Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0210156 · Physical Review A

Entangling power and operator entanglement in qudit systems

Xiaoguang Wang, Barry C. Sanders, Dominic William Berry

Published Apr 28, 2003• 85 citations
arXivHTMLPDF

Abstract

We establish the entangling power of a unitary operator on a general finite-dimensional bipartite quantum system with and without ancillas, and give relations between the entangling power based on the von Neumann entropy and the entangling power based on the linear entropy. Significantly, we demonstrate that the entangling power of a general controlled unitary operator acting on two equal-dimensional qudits is proportional to the corresponding operator entanglement if linear entropy is adopted as the quantity representing the degree of entanglement. We discuss the entangling power and operator entanglement of three representative quantum gates on qudits: the SUM, double SUM, and SWAP gates.

DOI
10.1103/physreva.67.042323

Related papers

Multipartite entangling power by von Neumann entropy

Xinyu Qiu, Zhiwei Song, Lin Chen

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 $n$-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 ${log}_{2}5$ ebits. Further we investigate the entangling power during the noisy implementation of operations.

arXiv:2410.15253•Physical Review A•Feb 7, 2025•1 citation

Quantifying Entangling Power of Controlled Unitary Gates

Ankur Haldar, Pankaj Agrawal, Prasenjit Deb

Applying controlled unitary gates to generate entanglement between qubits is a routine task in both quantum communication and computation. The existing tools for predicting how much entanglement a given gate can generate require either simulation of the entangling circuit or averaging over a distribution of inputs for a given controlled unitary gate. Here, we introduce a computable quantity $ζ$ that not only determines whether a controlled unitary gate generates entanglement, but also quantifies the entanglement for any specific input without requiring the construction of the output state. For two-qubit controlled unitary gates, we establish the physical conditions corresponding to the extremum values of the proposed quantity. Extending the dimension of control and target registers to arbitrary size through a generalized controlled unitary architecture, we derive a universal upper bound on $ζ$ and identify the conditions for its saturation. Later we establish functional relation between the quantity and other known quantities, such as purity, normalized linear entropy, and von Neumann entropy. Finally, we compare our results with previous research work and show that the quantity proposed in this work achieves the previously known optimal values of entangling power of controlled unitary gates for some specific dimensions of target and control registers.

quant-pharXiv:2608.21105•Aug 24, 2026•0 citations

Entangling power of symmetric two-qubit quantum gates and three-level operations

Diego Morachis Galindo, Jesús A. Maytorena

The capacity of a quantum gate to produce entangled states on a bipartite system is quantified in terms of the entangling power. This quantity is defined as the average of the linear entropy of entanglement of the states produced after applying a quantum gate over the whole set of separable states. Here we focus on symmetric two-qubit quantum gates, acting on the symmetric two-qubit space, and calculate the entangling power in terms of the appropriate local invariant. A geometric description of the local equivalence classes of gates is given in terms of the $\text{su}(3)$ Lie algebra root vectors. These vectors define a primitive cell with hexagonal symmetry on a plane, and through the Weyl group the minimum area on the plane containing the whole set of locally equivalent quantum gates is identified. We give conditions to determine when a given quantum gate produces maximally entangled states from separable ones (perfect entanglers). We find that these gates correspond to one-fourth of the whole set of locally distinct quantum gates. The formalism developed here is applicable to general three-level systems. Via the Majorana representation, qutrit transformations can be regarded as having entangling power and hence classified as perfect and nonperfect entanglers and be grouped into local-equivalence classes of the associated symmetric two-qubit space. The results are illustrated by an anisotropic Heisenberg model, the Lipkin-Meshkov-Glick model, and two coupled quantized oscillators with cross-Kerr interaction, which we use to obtain three-level gates relevant in qutrit quantum computation.

arXiv:2107.12653•Physical Review A•Jan 3, 2022•11 citations

Entangling and assisted entangling power of bipartite unitary operations

Lin Chen, Li Zhi Yu

Nonlocal unitary operations can create quantum entanglement between distributed particles, and the quantification of created entanglement is a hard problem. It corresponds to the concepts of entangling and assisted entangling power when the input states are, respectively, product and arbitrary pure states. We analytically derive them for Schmidt-rank-two bipartite unitary and some complex bipartite permutation unitaries. In particular, the entangling power of permutation unitary of Schmidt rank three can take only one of two values: ${log}_{2}9\ensuremath{-}16/9$ or ${log}_{2}3$ ebits. The entangling power, assisted entangling power, and disentangling power of $2\ifmmode\times\else\texttimes\fi{}{d}_{B}$ permutation unitaries of Schmidt rank four are all 2 ebits. These quantities are also derived for generalized Clifford operators. We further show that any bipartite permutation unitary of Schmidt rank greater than two has entangling power greater than 1.223 ebits. We construct the generalized controlled-not (cnot) gates whose assisted entangling power reaches the maximum. We quantitatively compare the entangling power and assisted entangling power for general bipartite unitaries and their connection to the disentangling power by proposing a probabilistic protocol for implementing bipartite unitaries.

