Adaptive compressive tomography: A numerical study

Abstract

We perform several numerical studies for our recently published adaptive compressive tomography scheme [D. Ahn et al. Phys. Rev. Lett. 122, 100404 (2019)], which significantly reduces the number of measurement settings to unambiguously reconstruct any rank-deficient state without any a priori knowledge besides its dimension. We show that both entangled and product bases chosen by our adaptive scheme perform comparably well with recently-known compressed-sensing element-probing measurements, and also beat random measurement bases for low-rank quantum states. We also numerically conjecture asymptotic scaling behaviors for this number as a function of the state rank for our adaptive schemes. These scaling formulas appear to be independent of the Hilbert space dimension. As a natural development, we establish a faster hybrid compressive scheme that first chooses random bases, and later adaptive bases as the scheme progresses. As an epilogue, we reiterate important elements of informational completeness for our adaptive scheme.

Modern compressive tomography for quantum information science

“…Thirdly, provable CS-QST measurements with known tight scaling for r d are restricted to highly specific measurements, two examples include the random Pauli-observable expectation 21 and element-probing measurements. 23 Since 2019, we reported several works [29][30][31][32][33][34][35] that investigate the problem of compressive tomography in different perspectives that are completely independent from CS. Essentially, we developed a self-consistent informational completeness certification (ICC) routine that can decisively state whether a given set of measurements and data is informationally complete (IC); that is, whether a unique reconstruction of the unknown quantum object is possible.…”

Objective compressive quantum process tomography

“…The standard ancilla-free framework shall be considered here, which consists of input states (ρ IN ) and outputstate von Neumann basis measurements that can be feasibly implemented in practice. Our scheme comprises an adaptive compressive (state) tomography (ACT) protocol [40,41] that reconstructs the unknown output states (ρ OUT = Φ[ρ IN ]) from adaptively chosen bases, and an informational completeness certification that determines whether the process estimator ρ Φ (distinguished with a hat from the true process operator ρ Φ ) is uniquely characterized by the accumulated dataset or not. This can be achieved with semidefinite programs [42,43].…”

“…1). The first component is the ACT scheme [40,41] that chooses a compressive sequence of optimal von Neumann measurements to efficiently characterize every output state. In every step, it first certifies if the accumulated data uniquely characterize, say, ρ OUT after feeding ρ IN to Φ.…”

“…Otherwise the next optimal measurement is chosen as the eigenbasis of the minimum-von-Neumann-entropy (minENT) estimator over C (ρ) . This generally leads to very efficient ACT convergences [41,49,50].…”

Universal Compressive Characterization of Quantum Dynamics

“…This is the adaptive measurement stage responsible for generating U. Since our χ of interest is rank deficient, we can turn ACQPT into a compressive scheme by employing an effective low-rank guiding prescription: After ICC, an estimator with the minimum von Neumann entropy (minENT) is chosen from C [46][47][48][49]. The next optimal U for the χ rotation would be the one that diagonalizes this estimator to be the eigenvalues in descending order.…”