Simulating all quantum measurements using only projective measurements and postselection

We report an alternative scheme for implementing generalized quantum measurements that does not require the usage of an auxiliary system. Our method utilizes solely (a) classical randomness and postprocessing, (b) projective measurements on a relevant quantum system, and (c) postselection on nonobserving 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 generalized measurements for unambiguous state discrimination. Finally, we test our scheme experimentally on an 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 an auxiliary system.

Figure 1

The idea of simulation of a generalized measurement with projective measurements and postselection. The protocol consists of four stages: (i) classical randomization, (ii) projective measurements performed on a relevant quantum system, (iii) postprocessing of obtained outcomes, and finally (iv) postselection on nonobserving the last outcome.

Figure 2

Quantum circuit needed to implement POVM using Naimark's construction method. Additional Hadamard gates at the center are needed due to one-way connectivity of cnot gates on IBM hardware.

Figure 3

Quantum circuit needed to implement POVM using our scheme.