Operational applications of the diamond norm and related measures in quantifying the non-physicality of quantum maps

Abstract

Although quantum channels underlie the dynamics of quantum states, maps which are not physical channels — that is, not completely positive — can often be encountered in settings such as entanglement detection, non-Markovian quantum dynamics, or error mitigation. We introduce an operational approach to the quantitative study of the non-physicality of linear maps based on different ways to approximate a given linear map with quantum channels. Our first measure directly quantifies the cost of simulating a given map using physically implementable quantum channels, shifting the difficulty in simulating unphysical dynamics onto the task of simulating linear combinations of quantum states. Our second measure benchmarks the quantitative advantages that a non-completely-positive map can provide in discrimination-based quantum games. Notably, we show that for any trace-preserving map, the quantities both reduce to a fundamental distance measure: the diamond norm, thus endowing this norm with new operational meanings in the characterisation of linear maps. We discuss applications of our results to structural physical approximations of positive maps, quantification of non-Markovianity, and bounding the cost of error mitigation.

Fundamental limits of quantum error mitigation

“…Let us also consider the 𝑑-dimensional depolarizing noise D 𝑑 𝜖 (𝜌) = (1 − 𝜖) 𝜌 + 𝜖I/𝑑. The optimal overhead coefficient for this noise was obtained as [42][43][44]…”

Noise effects on purity and quantum entanglement in terms of physical implementability

“…and TP [13,23]. The sampling cost for implementing an HPTP map N with the Monte Carlo method is characterized by its physical implementability [13,15], defined as…”

“…The quasi-probability method [41,42] for quantum error mitigation and its variants [23,[43][44][45] involve the simulation of the inverse of noise channels with physically implementable quantum channels. The sampling cost for implementing an HPTP map N is characterized by its physical implementability [13,15], defined as…”

“…Here, we try to characterize how much entanglement a noise channel destroys with a universal parameter. We believe that the physical implementability of the noise inverse [13], which represents the sampling cost to implement a linear map [13][14][15] constructed from quantum resource theories [16][17][18], is a prime candidate. Intuitively, the harder to implement the noise inverse, the more destructive the noise itself to quantum entanglement.…”

Quantum Approximation of Normalized Schatten Norms and Applications to Learning

“…In Gilchrist et al's work [52], for example, a similarity metric that is blind to input unitary operations was studied and validated, but the estimation of the metric is inefficient because it requires an exponential number of quantum states. Other metrics such as the diamond norm [53,54] have been conceptualized to distinguish quantum operations, but they heavily rely on the classical information of the input unitaries such as their eigen-decompositions.…”