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arXiv:1804.01222 · Journal of Physics A Mathematical and Theoretical

Entangling power of two-qubit unitary operations

Yi Shen, Lin Chen

Published Aug 3, 2018• 8 citations
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Abstract

Abstract The entangling power of a bipartite unitary operation shows the maximum created entanglement with the product input states. For an arbitrary two-qubit unitary operation, it is sufficient to consider its normalized operation U with parameters and c 3 . We show how to compute the entangling power of U when . In particular we construct the analytical expressions of entangling power of such U for two examples. We also formulate the entangling power of bipartite unitary operations of Schmidt rank two for any dimensions.

DOI
10.1088/1751-8121/aad7cb

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Abstract The entangling power of a bipartite unitary operation shows the maximum created entanglement with the product input states. For an arbitrary two-qubit unitary operation, it is sufficient to consider its normalized operation U with parameters and c 3 . We show how to compute the entangling power of U when . In particular we construct the analytical expressions of entangling power of such U for two examples. We also formulate the entangling power of bipartite unitary operations of Schmidt rank two for any dimensions.
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 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.
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 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.
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.
We consider two capacity quantities associated with bipartite unitary gates: the entangling and the disentangling power. Here, we prove that these capacities are different in general by constructing an explicit example of a qubit-qutrit unitary whose entangling power is maximal (2 ebits), but whose disentangling power is strictly less. A corollary is that there can be no unique ordering for unitary gates in terms of their ability to perform nonlocal tasks. Finally, we show that in large dimensions, almost all bipartite unitaries have entangling and disentangling capacities close to maximal.