Orthogonalization of partly unknown quantum states

A quantum analog of the fundamental classical not gate is a quantum gate that would transform any input qubit state onto an orthogonal state. Intriguingly, this universal NOT gate is forbidden by the laws of quantum physics. This striking phenomenon has far-reaching implications concerning quantum information processing and encoding information about directions and reference frames into quantum states. It also triggers the question of under what conditions the preparation of quantum states orthogonal to input states becomes possible. Here we report on experimental demonstration of orthogonalization of partly unknown single- and two-qubit quantum states. A state orthogonal to an input state is conditionally prepared by quantum filtering, and the only required information about the input state is a mean value of a single arbitrary operator. We show that perfect orthogonalization of partly unknown two-qubit entangled states can be performed by applying the quantum filter to one of the qubits only.

Figure 1

Orthogonalization of partly unknown single-qubit states. (a) Attenuation of amplitude of state $|0\rangle$. (b) Unitary $\pi$ phase shift. For details, see text.

Figure 2

Experimental setup for orthogonalization of (a) single-qubit and (b) two-qubit states. NLC, nonlinear crystal; SMF, single-mode fiber; BD, calcite beam displacer; PPBS, partially polarizing beam splitter; PBS, polarizing beam splitter; HWP, half-wave plate; QWP, quarter-wave pate, D, single-photon detector; DB, detection block consisting of a HWP, QWP, PBS, and two single-photon detectors.

Figure 3

Overlap $F$ between input and orthogonalized single-qubit states. The results are shown for the two approaches where $\langle {\sigma }_Z\rangle$ is determined either (a) from the theoretical knowledge of the prepared input state or (b) from measurements on the input state in the $H/V$ basis.

Figure 4

Minimum average overlap $F_{\min }$ between input and output single-qubit states that is achievable by deterministic operations when $\theta$ is known.

Figure 5

Success probability of orthogonalization is plotted as a function of $\theta$. The solid line represents theoretical dependence, and symbols indicate experimental results for single-qubit (blue circles) and two-qubit (red triangles) states. Results are shown for both methods of determination of $\langle {\sigma }_Z\rangle$, as in Fig. 3.