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The superposition invariance of unitary operators and maximally entangled state

Xin-Wei Zha, Ning Miao, Xiaoyuan Yu

Published Dec 18, 2019• 0 citations
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Abstract

In this paper, we study unitary operators and the superposition of unitary operators. We calculate the the superposition of unitary operators and find that some unitary operators superposition is also unitary operator. Furthermore, via this property, we discuss the set of orthogonal maximally entangled states. For 2,3,4,5-qubit, we introduce the complete sets of orthogonal maximally entangled states. We find that orthogonal basis of maximally entangled states can be divided into k subspaces. It is shown that some entanglement properties of superposed state in every subspace are invariant.

DOI
10.1117/12.2538451

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In this paper, we study unitary operators and the superposition of unitary operators. We calculate the the superposition of unitary operators and find that some unitary operators superposition is also unitary operator. Furthermore, via this property, we discuss the set of orthogonal maximally entangled states. For 2,3,4,5-qubit, we introduce the complete sets of orthogonal maximally entangled states. We find that orthogonal basis of maximally entangled states can be divided into k subspaces. It is shown that some entanglement properties of superposed state in every subspace are invariant.
We demonstrate that any pure bipartite state of two qubits may be decomposed into a superposition of a maximally entangled state and an orthogonal factorizable one. Although there are many such decompositions, the weights of the two superposed states are, remarkably, unique. We propose a measure of entanglement based on this decomposition. We also demonstrate that this measure is connected to three measures of entanglement previously set forth: maximal violation of Bell's inequality, concurrence, and two-particle visibility.
We investigate universal entanglement properties inherent to superpositions of randomized states. We find that an mmm-fold superposition of typical states may be classified into two distinct entanglement classes via the 2nd Rényi entropy density s2s_2s2​. The maximally entangled regime is defined by s2∼ln⁡(2)s_2 \sim \ln (2)s2​∼ln(2), for which superposition adds no additional entanglement. The sub-maximally entangled regime, s2<ln⁡2s_2<\ln 2s2​<ln2, instead constrains the reduced density matrices of independent components to be orthogonal in the thermodynamic limit, which fixes the entanglement of the superposition to a logarithmic enhancement ΔS(m)=ln⁡(m)ΔS(m)=\ln (m)ΔS(m)=ln(m). As a consequence, an exponentially large number of superpositions is required to transition from the sub-maximally entangled class to maximal entanglement. We explicitly calculate s2s_2s2​ and the logarithmic enhancement, and demonstrate orthogonality for two canonical examples of the sub-maximally entangled regime (superpositions of pure Gaussian states and of random matrix-product states). We also examine the entanglement of superpositions of random stabilizer states, and discuss their relaxation to the Haar limit.