Jonckheere‐Terpstra test for nonclassical error versus log‐sensitivity relationship of quantum spin network controllers

Abstract

Summary The selective information transfer in spin ring networks by energy landscape shaping control has the property that the error, 1‐prob, where prob is the transfer success probability, and the sensitivity of the error to spin coupling uncertainties are statistically increasing across a family of controllers of increasing error. The need for a statistical hypothesis testing of a concordant trend is made necessary by the noisy behavior of the sensitivity versus the error as a consequence of the optimization of the controllers in a challenging error landscape. Here, we examine the concordant trend between the error and another measure of performance, ie, the logarithmic sensitivity, used in robust control to formulate a well‐known fundamental limitation. Contrary to error versus sensitivity, the error‐versus‐logarithmic‐sensitivity trend is less obvious because of the amplification of the noise due to the logarithmic normalization. This results in the Kendall τ test for rank correlation between the error and the log sensitivity to be somewhat pessimistic with marginal significance level. Here, it is shown that the Jonckheere‐Terpstra test, because it tests the alternative hypothesis of an ordering of the medians of some groups of log sensitivity data, alleviates this statistical problem. This identifies cases of concordant trend between the error and the logarithmic sensitivity, ie, a highly anticlassical feature that goes against the well‐known sensitivity versus the complementary sensitivity limitation.

Structured Singular Value Analysis for Spintronics Network Information Transfer Control

“…We have shown that the reassuring good sensitivity properties of high fidelity excitation transport controllers for spintronic networks demonstrated in [15], [23] can be extended to larger variation, so that such controllers can be objectively referred to as "robust." Both classical differential sensitivity and µ analyses reveal a nonclassical crossover region in the space of controllers where fidelity, classical sensitivity, and robustness as quantified by the µ-function all deteriorate-a rather surprising observation that contradicts the classical limitations on achievable performance.…”

“…The problem is that breaking the Schrödinger equation into two parts introduces a control u that is artificial, that does not have physical existence, but that nevertheless exists mathematically. Even though there is no physical closed-loop backward measurement signal flow, the "virtual" feedback structure, even somewhat "hidden," has been shown to endow the system with good differential sensitivity properties relative to spin coupling uncertainties [15], [23]. This is a property certainly consistent with measurement feedback.…”

“…We consider homogeneous chains or rings of N spins with XX or Heisenberg couplings [11], [12], [13], [14], [15], [20]. Each spin can be up | ↑ or down | ↓ .…”

“…This is a puzzling observation given that traditionally control predicts that one cannot simultaneously achieve small logarithmic sensitivity to uncertain parameters and small error, here defined as the departure from the fidelity upper bound. This puzzle, explained in [15], already points to quantum control deviating from classical control, in that the latter requires a physical measurement feedback process whereas the former can be accomplished by field mediation. EAJ Here, we bring one more component to this broader study: robustness against large rather than differential uncertainties using µanalysis of modern robust multivariable theory [24].…”

Robustness of Energy Landscape Control for Spin Networks Under Decoherence

“…When the transfer time is long compared to the time required for the system to reach a steady state, it is useful to consider the sensitivity of the asymptotic probability of transfer (squared fidelity) p ∞ = OUT| ρ ∞ |OUT and compute the log-sensitivity in the same manner as [12]. Using the perturbed, controlled Hamiltonian H D = H + D + δS H D where S H D indicates the (certain) structure of the perturbation and δ its (uncertain) strength, we have…”