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Physica Scripta

Entropic characterization of stabilizer states and magic states

Huihui Li, Shunlong Luo, Yue Zhang

Published Feb 12, 2024• 6 citations
Publisher

Abstract

Abstract Quantum states with minimum or maximum uncertainty are of special significance due to their extreme properties. Celebrated examples are coherent states induced from certain Lie groups and intelligent states for various uncertainty relations. In this work, by virtue of the Maassen-Uffink entropic uncertainty relation, we introduce an entropic quantifier of uncertainty and use it to characterize several important families of states in the stabilizer formalism of quantum computation. More specifically, we show that the stabilizer states and T-type magic states stand at the two extremes of the entropic quantifier of uncertainty: The former are precisely the minimum entropic uncertainty states, while the latter are precisely the maximum entropic uncertainty states. Moreover, interpolating between the above two extremes, the H-type magic states are the saddle points of the entropic quantifier of uncertainty. These entropic characterizations reveal some intrinsic features of stabilizer states, H- and T-type magic states, and cast novel light on the resource-theoretic viewpoint of regarding the stabilizer states as free states and the T-type magic states as the most precious source states in the stabilizer quantum theory.

DOI
10.1088/1402-4896/ad28a8

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Abstract Quantum states with minimum or maximum uncertainty are of special significance due to their extreme properties. Celebrated examples are coherent states induced from certain Lie groups and intelligent states for various uncertainty relations. In this work, by virtue of the Maassen-Uffink entropic uncertainty relation, we introduce an entropic quantifier of uncertainty and use it to characterize several important families of states in the stabilizer formalism of quantum computation. More specifically, we show that the stabilizer states and T-type magic states stand at the two extremes of the entropic quantifier of uncertainty: The former are precisely the minimum entropic uncertainty states, while the latter are precisely the maximum entropic uncertainty states. Moreover, interpolating between the above two extremes, the H-type magic states are the saddle points of the entropic quantifier of uncertainty. These entropic characterizations reveal some intrinsic features of stabilizer states, H- and T-type magic states, and cast novel light on the resource-theoretic viewpoint of regarding the stabilizer states as free states and the T-type magic states as the most precious source states in the stabilizer quantum theory.
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].
Entropic uncertainty relations express the quantum mechanical uncertainty principle by quantifying uncertainty in terms of entropy. Central questions include the derivation of lower bounds on the total uncertainty for given observables, the characterization of observables that allow strong uncertainty relations, and the construction of such relations for the case of several observables. We demonstrate how the stabilizer formalism can be applied to these questions. We show that the Maassen–Uffink entropic uncertainty relation is tight for the measurement in any pair of stabilizer bases. We compare the relative strengths of variance-based and various entropic uncertainty relations for dichotomic anticommuting observables.
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.
In most stabilizer-based quantum computing schemes, so-called magic states are a necessary resource for implementing non-transversal quantum gates. With the resource theory of magic, it is possible to analyze and quantify the generation of the non-stabilizer states. The relative entropy is a measure used in various resource theories. For single qubits, we characterize magic states and their closest stabilizer states by applying analytical results known from the relative entropy of entanglement and show that the magic states and their closest stabilizer states are arranged symmetrically around the states at the centers of the faces of the stabilizer octahedron. For tensor products of single-qubit states, we prove analytically that the relative entropy of magic is nonadditive in almost all cases.
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.
The entropic uncertainty principle in the form proven by Maassen and Uffink yields a fundamental inequality that is prominently used in many places all over the field of quantum information theory. In this work, we provide a family of versatile generalizations of this relation. Our proof methods build on a deep connection between entropic uncertainties and interpolation inequalities for the doubly stochastic map that links probability distributions in two measurements bases. In contrast to the original relation, our generalization also incorporates the von Neumann entropy of the underlying quantum state. These results can be directly used to bound the extractable randomness of a source independent QRNG in the presence of fully quantum attacks, to certify entanglement between trusted parties, or to bound the entanglement of a system with an untrusted environment.