Diagonal unitary entangling gates and contradiagonal quantum states

Abstract

Nonlocal properties of an ensemble of diagonal random unitary matrices of order $N^2$ are investigated. The average Schmidt strength of such a bipartite diagonal quantum gate is shown to scale as $\log N$, in contrast to the $\log N^2$ behavior characteristic to random unitary gates. Entangling power of a diagonal gate $U$ is related to the von Neumann entropy of an auxiliary quantum state $\rho =A{A}^{†}/N^2$, where the square matrix $A$ is obtained by reshaping the vector of diagonal elements of $U$ of length $N^2$ into a square matrix of order $N$. This fact provides a motivation to study the ensemble of non-hermitian unimodular matrices $A$, with all entries of the same modulus and random phases and the ensemble of quantum states $\rho$, such that all their diagonal entries are equal to $1/N$. Such a state is contradiagonal with respect to the computational basis, in sense that among all unitary equivalent states it maximizes the entropy copied to the environment due to the coarse graining process. The first four moments of the squared singular values of the unimodular ensemble are derived, based on which we conjecture a connection to a recently studied combinatorial object called the "Borel triangle". This allows us to find exactly the mean von Neumann entropy for random phase density matrices and the average entanglement for the corresponding ensemble of bipartite pure states.

arXiv comment14 pages, 6 figures

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• Coherifying quantum channels
• Distinguishability of generic quantum states
• Entanglement in a dephasing model and many-body localization
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• Entanglement measures of bipartite quantum gates and their thermalization under arbitrary interaction strength
• Universal scaling of spectral fluctuation transitions for interacting chaotic systems
• Certainty relations, mutual entanglement and non-displacable manifolds
• Bipartite unitary gates and billiard dynamics in the Weyl chamber
• Eigenstate entanglement between quantum chaotic subsystems: universal transitions and power laws in the entanglement spectrum
• Particle fluctuations and the failure of simple effective models for many-body localized phases
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• Diagonal unitary and orthogonal symmetries in quantum theory II: Evolution operators
• Quantum coherence controls the nature of equilibration in coupled chaotic systems
• Dual unitaries as maximizers of the distance to local product gates
• Quantum tomography with random diagonal unitary maps and statistical bounds on information generation using random matrix theory
• Spectral statistics of the uni-modular ensemble
• Counting with Borel's Triangle
• Characterizations of the Borel triangle and Borel polynomials