Multilevel superconducting circuits as two-qubit systems: Operations, state preparation, and entropic inequalities

We theoretically study operations with a four-level superconducting circuit as a two-qubit system. Using a mapping on a two-qubit system, we show how to implement iswap gates and Hadamard gates through pulses on transitions between particular pairs of energy levels. Our approach allows one to prepare pure two-qubit entangled states with desired form of reduced density matrices of the same purity and, in particular, arbitrary identical reduced states of qubits. We propose using schemes for the Hadamard gate and two-qubit entangled states with identical reduced density matrices in order to verify $logN$ inequalities for Shannon and Rényi entropies for the considered noncomposite quantum system.

Figure 1

Isomorphic correspondence between a qudit state with $d=4$ and a two-qubit system. Transition frequencies between levels correspond to the multilevel superconducting circuit investigated in Ref. [60].

Figure 2

In (a) realization of rotation around the $y$ axis in the Bloch sphere of the corresponding two-dimensional Hilbert subspace spanned by vectors $|j\rangle$ and $|k\rangle$ (blue arrow) using only operators of rotations around the $x$ axis (black arrows) applied to different couples of three-level set $\left\{\right|j\rangle ,|k\rangle ,|l\rangle \}$ [see Eq. (5)]. In (b)–(e) we show the implementation of various gates via sequences of $\theta$ pulses (4) applied to different two-level transitions. The duration of each pulse is denoted by the value near the corresponding arrow. We note that in the figure, time is directed from left to right, i.e., the order of pulses is opposite to that in Eqs. (5), (6), and (8)–(10).

Figure 3

The suggested scheme of preparation of a pure two-qubit state $|\psi \rangle$ (15) with reduced states having the same purity (length of Bloch radius vector): in (a) reduced states of qubits in Bloch sphere representation; in (b) sequence of pulses applied to a four-level superconducting circuit that prepares $|\psi \rangle$ from the ground state $|0\rangle$, where the black arrows stands for $x$ rotations (4) and blue for $y$ rotations (5).