantum processes in cases where the system model is not known. arXiv:1909.09980v1 [quant-ph]

Pulsed characteristic-function measurement of a thermalizing harmonic oscillator

“…The situation changes appreciably in systems where such a direct access is not attainable. In this kind of scenario, an established method is to employ a readily accessible measurement probe [9][10][11][12][13][14][15], often given by two-level quantum systems. For example, such a procedure has been employed in a well-known probe-based direct measurement of the Wigner function [9].…”

Drawing together control landscape and tomography principles

“…Turning to tomography of a generic state, it follows that expectation value measurements of H c at times T * yield for almost all control fields an informationally complete measurement record. Unitary invariance of the Haar measure then immediately implies that information completeness generically holds when observables that are unitarily conjugate to H c , i.e., M = V † H c V with V ∈ SU(d), are considered [42]. Both results together can be leveraged to find control fields in a learning fashion when system access is limited, for instance, to a single qubit only [54].…”

“…That is, observables that are not directly accessible in the laboratory are typically created by rotating accessible observables into the desired ones by control fields or gates sequences [1,[35][36][37][38]. Alternatively, one can infer the state of such driven systems directly from the time traces of accessible observables [1,[39][40][41][42]. In this case, generic state reconstruction is possible when the obtained data is informationally complete.…”

“…Instead of deterministically creating observables, a set of observables randomly created through Haar random unitary transformations can also be used, as this almost always guarantees information completeness [41,[43][44][45]. Since a random control field can create a Haar random unitary evolution [46], applying a random field and measuring the time trace of a single observable almost always yields informationally complete measurement data [42]. This eliminates the need to optimize control pulses for rotating observables, as almost all control fields allow for complete quantum state tomography.…”

“…In each iteration step i a new field f (i+1) is determined by adding a random field to infer J and ∇ f J through expectation value measurements of an observable M at time points {t1, • • • , t d 2 −1 }. This procedure allows for reconstructing the state of the system created by f (i) , and thereby J, by directly inverting M or solving a least squares problem [42]. If needed, varying the applied field by δf then allows for evaluating the gradient ∇ f J in a similar fashion, e.g.…”