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arXiv:2112.05102 · SciPost Physics

Maximum entanglement of mixed symmetric states under unitary transformations

Eduardo Serrano-Ensástiga, John Martin

Published Sep 28, 2023• 6 citations
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Abstract

We study the maximum entanglement that can be produced by a global unitary transformation for systems of two and three qubits constrained to the fully symmetric states. This restriction to the symmetric subspace appears naturally in the context of bosonic or collective spin systems. We also study the symmetric states that remain separable after any global unitary transformation, called symmetric absolutely separable (SAS) states, or absolutely classical for spin states. The results for the two-qubit system are deduced analytically. In particular, we determine the maximal radius of a ball of SAS states around the maximally mixed state in the symmetric sector, and the minimal radius of a ball that contains the set of SAS states. As an application of our results, we also analyse the temperature dependence of the maximum entanglement that can be obtained from the thermal state of a spin-1 system with a spin-squeezing Hamiltonian. For the symmetric three-qubit case, our results are mostly numerical, and we conjecture a 3-parameter family of states that achieves the maximum negativity in the unitary orbit of any mixed state. In addition, we derive upper bounds, apparently tight, on the radii of balls containing only/all SAS states.

DOI
10.21468/scipostphys.15.3.120

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We study the maximum entanglement that can be produced by a global unitary transformation for systems of two and three qubits constrained to the fully symmetric states. This restriction to the symmetric subspace appears naturally in the context of bosonic or collective spin systems. We also study the symmetric states that remain separable after any global unitary transformation, called symmetric absolutely separable (SAS) states, or absolutely classical for spin states. The results for the two-qubit system are deduced analytically. In particular, we determine the maximal radius of a ball of SAS states around the maximally mixed state in the symmetric sector, and the minimal radius of a ball that contains the set of SAS states. As an application of our results, we also analyse the temperature dependence of the maximum entanglement that can be obtained from the thermal state of a spin-1 system with a spin-squeezing Hamiltonian. For the symmetric three-qubit case, our results are mostly numerical, and we conjecture a 3-parameter family of states that achieves the maximum negativity in the unitary orbit of any mixed state. In addition, we derive upper bounds, apparently tight, on the radii of balls containing only/all SAS states.
In this work, a variation of the problem originally solved by Verstraete, Audenaert, and De Moor [Phys. Rev. A 64, 012316 (2001)] on what is the maximum entanglement that can be created in a two-qubit system by a global unitary transformation is considered and solved when permutation invariance in the state is imposed. The additional constraint of permutation symmetry appears naturally in the context of bosonic systems or spin states. We also characterise symmetric two-qubit states that remain separable after any global unitary transformation, called symmetric absolutely separable states (SAS), or absolutely classical for spin states. This allows us to determine the maximal radius of a ball of SAS states around the maximally mixed state in the symmetric sector, and the minimal radius of a ball that includes the set of SAS states. For three-qubit systems, a necessary condition for absolute separability of symmetric states is given, which leads us to upper bounds on the ball radii similar to those studied for the two-qubit system.
This paper deals with the radius of the largest ball of separable mixed states around the maximally mixed state for multiqubit systems. This radius determines how close entangled states can be to the maximally mixed state. In Aubrun and Szarek (e-print arXiv:quant-ph/0503221) an upper bound on the radius was given, while in Gurvits and Barnum (e-print arXiv:quant-ph/0409095) a lower bound was provided. In this paper we improve both the upper and the lower bound, bringing the ratio of these bounds down to a constant $c=\sqrt{34∕27}\ensuremath{\approx}1.122$, as opposed to a term of order mlogm\sqrt{m\phantom{\rule{0.2em}{0ex}}\mathrm{log}\phantom{\rule{0.2em}{0ex}}m}mlogm​ for the best bounds known previously, where mmm is the number of qubits in the system. We construct concrete examples of separable states on the boundary to entanglement which realize the upper bounds. As a by-product, we compute the radii of the largest balls that fit into the projective tensor product of three and four unit balls in R3{\mathbb{R}}^{3}R3 and in the projective tensor product of an arbitrary number of unit balls in Rn{\mathbb{R}}^{n}Rn for n=2n=2n=2, 4, and 8.
We introduce the concept of ``absolutely classical'' spin states, in analogy to absolutely separable states of bipartite quantum systems. Absolutely classical states are states that remain classical (i.e., a convex sum of projectors on coherent states of a spin jjj) under any unitary transformation applied to them. We investigate the maximal size of the ball of absolutely classical states centered on the maximally mixed state and derive a lower bound for its radius as a function of the total spin quantum number. We also obtain a numerical estimate of this maximal radius and compare it to the case of absolutely separable states.
We consider mixed states of two qubits and show under which global unitary operations their entanglement is maximized. This leads to a class of states that is a generalization of the Bell states. Three measures of entanglement are considered: entanglement of formation, negativity, and relative entropy of entanglement. Surprisingly all states that maximize one measure also maximize the others. We give a complete characterization of these generalized Bell states and prove that these states for fixed eigenvalues are all equivalent under local unitary transformations. Furthermore we characterize all nearly entangled states closest to the maximally mixed state and derive a lower bound on the volume of separable mixed states.
We propose mixed states in two qubits that have a property that the amount of entanglement of these states cannot be increased by any unitary transformation. The property is proven when the rank of the states is less than 4, and confirmed numerically in the other general cases. The corresponding entanglement of formation specified by its eigenvalues gives an upper bound of that for density matrices with the same eigenvalues. Further, as a simple application of the upper bound of the entanglement of formation, we analyze the entanglement of formation of the state generated by a decohered controlled-NOT gate in the spin-boson model.
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 paper, we develop criteria for detecting SAS states of any number of qubits. Our approach is based on the Glauber-Sudarshan PPP 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.