Absolute separability witnesses for symmetric multiqubit states

The persistent separability of certain quantum states, known as symmetric absolutely separable (SAS), under symmetry-preserving global unitary transformations is of key significance in the context of quantum resources for bosonic systems. In this work, we develop criteria for detecting SAS states of any number of qubits. Our approach is based on the Glauber-Sudarshan $P$ representation for finite-dimensional quantum systems. We introduce three families of SAS witnesses, one linear and two nonlinear in the eigenvalues of the state, formulated respectively as an algebraic inequality or a quadratic optimization problem. These witnesses are capable of identifying more SAS states than previously known counterparts. We also explore the geometric properties of the subsets of SAS states detected by our witnesses, shedding light on their distinctions.Comment: 16 pages, 5 figure

Maximum entanglement of mixed symmetric states under unitary transformations

editorial reviewedThe problem studied by Verstraete, Audenaert and De Moor in [1] -about which global unitary operations maximize the entanglement of a bipartite qubit system- is revisited, extended and solved when permutation symmetry on the qubits is imposed [2]. This condition appears naturally in bosonic systems or spin systems [3]. We fully characterize the set of absolutely separable symmetric states (SAS) for two qubits and provide fairly tight bounds for three qubits. In particular, we find the maximal radius of a ball of SAS states around the maximally mixed state in the symmetric sector, and the minimum radius of a ball that includes the set of SAS states, for both two and three qubits. [1] F. Verstraete, K. Audenaert, and B. De Moor, Phys. Rev. A, 64, 012316, (2001). [2] J. Martin and E. Serrano-Ensástiga, arXiv:2112.05102. [3] O. Giraud, P. Braun, and D. Braun, Phys. Rev. A, 78, 042112, (2008)

Quantum metrology of rotations with mixed spin states

The efficiency of a quantum metrology protocol can be considerably reduced by the interaction of a quantum system with its environment, resulting in a loss of purity and, consequently, a mixed state for the probing system. In this paper we examine the potential of mixed spin-$j$ states to achieve sensitivity comparable, and even equal, to that of pure states in the measurement of infinitesimal rotations about arbitrary axes. We introduce the concept of mixed optimal quantum rotosensors based on a maximization of the Fisher quantum information and show that it is related to the notion of anticoherence of spin states and its generalization to subspaces. We present several examples of anticoherent subspaces and their associated mixed optimal quantum rotosensors. We also show that the latter maximize negativity for specific bipartitions, reaching the same maximum value as pure states. These results elucidate the interplay between quantum metrology of rotations, anticoherence and entanglement in the framework of mixed spin states

Optimal quantum states for quantum information and how to prepare them

The quantum correlations in systems of indistinguishable particles are central resources for quantum technology applications. However, the efficiency of a quantum protocol may be reduced considerably by the surrounding environment that, consequently, turns the probing system into a mixed state. In this talk, we present the optimal states that achieve the maximum entanglement attainable for a multiqubit bosonic system with a fixed spectrum, and how to prepare them via a unitary transformation [1]. In particular, we study the maximum entanglement attainable for systems with two and three qubits. Along the same lines, we study multiqubit states with little or no use in quantum information applications which are absolutely separable (non-entangled) after any unitary transformation [2]. Lastly, as an application in quantum-enhanced metrology, we characterize the most susceptible mixed states to estimate an infinitesimal rotation over an arbitrary axis, called optimal quantum rotosensors [3]. These mixed states can achieve the same sensitivity as optimal pure states, and the quantum property called anticoherence comes into play.[1] Absolute separability witnesses for symmetric multiqubit states, ESE, J. Denis and J. Martin, Phys. Rev. A 109, 022430 (2024). [2] Maximum entanglement of mixed symmetric states under unitary transformations, ESE and J. Martin, SciPost Phys. 15, 120 (2023). [3] Quantum metrology with mixed spin states, ESE, J. Martin and C.Chryssomalakos, arXiv.2404.15548 (2024)

Symmetric Multiqudit States: Stars, Entanglement, Rotosensors

A constellation of $N=d-1$ Majorana stars represents an arbitrary pure quantum state of dimension $d$ or a permutation-symmetric state of a system consisting of $n$ qubits. We generalize the latter construction to represent in a similar way an arbitrary symmetric pure state of $k$ subsystems with $d$ levels each. For $d\geq 3$, 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

Majorana representation for mixed states

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 $\rho$. 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 [1]