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arXiv:2608.21105

Quantifying Entangling Power of Controlled Unitary Gates

Ankur Haldar, Pankaj Agrawal, Prasenjit Deb

quant-phSubmitted Aug 21, 2026• Announced Aug 24, 2026• 0 citations
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Abstract

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.

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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.
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.
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.
We perform optimal-control-theory calculations to determine the minimum number of two-qubit controlled-not (cnot) gates needed to perform quantum state preparation and unitary operator synthesis for few-qubit systems. By considering all possible gate configurations, we determine the maximum achievable fidelity as a function of quantum circuit size. This information allows us to identify the minimum circuit size needed for a specific target operation and enumerate the different gate configurations that allow a perfect implementation of the operation. We find that there are a large number of configurations that all produce the desired result, even at the minimum number of gates. We also show that the number of entangling gates can be reduced if we use multiqubit entangling gates instead of two-qubit cnot gates, as one might expect based on parameter counting calculations. In addition to treating the general case of arbitrary target states or unitary operators, we apply the numerical approach to the special case of synthesizing the multiqubit Toffoli gate. This approach can be used to investigate any other specific few-qubit task and provides insight into the tightness of different bounds in the literature.
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.
In this paper, we study the optimal simulation of the three-qubit unitary using two-qubit gates. First, we completely characterize the two-qubit gate cost of simulating the Deutsch gate (controlled-controlled gate) by generalizing our result on the two-qubit cost of the Toffoli gate. The function of any Deutsch gate is simply a three-qubit controlled-unitary gate and can be intuitively explained as follows: The gate outputs the states of the two control qubits directly, and applies the given one-qubit unitary uuu on the target qubit only if both the states of the control qubits are $|1\ensuremath{\rangle}$. Previously, it was only known that five two-qubit gates are sufficient for implementing such a gate [Sleator and Weinfurter, Phys. Rev. Lett. 74, 4087 (1995)]. We show that if the determinant of uuu is 1, four two-qubit gates are optimal. Otherwise, five two-qubit gates are required. For the Fredkin gate (the controlled-swap gate), we prove that five two-qubit gates are necessary and sufficient, which settles the open problem introduced in Smolin and DiVincenzo [Phys. Rev. A 53, 2855 (1996)].
We study two distinct diagnostics of bipartite quantum channels: the average fidelity between a desired state and noisy output across different classes of input states, and generation of entanglement from pure product input states, quantified by well-defined entangling power. These diagnostics are generally independent, but together they provide a more complete description of bipartite channels. We show that the fidelity averaged over any local-unitary orbit with fixed Schmidt coefficients for equal local dimensions is completely determined by the average input-output fidelity and its restriction to product inputs. We also introduce concurrence- and negativity-based entangling power for two-qubit channels, prove their convexity and monotonicity under local postprocessing, and show that, unlike the previously proposed linear-entropy quantity, they vanish for all separable channels. Representative separable and nonseparable channels are investigated. Finally, we generalize our definitions of entangling power to nonproduct inputs, and provide an analytical lower bound for the concurrence-based entanglement variation, showing its effectiveness with specific examples.