Measures of operator entanglement of two-qubit gates

Entanglement properties of bipartite unitary operators are studied via their local invariants, namely the entangling power e p and a complementary quantity, the gate typicality g t. We characterize the boundaries of the set K 2 representing all two-qubit gates projected onto the plane (e p , g t) showing that the fractional powers of the SWAP operator form a parabolic boundary of K 2 , while the other bounds are formed by two straight lines. In this way, a family of gates with extreme properties is identified and analyzed. We also show that the parabolic curve representing powers of SWAP persists in the set K N for gates of higher dimensions (N > 2). Furthermore, we study entanglement of bipartite quantum gates applied sequentially n times, and we analyze the influence of interlacing local unitary operations, which model generic Hamiltonian dynamics. An explicit formula for the entangling power of a gate applied n times averaged over random local unitary dynamics is derived for an arbitrary dimension of each subsystem. This quantity shows an exponential saturation to the value predicted by the random matrix theory, indicating "thermalization" in the entanglement properties of sequentially applied quantum gates that can have arbitrarily small, but nonzero, entanglement to begin with. The thermalization is further characterized by the spectral properties of the reshuffled and partially transposed unitary matrices.

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“…The operator entanglements E (U ) and E (U S) can be written in terms of local invariants G 1 and G 2 as follows [67]:…”

Section: The Weyl Chamber and Various Gatesmentioning

confidence: 99%

Entanglement measures of bipartite quantum gates and their thermalization under arbitrary interaction strength

Jonnadula

Mandayam

Życzkowski

et al. 2020

Phys. Rev. Research

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“…If α 1 is selected at random, with a flat measure in [0, π/4], the probability of such a case is equal zero. The above argument can also be applied to the case of continuous time [24,28], which leads to the continuous flow V (t). For instance, α( √ V ) = 1 2 α(V ) corresponds to the position of the gate at t = 1/2.…”

Section: Dynamics Of Two-qubit Gatesmentioning

confidence: 99%

Bipartite unitary gates and billiard dynamics in the Weyl chamber

2018

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Long time behavior of a unitary quantum gate U, acting sequentially on two subsystems of dimension N each, is investigated. We derive an expression describing an arbitrary iteration of a two-qubit gate making use of a link to the dynamics of a free particle in a 3D billiard. Due to ergodicity of such a dynamics an average along a trajectory V t stemming from a generic two-qubit gate V in the canonical form tends for a large t to the average over an ensemble of random unitary gates distributed according to the flat measure in the Weyl chamber -the minimal 3D set containing points from all orbits of locally equivalent gates. Furthermore, we show that for a large dimension N the mean entanglement entropy averaged along a generic trajectory coincides with the average over the ensemble of random unitary matrices distributed according to the Haar measure on U(N 2 ).

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“…The entanglement property of a unitary operator has been studied from different aspects, such as entangling power 17 18 19 20 21 22 , entanglement measure 18 23 24 , entangling capacity 25 26 27 28 29 30 , and entanglement-changing power 31 . Entangling power of a joint unitary operation is defined as the mean entanglement (linear entropy) produced by applying the joint unitary operation on a given distribution of pure product states 17 .…”

mentioning

confidence: 99%

Matching relations for optimal entanglement concentration and purification

Kong

Xia

Yang

et al. 2016

Sci Rep

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The bilateral controlled NOT (CNOT) operation plays a key role in standard entanglement purification process, but the CNOT operation may not be the optimal joint operation in the sense that the output entanglement is maximized. In this paper, the CNOT operations in both the Schmidt-projection based entanglement concentration and the entanglement purification schemes are replaced with a general joint unitary operation, and the optimal matching relations between the entangling power of the joint unitary operation and the non-maximal entangled channel are found for optimizing the entanglement in- crement or the output entanglement. The result is somewhat counter-intuitive for entanglement concentration. The output entanglement is maximized when the entangling power of the joint unitary operation and the quantum channel satisfy certain relation. There exist a variety of joint operations with non-maximal entangling power that can induce a maximal output entanglement, which will greatly broaden the set of the potential joint operations in entanglement concentration. In addition, the entanglement increment in purification process is maximized only by the joint unitary operations (including CNOT) with maximal entangling power.

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Measures of operator entanglement of two-qubit gates

Cited by 12 publications

References 14 publications

Entanglement measures of bipartite quantum gates and their thermalization under arbitrary interaction strength

Entanglement measures of bipartite quantum gates and their thermalization under arbitrary interaction strength

Bipartite unitary gates and billiard dynamics in the Weyl chamber

Matching relations for optimal entanglement concentration and purification

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