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arXiv:1604.05788 · Physical Review A

Entangling and assisted entangling power of bipartite unitary operations

Lin Chen, Li Zhi Yu

Published Aug 8, 2016• 16 citations
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Abstract

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.

DOI
10.1103/physreva.94.022307

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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: log29\ensuremath−16/9{log}_{2}9\ensuremath{-}16/9log2​9\ensuremath−16/9 or log23{log}_{2}3log2​3 ebits. The entangling power, assisted entangling power, and disentangling power of 2\ifmmode×\else\texttimes\fidB2\ifmmode\times\else\texttimes\fi{}{d}_{B}2\ifmmode×\else\texttimes\fidB​ 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.
It is known that any bipartite unitary operator of Schmidt rank three is equivalent to a controlled unitary under local unitaries. We propose a standard form of such operators. Using the form we improve the upper bound for the entanglement cost to implement such operators under local operations and classical communications (LOCC), and provide a corresponding protocol. A part of our protocol is based on a recursive-control protocol which is helpful for implementing other unitary operators. We show that any bipartite permutation unitary of Schmidt rank three can be implemented using LOCC and two ebits. We give two protocols for implementing bipartite permutation unitaries of any Schmidt rank rrr, and showed that one of the protocol uses O(r)O(r)O(r) ebits of entanglement and O(r)O(r)O(r) bits of classical communication, while these two types of costs for the other protocol scale as O(rlog⁡r)O(r\log r)O(rlogr) but the actual values are smaller for all r<1100r<1100r<1100. Based on this we obtain upper bounds of the number of nonlocal CNOT gates needed to implement bipartite classical reversible maps using classical circuits under two different conditions. We also quantify the entangling power of bipartite permutation unitaries of Schmidt rank two and three. We show that they are respectively 111 ebit and some value between log⁡29−16/9\log_2 9 - 16/9log2​9−16/9 and log⁡23\log_2 3log2​3 ebits.
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.
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.
It is known that any bipartite unitary operator of Schmidt rank 3 is equivalent to a controlled unitary under local unitaries. We propose a standard form of such operators. Using the form we improve the upper bound for the entanglement cost to implement such operators under local operations and classical communications (LOCC), and provide a corresponding protocol. A part of our protocol is based on a recursive-control protocol which is helpful for implementing other unitary operators. We show that any bipartite permutation unitary of Schmidt rank 3 can be implemented using LOCC and two ebits. We give two protocols for implementing bipartite permutation unitaries of any Schmidt rank rrr and showed that one of the protocol uses O(r)O(r)O(r) ebits of entanglement and O(r)O(r)O(r) bits of classical communication, while these two types of costs for the other protocol scale as O(rlogr)O(rlogr)O(rlogr) but the actual values are smaller for all r<1100r<1100r<1100. Based on this we obtain upper bounds of the number of nonlocal controlled-not gates needed to implement bipartite classical reversible maps using classical circuits under two different conditions. We also quantify the entangling power of bipartite permutation unitaries of Schmidt ranks 2 and 3. We show that they are respectively 1 ebit and some value between log29−−16/9{log}_{2}9--16/9log2​9−−16/9 and log23{log}_{2}3log2​3 ebits.
Implementing nonlocal unitary operators is an important and hard question in quantum computing and cryptography. We show that any bipartite nonlocal unitary operator of Schmidt rank three on the $({d}_{A}\ifmmode\times\else\texttimes\fi{}{d}_{B})$-dimensional system is locally equivalent to a controlled unitary when dA{d}_{A}dA​ is at most three. This operator can be locally implemented assisted by a maximally entangled state of Schmidt rank r=mindA2,dBr=\mathrm{min}{{d}_{A}^{2},{d}_{B}}r=mindA2​,dB​. We further show that stochastic-equivalent nonlocal unitary operators are indeed locally equivalent, and propose a sufficient condition on which nonlocal and controlled unitary operators are locally equivalent. We also provide the solution to a special case of a conjecture on the ranks of multipartite quantum states.
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.