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Physica Scripta

Characterization of stabilizier states and magic states in terms of Tsallis and Rényi entropies for qubit systems

Jiayu He, Bowen Wang, Shuangshuang Fu

Published Dec 10, 2024• 1 citation
Publisher

Abstract

Abstract Uncertainty relations are fundamental in quantum mechanics, distinguishing it from classical physics by setting limits on the precision of incompatible measurements. For qubit systems, considering the three incompatible Pauli observables, we propose two entropic quantifiers of uncertainty exploiting the Tsallis entropies and Rényi entropies. We analyze the minimum and maximum uncertainty states and uncover their close connections with some important families of states in the stabilizer formalism. Explicitly, our findings show that for the two entropic quantifiers of uncertainties with suitable parameters, the stabilizer states correspond to the minimum uncertainty states, while the T-type magic states correspond to the maximum uncertainty states. Additionally, we identify that the H-type magic states serve as the saddle points in the Tsallis entropy framework. These results expand the recent proposed characterizations of stabilizer and magic states via the Shannon entropic uncertainty relation [Li et al 2024 Phys. Scr. 99, 035117].

DOI
10.1088/1402-4896/ad9d09

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Abstract We introduce the notion of (renormalized) quantum uncertainty and reveal its basic features. In terms of this quantity, we completely characterize the minimum and maximum quantum uncertainty states for qubit systems involving Pauli matrices. It turns out that the minimum quantum uncertainty states consist of both certain pure states and certain mixed states, in sharp contrast to the case of conventional Heisenberg uncertainty relation. The maximum quantum uncertainty states are H-type magic states arising from the stabilizer formalism of quantum computation, and can be obtained from minimum quantum uncertainty states via the T-gate.
We obtain uncertainty and certainty relations of state-independent form for the three Pauli observables with use of the Rényi entropies of order α ∈ (0; 1]. It is shown that these entropic bounds are tight in the sense that they are always reached with certain pure states. A new result is the condition for equality in Rényi-entropy uncertainty relations for the Pauli observables. Upper entropic bounds in the pure-state case are also novel. Combining the presented bounds leads to a band, in which the rescaled average Rényi α-entropy ranges for a pure measured state. A width of this band is compared with the Tsallis formulation derived previously.
We study two operational approaches to quantifying incompatibility that depart significantly from the well known entropic uncertainty relation (EUR) formalism. Both approaches result in incompatibility measures that yield non-zero values even when the pair of incompatible observables commute over a subspace, unlike EURs which give a zero lower bound in such cases. Here, we explicitly show how these measures go beyond EURs in quantifying incompatibility: For any set of quantum observables, we show that both incompatibility measures are bounded from below by the corresponding EURs for the Tsallis (T2T_{2}T2​) entropy. We explicitly evaluate the incompatibility of a pair of qubit observables in both operational scenarios. We also obtain an efficiently computable lower bound for the mutually incompatibility of a general set of observables.
Abstract Entropic uncertainty relations are studied from the conceptual viewpoint and in the context of various applications. Providing as tight inequalities as possible is thus an important issue. Measurements of special types are indispensable in quantum information science. Their structure allows us to estimate the index of coincidence. Uncertainty relations follow an estimation of entropies at the given index of coincidence. Relations of such a kind are a known issue in information theory. It concerns links between different information measures. Information diagrams are a tool to study relations between two utilized characteristics. A search for improved inequalities uses the index of coincidence as the abscissa and the entropy of interest as the ordinate. This approach before dealt with standard information functions assigned to the Shannon entropy. But generalized entropies often reveal new links between information measures. This paper applies the method of information diagrams to the Rényi and Tsallis entropies. Such entropies and related information functions have found use in quantum information theory. Improved estimates of the Tsallis and Rényi entropies at the given index of coincidence are obtained. Hence, new entropic uncertainty relations for specific types of measurements follow. They include mutually unbiased bases, symmetric informationally complete measurements, and equiangular tight frames.
The stabilizer states and symmetric informationally complete positive operator valued measure (SIC-POVM) fiducial states are two important families of states in the stabilizer formalism of fault-tolerant quantum computation. They have been demonstrated to be on two extremes with respect to some quantifiers of magic. The present work is devoted to a further study of these two families of quantum states. For a given quantum state in a ddd-dimensional quantum system, employing the generalized variance and the generalized Wigner-Yanase skew information for the set of discrete Heisenberg-Weyl displacement operators, we propose two quantifiers of uncertainty defined in the product form. We reveal that the extremal states of the uncertainty quantifiers are also closely related to the above-mentioned families of states. Explicitly, for prime-dimensional quantum systems, we prove that the minimum uncertainty states correspond to the stabilizer states. While for arbitrary dimensional quantum system, the maximum uncertainty states are the SIC-POVM fiducial states (assuming their existence). For the quantifier of total uncertainty, the conclusions hold under the restriction that the quantum states are pure, while for the quantifier of quantum uncertainty, the conclusions hold for general quantum states. The findings in this work shed light on the characterization of important families of states in the stabilizer formalism of quantum computation, which reveal the close connections between the stabilizer formalism and uncertainty.