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arXiv:1909.07709 · Journal of Physics A Mathematical and Theoretical

Entangling power of multipartite unitary gates

Tomasz Linowski, Grzegorz Rajchel-Mieldzioć, Karol Życzkowski

Published Feb 10, 2020• 16 citations
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Abstract

Abstract We study the entangling properties of multipartite unitary gates with respect to the measure of entanglement called one-tangle. Putting special emphasis on the case of three parties, we derive an analytical expression for the entangling power of an n-partite gate as an explicit function of the gate, linking the entangling power of gates acting on the n-partite Hilbert space of dimension to the entanglement of pure states in the Hilbert space of dimension . Furthermore, we evaluate its mean value averaged over the unitary and orthogonal groups, analyze the maximal entangling power and relate it to the absolutely maximally entangled (AME) states of a system with 2n parties. Finally, we provide a detailed analysis of the entangling properties of the three-qubit unitary and orthogonal gates.

DOI
10.1088/1751-8121/ab749a

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•Sep 17, 2019•1 citation
Abstract We study the entangling properties of multipartite unitary gates with respect to the measure of entanglement called one-tangle. Putting special emphasis on the case of three parties, we derive an analytical expression for the entangling power of an n-partite gate as an explicit function of the gate, linking the entangling power of gates acting on the n-partite Hilbert space of dimension to the entanglement of pure states in the Hilbert space of dimension . Furthermore, we evaluate its mean value averaged over the unitary and orthogonal groups, analyze the maximal entangling power and relate it to the absolutely maximally entangled (AME) states of a system with 2n parties. Finally, we provide a detailed analysis of the entangling properties of the three-qubit unitary and orthogonal gates.
A concise introduction to quantum entanglement in multipartite systems is presented. We review entanglement of pure quantum states of three--partite systems analyzing the classes of GHZ and W states and discussing the monogamy relations. Special attention is paid to equivalence with respect to local unitaries and stochastic local operations, invariants along these orbits, momentum map and spectra of partial traces. We discuss absolutely maximally entangled states and their relation to quantum error correction codes. An important case of a large number of parties is also analysed and entanglement in spin systems is briefly reviewed.
The ability to reach a maximally entangled state from a separable one through the use of a two-qubit unitary operator is analyzed for mixed states. This extension from the known case of pure states shows that there are at least two families of gates which are able to give maximum entangling power for all values of purity. It is notable that one of this gates coincides with a maximum discording one. We give analytical proof that such gate is indeed a perfect entangler at all purities and give numerical evidence for the existence of the second one. Furthermore, we find that there are other gates, many in fact, which are perfect entanglers for a restricted range of purities. This highlights the fact that many perfect entangler gates could in principle be found if a thorough analysis of the full parameter space is performed.
We analyze the entangling capabilities of unitary transformations U acting on a bipartite ${(d}_{1}\ifmmode\times\else\texttimes\fi{}{d}_{2})$-dimensional quantum system. To this aim we introduce an entangling power measure e(U)e(U)e(U) given by the mean linear entropy produced acting with U on a given distribution of pure product states. This measure admits a natural interpretation in terms of quantum operations. For a uniform distribution explicit analytical results are obtained using group-theoretic arguments. The behavior of the features of e(U)e(U)e(U) as the subsystem dimensions d1{d}_{1}d1​ and d2{d}_{2}d2​ are varied is studied both analytically and numerically. The two-qubit case d1=d2=2{d}_{1}{=d}_{2}=2d1​=d2​=2 is argued to be peculiar.
Abstract Given a quantum gate U acting on a bipartite quantum system, its maximum (average, minimum) entangling power is the maximum (average, minimum) entanglement generation with respect to certain entanglement measures when the inputs are restricted to be product states. In this paper, we mainly focus on the ‘weakest’ one, i.e. the minimum entangling power, among all these entangling powers. We show that, by choosing the entropy of entanglement or Schmidt rank as entanglement measure, even the ‘weakest’ entangling power is generically very close to the maximum possible value of the entanglement measure. In other words, maximum, average and minimum entangling powers are generically close. We then study minimum entangling power with respect to other Lipschitiz-continuous (for the Hilbert space distance) entanglement measures and generalize our results to multipartite quantum systems.
Quantifying the entanglement generation of a multipartite unitary operation is a key problem in quantum information processing. We introduce the definition of multipartite entangling, assisted entangling, and disentangling power, which is a natural generalization of the bipartite ones. We show that they are assumed at a specified quantum state. We analytically derive the entangling power of Schmidt-rank-two multiqubit unitary operations by the minimal convex sum of modulo-one complex numbers. Besides we show the necessary and sufficient condition that the assisted entangling power of Schmidt-rank-two unitary operations reaches the maximum. We investigate the widely used multiqubit gates, for example, the entangling power of CCZS gate, the entangling and assisted entangling power of the nnn-qubit Toffoli gate. The entangling power of the three-qubit Fredkin gate is two ebits, and that of the four-qubit Fredkin gate is in two to log25{log}_{2}5log2​5 ebits. Further we investigate the entangling power during the noisy implementation of operations.
The change of the entangling power of nnn fixed bipartite unitary gates, describing interactions, when interlaced with local unitary operators describing monopartite evolutions, is studied as a model of the entangling power of generic Hamiltonian dynamics. A generalization of the local unitary averaged entangling power for arbitrary subsystem dimensions is derived. This quantity shows an exponential saturation to the random matrix theory (RMT) average of the bipartite space, indicating thermalization of quantum gates that could otherwise be very non-generic and have arbitrarily small, but nonzero, entanglement. The rate of approach is determined by the entangling power of the fixed bipartite unitary, which is invariant with respect to local unitaries. The thermalization is also studied numerically via the spectrum of the reshuffled and partially transposed unitary matrices, which is shown to tend to the Girko circle law expected for random Ginibre matrices. As a prelude, the entangling power epe_pep​ is analyzed along with the gate typicality gtg_tgt​ for bipartite unitary gates acting on two qubits and some higher dimensional systems. We study the structure of the set representing all unitaries projected into the plane (ep,gt)(e_p, g_t)(ep​,gt​) and characterize its boundaries which contains distinguished gates including Fourier gate, CNOT and its generalizations, swap and its fractional powers. In this way, a family of gates with extreme properties is identified and analyzed. We remark on the use of these operators as building blocks for many-body quantum systems.