Quantum noise reduction using DFS

Decoherence-free subspace (DFS) is a subspace of the Hilbert space of multiple physical qubits that do not sense the decoherence of the qubits. Therefore, if we can encode the quantum information of a logical qubit into this subspace, the quantum information is free from physical decoherence during transmission over noisy channel or keeping it in a quantum memory surrounded by noisy environment. The difference between DFS-based noise reduction and so called quantum error correction codes (QECC) is that QECC can correct any kind of error if the complicated system is truly realized with highly reliable quantum circuits, while the implementation of DFS-based noise reduction is rather simple if we do not intend to correct all possible errors but only those errors caused by practical decoherence.

As a simple example, let we use two photons (i.e. two polarization qubits) having a small time lag to encode a qubit information. We define state "0" when both photons have the same polarization, and state "1" to be the case when the two photons have orthogonal polarizations. Now, after transmission over a refractive-index-fluctuating channel such as an optical fiber or air with turbulence, the phase difference between two main polarizations will fluctuate in an uncontrollable manner. The relationship of "same polarization" or "orthogonal polarization" is, however, unchanged, assuming that the time lag is small enough in comparison with the frequency bandwidth of the refractive index fluctuation. This assumption is valid in the usual cases, and we refer to this kind of noise as "collective noise."

In this example, the Hilbert space of a qubit, whose bases are H (horizontal polarization) and V (vertical polarization), is enlarged to four dimension, HH, HV, VH, and VV by using two photons. The docoherence-free subspace in this case is the space spanned by HV and VH. Since the phase noise phi_H and phi_V are summed both for HV and VH, the phase noise is cancelled in this subspace, and thus, the subspace does not feel the phase noise. Note that the phase noise is doubled in the other subspace spanned HH and VV. This means that if we pick up only the case that "the two photons have orthogonal polarizations" without determining each polarization, we can successfully pick up the quantum information.

When this scheme is used for photons A1 and A2 that are entangled with B1 and B2, respectively, then this scheme becomes "entanglement distillation" under collective noise. We did this experiment for the first time, which was published in and its theoretical proposal was published in We generalized this entanglement distillation (having no encoded classical information actually) to a qubit information protection. The problem in this case is that you cannot prepare the two photons by yourself, but someone (or the previous subroutine) will give you a physical qubit, and you must transfer the quantum information (without looking into it) into the DFS of the two photons: one is the given photon and the other is the ancilla photon you prepared. This is not trivial at all, and the proposal is found in In the experiment in Nature(2003), we used a liquid crystal retarder to simulate a real optical fiber. The experiment for the qubit information protection has been done using a real optical fiber, resulting in This scheme is newly generalized to entanglement distribution in the scope of quantum repeater: The drawback of this system is that the efficiency goes down as the square of the transmission T since both photons need to reach the receiver side. (Here we point out that this scheme is still far better than the quantum error correction, which suffers from even stringent efficiency deterioration as T^(-5) or T^(-7) or T^(-9) for the famous three error correction codes.) We proposed a new scheme to use the ancilla photon that counter propagate with the signal photon. The experiment has been successfully done in

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