Achieving perfect control over the parameters defining a quantum gate is, in general, a very challenging task, and at the same time, environmental interactions can introduce disturbances to the initial states as well. Here we address the problem of how the imperfections in unitaries and noise present in the input states affect the entanglement-generating power of a given quantum gate -- we refer to it as imperfect (noisy) entangling power. We observe that, when the parameters of a given unitary are chosen randomly from a Gaussian distribution centered around the desired mean, the quenched average entangling power -- averaged across multiple random samplings -- exhibits intriguing behavior like it may increase or show nonmonotonic behavior with the increase of disorder strength for certain classes of diagonal unitary operators. For arbitrary unitary operators, the quenched average power tends to stabilize, showing almost constant behavior with variation in the parameters instead of oscillating. Our observations also reveal that, in the presence of a local noise model, the input states that maximize the entangling power of a given unitary operator differ considerably from the noiseless scenario. Additionally, we report that the rankings among unitary operators according to their entangling power in the noiseless case change depending on the noise model and noise strength.
Achieving perfect control over the parameters defining a quantum gate is, in general, a very challenging task, and at the same time, environmental interactions can introduce disturbances to the initial states as well. Here we address the problem of how the imperfections in unitaries and noise present in the input states affect the entanglement-generating power of a given quantum gate -- we refer to it as imperfect (noisy) entangling power. We observe that, when the parameters of a given unitary are chosen randomly from a Gaussian distribution centered around the desired mean, the quenched average entangling power -- averaged across multiple random samplings -- exhibits intriguing behavior like it may increase or show nonmonotonic behavior with the increase of disorder strength for certain classes of diagonal unitary operators. For arbitrary unitary operators, the quenched average power tends to stabilize, showing almost constant behavior with variation in the parameters instead of oscillating. Our observations also reveal that, in the presence of a local noise model, the input states that maximize the entangling power of a given unitary operator differ considerably from the noiseless scenario. Additionally, we report that the rankings among unitary operators according to their entangling power in the noiseless case change depending on the noise model and noise strength.
Quantifying the entangling capability of quantum operations is a fundamental task in quantum information science. Traditionally, this capability is measured by the entangling power (EP), defined as the average entanglement generated when a quantum operation acts uniformly on all possible product states. However, EP alone cannot capture the intricate input-state-dependent nature of entanglement generation. To address this, we define a complementary metric -- entangling power deviation (EPD) -- as the standard deviation of entanglement generated over all product input states, thereby capturing the multifaceted nature of entangling behavior. We develop a general group-theoretical framework that yields closed-form expressions for both EP and EPD. Our analysis shows that any nontrivial entangling operation necessarily exhibits input-state dependence: nonzero EP implies a nonzero EPD. By analyzing representative two-qubit gates, we show that the gates with identical EP can exhibit markedly different EPD values, illustrating that the nature of entanglement generation can significantly differ depending on the gate functionality. Extending our framework to a class of generalized controlled-unitary operations acting on bipartite Hilbert spaces of arbitrary dimensions, we further analyze the interplay between the entangling strength and uniformity, as quantified by EP and EPD. Moreover, we uncover a subtle dimension-parity-dependent behavior in entanglement generation, which EP alone fails to detect. These findings highlight EPD as an indispensable diagnostic tool -- one that, alongside EP, provides a deeper and more complete characterization of the entangling structure.
It is demonstrated here that local dynamics have the ability to strongly modify the entangling power of unitary quantum gates acting on a composite system. The scenario is common to numerous physical systems, in which the time evolution involves local operators and nonlocal interactions. To distinguish between distinct classes of gates with zero entangling power we introduce a complementary quantity called gate typicality and study its properties. Analyzing multiple, say n, applications of any entangling operator, U, interlaced with random local gates we prove that both investigated quantities approach their asymptotic values in a simple exponential form. These values coincide with the averages for random nonlocal operators on the full composite space and could be significantly larger than that of Un. This rapid convergence to equilibrium, valid for subsystems of arbitrary size, is illustrated by studying multiple actions of diagonal unitary gates and controlled unitary gates.
We study quantum dynamics through the lens of entanglement generation and characterize the underlying unitary evolution by the distinct signatures it imprints on the time-dependent entangling power. For a unitary operator, we characterize ergodicity by the equality between its long-time-averaged entangling power and the Haar-averaged linear entropy. We define mixing more stringently as the convergence of the time-dependent entangling power itself to the Haar value at long times. Within this framework, we establish the ergodic hierarchy of dynamical behavior, showing in particular that mixing implies ergodicity, whereas ergodicity does not necessarily imply mixing. As an application, we find that two-qubit unitary gates are neither ergodic nor mixing: their long-time-averaged entangling power can take only four discrete values, none of which coincides with the Haar average. We then investigate many-body dynamics using the kicked Ising chain and find that the long-time-averaged entangling power converges to the Haar value in both integrable and nonintegrable cases, indicating ergodicity. Remarkably, however, the nonintegrable chain exhibits mixing, whereas the integrable chain, despite being ergodic, is demonstrably nonmixing. We also introduce a Lyapunov-like exponent to characterize the rate at which the time-dependent entangling power approaches its saturation value. We find that this exponent increases systematically with the degree of integrability breaking in the many-body system. Our results establish entanglement generation as a useful framework for characterizing dynamical systems and reveal qualitatively different signatures of integrability beyond conventional diagnostics.
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.
The change of the entangling power of n 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 ep is analyzed along with the gate typicality gt 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) 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.
Nonlocal properties of an ensemble of diagonal random unitary matrices of order N2 are investigated. The average Schmidt strength of such a bipartite diagonal quantum gate is shown to scale as lnN, in contrast to the lnN2 behavior characteristic of random unitary gates. Entangling power of a diagonal gate U is related to the von Neumann entropy of an auxiliary quantum state $\ensuremath{\rho}=A{A}^{\ifmmode\dagger\else\textdagger\fi{}}/{N}^{2}$, where the square matrix A is obtained by reshaping the vector of diagonal elements of U of length N2 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 $\ensuremath{\rho}$, such that all their diagonal entries are equal to 1/N. Such a state is contradiagonal with respect to the computational basis, in the 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.