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arXiv:2306.00532 · Quantum

Orthonormal bases of extreme quantumness

Marcin Rudziński, Adam Burchardt, Karol Życzkowski

Published Jan 25, 2024• 8 citations
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Abstract

Spin anticoherent states acquired recently a lot of attention as the most "quantum" states. Some coherent and anticoherent spin states are known as optimal quantum rotosensors. In this work, we introduce a measure of quantumness for orthonormal bases of spin states, determined by the average anticoherence of individual vectors and the Wehrl entropy. In this way, we identify the most coherent and most quantum states, which lead to orthogonal measurements of extreme quantumness. Their symmetries can be revealed using the Majorana stellar representation, which provides an intuitive geometrical representation of a pure state by points on a sphere. Results obtained lead to maximally (minimally) entangled bases in the 2j+1 dimensional symmetric subspace of the 22j dimensional space of states of multipartite systems composed of 2j qubits. Some bases found are iso-coherent as they consist of all states of the same degree of spin-coherence.

DOI
10.22331/q-2024-01-25-1234

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Spin anticoherent states acquired recently a lot of attention as the most "quantum" states. Some coherent and anticoherent spin states are known as optimal quantum rotosensors. In this work, we introduce a measure of quantumness for orthonormal bases of spin states, determined by the average anticoherence of individual vectors and the Wehrl entropy. In this way, we identify the most coherent and most quantum states, which lead to orthogonal measurements of extreme quantumness. Their symmetries can be revealed using the Majorana stellar representation, which provides an intuitive geometrical representation of a pure state by points on a sphere. Results obtained lead to maximally (minimally) entangled bases in the 2j+1 dimensional symmetric subspace of the 22j dimensional space of states of multipartite systems composed of 2j qubits. Some bases found are iso-coherent as they consist of all states of the same degree of spin-coherence.
A constellation of $N=d\ensuremath{-}1$ Majorana stars represents an arbitrary pure quantum state of dimension ddd or a permutation-symmetric state of a system consisting of nnn qubits. We generalize the latter construction to represent in a similar way an arbitrary symmetric pure state of kkk subsystems with ddd levels each. For $d\ensuremath{\ge}3$, such states are equivalent, as far as rotations are concerned, to a collection of various spin states, with definite relative complex weights. Following Majorana's lead, we introduce a multiconstellation, consisting of the Majorana constellations of the above spin states, augmented by an auxiliary, ``spectator'' constellation, encoding the complex weights. Examples of stellar representations of symmetric states of four qutrits, and two spin-3/2 systems, are presented. We revisit the Hermite and Murnaghan isomorphisms, which relate multipartite states of various spins, number of parties, and even symmetries. We show how the tools introduced can be used to analyze multipartite entanglement and to identify optimal quantum rotosensors, i.e., pure states which are maximally sensitive to rotations around a specified axis, or averaged over all axes.
We investigate multiqubit permutation-symmetric states with maximal entropy of entanglement. Such states can be viewed as particular spin states, namely anticoherent spin states. Using the Majorana representation of spin states in terms of points on the unit sphere, we analyze the consequences of a point-group symmetry in their arrangement on the quantum properties of the corresponding state. We focus on the identification of anticoherent states (for which all reduced density matrices in the symmetric subspace are maximally mixed) associated with point-group-symmetric sets of points. We provide three different characterizations of anticoherence and establish a link between point symmetries, anticoherence, and classes of states equivalent through stochastic local operations with classical communication. We then investigate in detail the case of small numbers of qubits and construct infinite families of anticoherent states with point-group symmetry of their Majorana points, showing that anticoherent states do exist to arbitrary order.
Coherent and anticoherent states of spin systems up to spin j=2 are known to be optimal in order to detect rotations by a known angle but unknown rotation axis. These optimal quantum rotosensors are characterized by minimal fidelity, given by the overlap of a state before and after a rotation, averaged over all directions in space. We calculate a closed-form expression for the average fidelity in terms of anticoherent measures, valid for arbitrary values of the quantum number j . We identify optimal rotosensors (i) for arbitrary rotation angles in the case of spin quantum numbers up to j=7/2 and (ii) for small rotation angles in the case of spin quantum numbers up to j=5 . The closed-form expression we derive allows us to explain the central role of anticoherence measures in the problem of optimal detection of rotation angles for arbitrary values of j .
The set of pure spin states with vanishing spin expectation value can be regarded as the set of the less coherent pure spin states. This set can be divided into a finite number of nested subsets on the basis of higher order moments of the spin operators. This subdivision relies on the notion of anticoherent spin state to order ttt: A spin state is said to be anticoherent to order ttt if the moment of order kkk of the spin components along any directions are equal for k=1,2,...,tk=1,2,...,tk=1,2,...,t. Most spin states are neither coherent nor anticoherent, but can be arbitrary close to one or the other. In order to quantify the degree of anticoherence of pure spin states, we introduce the notion of anticoherence measures. By relying on the mapping between spin-jjj states and symmetric states of 2j2j2j spin 1/21/21/2 (Majorana representation), we present a systematic way of constructing anticoherence measures to any order. We briefly discuss their connection with measures of quantum coherence. Finally, we illustrate our measures on various spin states and use them to investigate the problem of the existence of anticoherent spin states with degenerated Majorana points.
We look for optimal quantum rotosensors, i.e., quantum spin states that are optimal in detecting rotations by a given angle. The exact quantity to be minimized is the probability that the rotated state projects onto the original one, averaged uniformly over all rotation axes. We show analytically that, for small rotation angles, the solution is given by anticoherent states. Numerical analysis shows that, for spin s=1s=1s=1 and 3/23/23/2, there exists, in each case, a critical rotation angle beyond which the minimum is instead achieved by the coherent states, the transition between the two being discontinuous. For s=2s=2s=2, a third degenerate minimum appears in between the above two, consisting of pairs of antipodal points, regardless of the angle between their axes. We also determine the worst rotosensors, i.e., those states that maximize the above quantity; interestingly, the same set of states shows up but for different ranges of the rotation angle.
The efficiency of a quantum metrology protocol can be significantly diminished by the interaction of the system with its environment, leading to a loss of purity and, as a result, a mixed state for the probing system. An example is the measurement of a magnetic field through the rotation of a spin that is subject to decoherence due to its coupling to a surrounding spin or bosonic bath. In this work we define mixed optimal quantum rotosensors (OQRs) as mixed spin-jjj states that achieve maximum sensitivity to estimate infinitesimal rotations, when the rotation axis is unknown. We study two scenarios, where the probe states saturate the averaged fidelity or the averaged quantum Cram\'er-Rao bound, the latter giving the ultimate sensitivity. We find that mixed OQRs can achieve sensitivity equal to that of pure states and are obtained by mixing states from linear subspaces of anticoherent states. We present several examples of mixed OQRs and their associated anticoherent subspaces. We also show that OQRs maximize entanglement in a specific sense, preserving the known relation between entanglement and optimal rotation sensitivity for pure states, even in the context of mixed states. Our results highlight the interconnection between quantum metrology of rotations, anticoherence, and entanglement in mixed spin states.