We consider how much entanglement can be produced by a nonlocal two-qubit unitary operation,
UAB---the entangling capacity of
UAB. For a single application of
UAB, with no ancillas, we find the entangling capacity and show that it generally helps to act with
UAB on an entangled state. Allowing ancillas, we present numerical results from which we can conclude, quite generally, that allowing initial entanglement typically increases the optimal capacity in this case as well. Next, we show that allowing collective processing does not increase the entangling capacity if initial entanglement is allowed.
Abstract We study the optimal way of creating bi-partite entanglement using a general two-qubit in teraction. On the one hand, we analyze the entanglement capability of an arbitrary non-local Hamiltonian acting on two qubits. We explicitly calculate the state which maximizes the entangle ment produced per time step 81 during the non-local evolution. On the other hand, we determine the maximal amount of entanglement which can be produced by an arbitrary two-qubit gate. We also give the separable state which leads to the output state containing this amount of entanglement. Furthermore, we consider the situation where auxiliary systems are present. Finally, we determine the non-unitary processes which are able to create entanglement from an initially separable state of two systems.
Optimal implementation of quantum gates is crucial for designing a quantum computer. The necessary condition for optimal construction of a two-qubit unitary operation is obtained. It can be proved that the
B gate is the unique gate that can construct a two-qubit universal circuit with only two applications, i.e., this condition is also sufficient in the case of two applications of the elementary two-qubit gate. It is also shown that many perfect entanglers cannot simulate an arbitrary two-qubit gate with only three applications. We show how a two-qubit gate's construction power is related to the construction power of its mirror gate, and introduce a specific kind of two-qubit entangling gate, i.e., a super controlled gate, which can construct an arbitrary two-qubit gate by three applications.
Entanglement-changing power of an arbitrary two-qubit operation, including increasing and decreasing power, is investigated in this paper. We consider the maximal entanglement
Cmax and the minimal entanglement
Cmin of the states obtained by a given two-qubit unitary operation
Ud acting on arbitrary pure states with fixed entanglement
C0. We give the condition that the maximal entanglement
Cmax of the obtained states can be
1 and the minimal entanglement
Cmin can be
0. When the maximal entanglement
Cmax cannot be
1, we give the maximal value it can reach. When the minimal entanglement
Cmin cannot be
0, we give the minimal value it can reach. We think
Cmax and
Cmin represent the entanglement-changing power of two-qubit unitary operations.
In order to demonstrate nontrivial quantum computations experimentally, such as the synthesis of arbitrary entangled states, it will be useful to understand how to decompose a desired quantum computation into the shortest possible sequence of one-qubit and two-qubit gates. We contribute to this effort by providing a method to construct an optimal quantum circuit for a general two-qubit gate that requires at most 3 controlled-NOT (CNOT) gates and 15 elementary one-qubit gates. Moreover, if the desired two-qubit gate corresponds to a purely real unitary transformation, we provide a construction that requries at most 2 CNOT and 12 one-qubit gates. We then prove that these constructions are optimal with respect to the family of CNOT,
y-rotation,
z-rotation, and phase gates.
The work of Knill et. al. (2001) established the possibility of nondeterministic realization of certain quantum logic operations using linear optical elements, ancilla photons and postselection techniques.It was also shown that any discrete unitary operator acting on N optical modes can be implemented by a triangular multiport device constructed from a series of beam splitters and phase shifters (see work of Reck, Zeilinger et. al., 1994).Here, we consider the rectangular linear optical multiport that is used for the probabilistic realization of unitary transformations on n qubits.This kind of linear optical scheme is suitable for probabilistic realization of unitary operators using ancilla photons and projective measurements.Qubits are encoded into the bosonic states of optical modes in two possible polarizations, and a number of ancilla photons and photodetectors are used for postselection of the qubits' state, based on the output of the detectors.We derive a procedure of evolutionary operator calculation for schemes of the considered type and present algorithms for their efficient computation on symmetric state space.We also provide complexities for different algorithms for the computation of evolutionary operator and estimate demands of resources in each case.A destructive Toffoli gate, acting on three qubits, using one ancilla photon and a photodetector, is implemented using schemes of the presented type.