Simulating all quantum measurements using only projective measurements and postselection

Abstract

We report an alternative scheme for implementing generalized quantum measurements that does not require the usage of auxiliary system. Our method utilizes solely: (a) classical randomness and post-processing, (b) projective measurements on a relevant quantum system and (c) postselection on non-observing certain outcomes. The scheme implements arbitrary quantum measurement in dimension d with the optimal success probability 1/d. We apply our results to bound the relative power of projective and generalised measurements for unambiguous state discrimination. Finally, we test our scheme experimentally on IBM quantum processor. Interestingly, due to noise involved in the implementation of entangling gates, the quality with which our scheme implements generalized qubit measurements outperforms the standard construction using the auxiliary system.

Implementation of a general single-qubit positive operator-valued measure on a circuit-based quantum computer

“…For example, in reference [15] a quantum Fourier transform is used to implement a restricted class of projective POVMs. In references [16,17] a probabilistic method, based on classical randomness and post-selection, is proposed to implement projective POVMs. A deterministic method to perform a general POVM can be implemented using Neumark's dilation theorem [18,19], which states that a POVM of n elements can be performed as a projective measurement in a n-dimensional space.…”

Relationship between subjecting the qubit to dynamical decoupling and to a sequence of projective measurements

“…In fact, there are several pieces of evidence showing that the two above-mentioned classes of operations can provide similar information, provided that several sequences of measurements are together taken into account. Examples include observation that positive operator valued measures (POVMs) can be explained in terms of projective measurements [31,32]; derivation of noisy quantum channel decoding efficiency bound via the expansion over projective sequential measurements [33]; projective measurement(-preparations) reconstruction of non-Markovian dynamics over limited controls [34], and explanation of non-Markovian control in terms of alternative formalism of quantum stochastic process constructed from a set of measurement-preparation pairs [17]. For this work, the most interesting is the observation that a widely used [12,13,[35][36][37][38][39][40][41][42][43][44] dynamical decoupling based environmental noise spectroscopy scheme [12,13], can be emulated by a protocol in which only measurements are performed on the qubit [27][28][29], when the environment can be treated as a source of classical external noise acting on the qubit.…”

Implementation of POVMs by Projective Measurements and Postselection:Optimal Strategies and Applications to Unambiguous State Discrimination

“…In the recent work [1] it was proved that arbitrary quantum measurement (POVM) can be simulated by projective measurements if one allows for standard classical operations (randomization and post-processing), followed by postselection. The concrete algorithm of simulation was presented, with probability of success equal to the 1 d for d-dimensional quantum system.…”

“…First, we show what is the best possible projective-simulable measurement for USD without postselection. Then we present the strategy of simulation which in the task of USD performs better than 1 d bound from [1]. Finally, we formulate a problem of finding optimal strategy of simulation with postselection for a given quantum measurement.…”

“…We will denote set of n-outcome PM-simulable measurements on space of dimension d as SP (n, d). In recent work [1] the notion of PM-simulability was extended by the simulation of the POVM using projective measurements and postselection. To simulate n-outcome POVM M = (M 1 , M 2 , .…”