Optical implementation of a unitarily correctable code

Noise poses a challenge for any real-world implementation in quantum information science. The theory of quantum error correction deals with this problem via methods to encode and recover quantum information in a way that is resilient against that noise. Unitarily correctable codes are an error correction technique wherein a single unitary recovery operation is applied without the need for an ancilla Hilbert space. Here, we present an optical implementation of a nontrivial unitarily correctable code for a noisy quantum channel with no decoherence-free subspaces or noiseless subsystems. We show that recovery of our initial states is achieved with high fidelity $\left(\ge 0.97\right)$, quantitatively proving the efficacy of this unitarily correctable code.

Figure 1

(Color online) Experimental setup. (a) Source of polarization-entangled photon pairs. An ultraviolet diode laser is used to pump a pair of orthogonally oriented BiBO crystals to produce entangled photon pairs by spontaneous parametric down conversion. The precise state produced is determined by the polarization of the laser light and can be set using a half-wave plate. A pair of compensation crystals are used to compensate group-velocity mismatch in the down-conversion crystals. The light is collected into single-mode fibers. More details can be found in the text. (b) Experimental implementation of the noise model, correction, and tomography. A quarter-wave plate is used to adjust the phase of the entangled pairs; together with the HWP in the uv laser and the fiber-based polarization controllers, these operations are sufficient to prepare an arbitrary pure state in ${\mathcal{C}}_1$. When the noise is on, the photon pairs are subject to computer controlled anticorrelated phase errors through liquid-crystal variable phase retarders (LCVR); a decision as to which LCVR will fire is made by a pseudorandom number generator every 1 s. The state of the light is measured using quantum state tomography with a tomographically overcomplete set of measurements and the maximum likelihood procedure [22].

Figure 2

(Color online) Example set of experimentally measured two-photon density matrices (left: real parts, right: imaginary parts). (a) The initial state was chosen to have a nontrivial value of both angles in Eq. (5); the nearest pure state in ${\mathcal{C}}_1$ has angles $\theta =35.5°$ and $\varphi =46.5°$. (b) The noise affected state has its fidelity to the initial state reduced to 0.62; one can see that the primary effect of the noise is to flip the sign of the real and imaginary parts of the coherences rather than decohere the state. (c) The final state after both noise and correction. The fidelity with the initial state has been restored to 0.97 thus demonstrating the efficacy of the unitarily correctable code.

Figure 3

(Color online) Measured fidelity between initial and final states. The mixed state fidelity [24] was calculated on experimentally reconstructed density matrices between the initial states and noise-affected states (open circles, solid line theory) and the initial states and states affected by noise and unitary correction (closed, red circles). The data show that the effect of the noise depends strongly on the initial state. Initial states closest to the product states $|HH⟩$ and $|VV⟩$ are least affected while those close to the maximally entangled state $|{\varphi }^{+}⟩$ are driven to nearly orthogonal states. The correction restores the photons to states with fidelity of higher than 0.98 with the initial states in all cases.