arXiv:1604.05788•Physical Review A•Aug 8, 2016•16 citations

Entangling power of quantum evolutions

Paolo Zanardi A, Christof Zalka, Lara Faoro

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 $e(U)$ 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 $e(U)$ as the subsystem dimensions ${d}_{1}$ and ${d}_{2}$ are varied is studied both analytically and numerically. The two-qubit case ${d}_{1}{=d}_{2}=2$ is argued to be peculiar.

arXiv:quant-ph/0005031•Physical Review A•Aug 15, 2000•419 citations

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

Bhargavi Jonnadula, Prabha Mandayam, Karol Życzkowski +1 more

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.

arXiv:1909.08139•Physical Review Research•Oct 23, 2020•28 citations
We establish the entangling power of a unitary operator on a general finite-dimensional bipartite quantum system with and without ancillas, and give relations between the entangling power based on the von Neumann entropy and the entangling power based on the linear entropy. Significantly, we demonstrate that the entangling power of a general controlled unitary operator acting on two equal-dimensional qudits is proportional to the corresponding operator entanglement if linear entropy is adopted as the quantity representing the degree of entanglement. We discuss the entangling power and operator entanglement of three representative quantum gates on qudits: the SUM, double SUM, and SWAP gates.
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.
Applying controlled unitary gates to generate entanglement between qubits is a routine task in both quantum communication and computation. The existing tools for predicting how much entanglement a given gate can generate require either simulation of the entangling circuit or averaging over a distribution of inputs for a given controlled unitary gate. Here, we introduce a computable quantity ζζζ that not only determines whether a controlled unitary gate generates entanglement, but also quantifies the entanglement for any specific input without requiring the construction of the output state. For two-qubit controlled unitary gates, we establish the physical conditions corresponding to the extremum values of the proposed quantity. Extending the dimension of control and target registers to arbitrary size through a generalized controlled unitary architecture, we derive a universal upper bound on ζζζ and identify the conditions for its saturation. Later we establish functional relation between the quantity and other known quantities, such as purity, normalized linear entropy, and von Neumann entropy. Finally, we compare our results with previous research work and show that the quantity proposed in this work achieves the previously known optimal values of entangling power of controlled unitary gates for some specific dimensions of target and control registers.
The capacity of a quantum gate to produce entangled states on a bipartite system is quantified in terms of the entangling power. This quantity is defined as the average of the linear entropy of entanglement of the states produced after applying a quantum gate over the whole set of separable states. Here we focus on symmetric two-qubit quantum gates, acting on the symmetric two-qubit space, and calculate the entangling power in terms of the appropriate local invariant. A geometric description of the local equivalence classes of gates is given in terms of the su(3)\text{su}(3)su(3) Lie algebra root vectors. These vectors define a primitive cell with hexagonal symmetry on a plane, and through the Weyl group the minimum area on the plane containing the whole set of locally equivalent quantum gates is identified. We give conditions to determine when a given quantum gate produces maximally entangled states from separable ones (perfect entanglers). We find that these gates correspond to one-fourth of the whole set of locally distinct quantum gates. The formalism developed here is applicable to general three-level systems. Via the Majorana representation, qutrit transformations can be regarded as having entangling power and hence classified as perfect and nonperfect entanglers and be grouped into local-equivalence classes of the associated symmetric two-qubit space. The results are illustrated by an anisotropic Heisenberg model, the Lipkin-Meshkov-Glick model, and two coupled quantized oscillators with cross-Kerr interaction, which we use to obtain three-level gates relevant in qutrit quantum computation.
Nonlocal unitary operations can create quantum entanglement between distributed particles, and the quantification of created entanglement is a hard problem. It corresponds to the concepts of entangling and assisted entangling power when the input states are, respectively, product and arbitrary pure states. We analytically derive them for Schmidt-rank-two bipartite unitary and some complex bipartite permutation unitaries. In particular, the entangling power of permutation unitary of Schmidt rank three can take only one of two values: ${log}_{2}9\ensuremath{-}16/9$ or log23{log}_{2}3log2​3 ebits. The entangling power, assisted entangling power, and disentangling power of $2\ifmmode\times\else\texttimes\fi{}{d}_{B}$ permutation unitaries of Schmidt rank four are all 2 ebits. These quantities are also derived for generalized Clifford operators. We further show that any bipartite permutation unitary of Schmidt rank greater than two has entangling power greater than 1.223 ebits. We construct the generalized controlled-not (cnot) gates whose assisted entangling power reaches the maximum. We quantitatively compare the entangling power and assisted entangling power for general bipartite unitaries and their connection to the disentangling power by proposing a probabilistic protocol for implementing bipartite unitaries.
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.
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.