Quantum tomography protocols with positivity are compressed sensing protocols

Abstract

Characterising complex quantum systems is a vital task in quantum information science. Quantum tomography, the standard tool used for this purpose, uses a well-designed measurement record to reconstruct quantum states and processes. It is, however, notoriously inefficient. Recently, the classical signal reconstruction technique known as 'compressed sensing' has been ported to quantum information science to overcome this challenge: accurate tomography can be achieved with substantially fewer measurement settings, thereby greatly enhancing the efficiency of quantum tomography. Here we show that compressed sensing tomography of quantum systems is essentially guaranteed by a special property of quantum mechanics itself-that the mathematical objects that describe the system in quantum mechanics are matrices with non-negative eigenvalues. This result has an impact on the way quantum tomography is understood and implemented. In particular, it implies that the information obtained about a quantum system through compressed sensing methods exhibits a new sense of 'informational completeness.' This has important consequences on the efficiency of the data taking for quantum tomography, and enables us to construct informationally complete measurements that are robust to noise and modelling errors. Moreover, our result shows that one can expand the numerical tool-box used in quantum tomography and employ highly efficient algorithms developed to handle large dimensional matrices on a large dimensional Hilbert space. Although we mainly present our results in the context of quantum tomography, they apply to the general case of positive semidefinite matrix recovery.

Strictly-complete measurements for bounded-rank quantum-state tomography

“…Other schemes [4, 5, 8-10, 13, 14], not related to compressed sensing, construct specific measurements (POVMs) that accomplish bounded-rank QST, and some * baldwin4@unm.edu † ideutsch@unm.edu ‡ amirk@unm.edu of these protocols have been implemented experimentally [10,15]. In addition, some general properties of such measurements have been derived [11,12,14]. In bounded-rank QST (rank ≤ r), two notions of informational completeness become relevant [8,9,12]: "rankr complete" and "rank-r strictly-complete."…”

“…Strictlycomplete measurements are only possible due to the positive semidefinite property of the density matrix (simply referred to throughout as "positivity") for bounded-rank QST [8]. It has been shown [12,16,17] that strictcompleteness has implications for estimating the state of system in the presence of noise. Whereas the set of rank-r states is nonconvex, with rank-r strictly-complete measurements one can use convex optimization programs to estimate the state of the system [12,16,17].…”

“…Bounded-rank QST has been studied by a number of previous workers [4][5][6][7][8][9][10][11][12][13][14], and has been shown to require less resources than general QST. One approach is based on the compressed sensing methodology [6,7], where certain sets of randomly chosen measurements guarantee a robust estimation of low-rank states with high probability.…”

“…In addition, some general properties of such measurements have been derived [11,12,14]. In bounded-rank QST (rank ≤ r), two notions of informational completeness become relevant [8,9,12]: "rankr complete" and "rank-r strictly-complete." Rank-r complete measurements can distinguish a rank ≤ r state from any other rank ≤ r state, while rank-r strictlycomplete measurements can distinguish a rank ≤ r state from any other physical state, of any rank.…”

Optimized tomography of continuous variable systems using excitation counting

“…However, QST is often resource-consuming, involving preparation of a large number of identical unknown states and measurement of a large set of independent observables. For qubit systems, many techniques have been developed to reduce the cost of full state tomography, such as compressed sensing [1][2][3], permutationally invariant tomography [4], self-guided/adaptive tomography [5,6], matrix product states tomography [7]. In contrast, for continuous variable (CV) systems that also play an important role in quantum information, the standard techniques in use today are decades old, namely homodyne measurement [8,9] for optical photons and direct Wigner function measurement [10][11][12] for cavity QED.…”