We introduce the discording power of a unitary transformation, which assesses its capability to produce quantum discord, and analyze in detail the generation of discord by relevant classes of two-qubit gates. Our measure is based on the Cartan decomposition of two-qubit unitaries and on evaluating the maximum discord achievable by a unitary upon acting on classical-classical states at fixed purity. We find that there exist gates which are perfect discorders for any value of purity μ, and that they belong to a class of operators that includes the sqrt[SWAP]. Other gates, even those universal for quantum computation, do not possess the same property: the CNOT, for example, is a perfect discorder only for states with low or unit purity, but not for intermediate values. The discording power of a two-qubit unitary also provides a generalization of the corresponding measure defined for entanglement to any value of the purity.
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 ability to reach a maximally entangled state from a separable one through the use of a two-qubit unitary operator is analyzed for mixed states. This extension from the known case of pure states shows that there are at least two families of gates which are able to give maximum entangling power for all values of purity. It is notable that one of this gates coincides with a maximum discording one. We give analytical proof that such gate is indeed a perfect entangler at all purities and give numerical evidence for the existence of the second one. Furthermore, we find that there are other gates, many in fact, which are perfect entanglers for a restricted range of purities. This highlights the fact that many perfect entangler gates could in principle be found if a thorough analysis of the full parameter space is performed.
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.
Abstract We study the entangling properties of multipartite unitary gates with respect to the measure of entanglement called one-tangle. Putting special emphasis on the case of three parties, we derive an analytical expression for the entangling power of an n-partite gate as an explicit function of the gate, linking the entangling power of gates acting on the n-partite Hilbert space of dimension to the entanglement of pure states in the Hilbert space of dimension . Furthermore, we evaluate its mean value averaged over the unitary and orthogonal groups, analyze the maximal entangling power and relate it to the absolutely maximally entangled (AME) states of a system with 2n parties. Finally, we provide a detailed analysis of the entangling properties of the three-qubit unitary and orthogonal gates.
The entangling power of a unitary operator acting on a bipartite Hilbert space measures the entanglement it generates from product states, averaged over the inputs. A finite-dimensional unitary has a spectral decomposition
U=∑a=1neiθaPa, where
eiθa are the eigenvalues,
Pa the corresponding eigen-projectors, and
n is the number of distinct eigenvalues. After removing an overall phase, the entangling power is a function on the
(n−1)-torus of relative eigenphases at fixed spectral projectors. We prove that this function is stationary at all
2n−1 points on the torus where every relative phase is
0 or
π, which we define as \textit{corners}. Up to an overall phase,
U at each corner is a generalized reflection
R=I−2Q satisfying
R2=I, where
Q is the sum of spectral projectors whose relative phase is
π. At the corner the entangling power is expressed in terms of seven local-unitary invariants of
Q. A unitary gate
U can be realized as a corner of some projector family if and only if
U2∝I, a condition satisfied by many Clifford and non-Clifford gates. We illustrate the theorem with two-qubit gates,
SU(N) channel decompositions, and two-site spin chains, obtaining examples of minima, maxima, and saddle points. In addition, a corner that is a saddle point on the full phase torus can appear as a local maximum or minimum along different time-evolution trajectories.