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arXiv:1909.07740 · Physical Review A

Majorana representation for mixed states

Eduardo Serrano-Ensástiga, Daniel Braun

Published Feb 26, 2020• 25 citations
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Abstract

We generalize the Majorana stellar representation of spin-$s$ pure states to mixed states, and in general to any Hermitian operator, defining a bijective correspondence between three spaces: the spin-density matrices, a projective space of homogeneous polynomials of four variables, and a set of equivalence classes of points (constellations) on spheres of different radii. The representation behaves well under rotations by construction, and also under partial traces where the reduced density matrices inherit their constellation classes from the original state $\ensuremath{\rho}$. We express several concepts and operations related to density matrices in terms of the corresponding polynomials, such as the anticoherence criterion and the tensor representation of spin-$s$ states described in Giraud et al. [Phys. Rev. Lett. 114, 080401 (2015)].

DOI
10.1103/physreva.101.022332

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We generalize the Majorana stellar representation of spin-sss pure states to mixed states, and in general to any Hermitian operator, defining a bijective correspondence between three spaces: the spin-density matrices, a projective space of homogeneous polynomials of four variables, and a set of equivalence classes of points (constellations) on spheres of different radii. The representation behaves well under rotations by construction, and also under partial traces where the reduced density matrices inherit their constellation classes from the original state \ensuremathρ\ensuremath{\rho}\ensuremathρ. We express several concepts and operations related to density matrices in terms of the corresponding polynomials, such as the anticoherence criterion and the tensor representation of spin-sss states described in Giraud et al. [Phys. Rev. Lett. 114, 080401 (2015)].
We propose a generalization of the Bloch sphere representation for arbitrary spin states. It provides a compact and elegant representation of spin density matrices in terms of tensors that share the most important properties of Bloch vectors. Our representation, based on covariant matrices introduced by Weinberg in the context of quantum field theory, allows for a simple parametrization of coherent spin states, and a straightforward transformation of density matrices under local unitary and partial tracing operations. It enables us to provide a criterion for anticoherence, relevant in a broader context such as quantum polarization of light.
The Majorana stellar representation maps a pure spin-SSS state to 2S2S2S points on a sphere. This review develops it with entanglement as the organising principle, and two objects recur throughout: the constellation, and the permanent of the Gram matrix of its stars. The degeneracy pattern of the constellation is invariant under stochastic local operations and classical communication, so the integer partitions of NNN label a finite set of families of symmetric NNN-qubit states. That labelling is a coarse-graining rather than a classification, since from four distinct stars onwards each family carries continuous Möbius moduli. The permanent supplies what the pattern omits. Normalised by it, inter-star chordal distances give the concurrence and the three-tangle in closed form, and the same permanent governs the anomalous contribution to the Berry phase acquired under adiabatic cyclic evolution, so that a single quantity links static correlations to dynamical holonomy. We also fix the computational reach of the geometry. Overlaps of symmetric states are permanents of matrices of rank at most two and are polynomially computable, whereas measures defined by an optimisation over the sphere are not reached by that argument. Interest in stellar representations has resurged, but the literature remains dispersed, and no existing treatment develops the link between constellation geometry, multipartite entanglement, and geometric phases within a single framework. The same two objects organise the applications reviewed here, from extremal states in metrology and permutation-invariant codes to collective spin models and photonic constellations, together with extensions to mixed states and to continuous-variable systems through the stellar rank. Whether the anomalous phase admits a bound in terms of any entanglement monotone remains open.
While the Bloch sphere provides a canonical geometric representation for qubits, its generalization to pure qudit states (d>2d>2d>2) is not unique. This representation can be extended following two different approaches: increasing the number of points, as in the Majorana representation, or using a single point but in a space of larger dimension, as in the generalized Bloch representation. In this work, we compare these representations of pure qudit states by finding explicit analytical relations between them. In particular, a general formula of the Bloch vector in terms of the Majorana points of a given state is provided. We also analyze some suitable extensions of these representations in the case of pure permutation-symmetric two- and three-qubit states, deriving the corresponding relation between them. These analytical results allow us to study, from a geometric point of view, quantum correlations of special classes of two-qubit states.
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.
By describing the evolution of a quantum state with the trajectories of the Majorana stars on a Bloch sphere, Majorana’s stellar representation provides an intuitive geometric perspective to comprehend the quantum system with high-dimensional Hilbert space. However, the representation of a two-spin coupling system on a Bloch sphere has not been solved satisfactorily yet. Here, a practical method is presented to resolve the problem for the mixed-spin ( s , 1/2) system and describe the entanglement of the system. The system can be decomposed into two spins: spin-( s + 1/2) and spin-( s – 1/2) at the coupling bases, which can be regarded as independent spins. Besides, any pure state may be written as a superposition of two orthonormal states with one spin-( s + 1/2) state and the other spin-( s – 1/2) state. Thus, the whole initial state can be regarded as a state of a pseudo spin-1/2. In this way, the mixed spin decomposes into three spins. Therefore, the state can be represented by (2 s + 1) + (2 s – 1) + 1 = 4 s + 1 sets of stars on a Bloch sphere. Finally, some examples are given to show symmetric patterns on the Bloch sphere and unveil the properties of the high-spin system by analyzing the trajectories of the Majorana stars on the Bloch sphere.
By the Majorana representation, for any d>1d > 1d>1 there is a one-one correspondence between a quantum state of dimension ddd and d−1d-1d−1 qubits represented as d−1d-1d−1 points in the Bloch sphere. Using the theory of symmetry class of tensors, we present a simple scheme for constructing d−1d-1d−1 points on the Bloch sphere and the corresponding d−1d-1d−1 qubits representing a ddd-dimensional quantum state. Additionally, we demonstrate how the inner product of two ddd-dimensional quantum states can be expressed as a permanent of a matrix related to their (d−1)(d-1)(d−1)-qubit state representations. Extension of the result to mixed states is also considered.