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arXiv:quant-ph/0506020 · Open Systems & Information Dynamics

Decoherence-Free Subspaces and Subsystems for a Collectively Depolarizing Bosonic Channel

Jonathan L. Ball, Konrad Banaszek

Published May 31, 2005• 9 citations
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Abstract

We discuss the structure of decoherence-free subsystems for a bosonic channel affected by collective depolarization. A single use of the channel is defined as a transmission of a pair of bosonic modes. Collective depolarization consists in a random linear U(2) transformation of the respective mode operators, which is assumed to be identical for N consecutive uses of the channel. We derive a recursion formula that characterizes the dimensionality of available decoherence-free subsystems in such a setting.

DOI
10.1007/s11080-005-5723-1

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We discuss the structure of decoherence-free subsystems for a bosonic channel affected by collective depolarization. A single use of the channel is defined as a transmission of a pair of bosonic modes. Collective depolarization consists in a random linear U(2) transformation of the respective mode operators, which is assumed to be identical for N consecutive uses of the channel. We derive a recursion formula that characterizes the dimensionality of available decoherence-free subsystems in such a setting.
We present a technique to build, within a dissipative bosonic network, decoherence-free channels (DFCs): a group of normal-mode oscillators with null effective damping rates. We verify that the states protected within the DFC define the well-known decoherence-free subspaces (DFSs) when mapped back into the natural network oscillators. Therefore, our technique to build protected normal-mode channels turns out to be an alternative way to build DFSs, which offers advantages over the conventional method. It enables the computation of all the network-protected states at once, as well as leading naturally to the concept of the decoherence quasi-free subspace (DQFS), inside which a superposition state is quasi-completely protected against decoherence. The concept of the DQFS, weaker than that of the DFS, may provide a more manageable mechanism to control decoherence. Finally, as an application of the DQFSs, we show how to build them for quasi-perfect state transfer in networks of coupled quantum dissipative oscillators.
Properties of Bosonic linear (quasi-free) channels, in particular, Bosonic Gaussian channels with two types of degeneracy are considered. The first type of degeneracy can be interpreted as existence of noise-free canonical variables (for Gaussian channels it means that det⁡α=0\detα=0detα=0). It is shown that this degeneracy implies existence of (infinitely many) "direct sum decompositions" of Bosonic linear channel, which clarifies reversibility properties of this channel (described in arXiv:1212.2354) and provides explicit construction of reversing channels. The second type of degeneracy consists in rank deficiency of the operator describing transformations of canonical variables. It is shown that this degeneracy implies existence of (infinitely many) decompositions of input space into direct sum of orthogonal subspaces such that the restriction of Bosonic linear channel to each of these subspaces is a discrete classical-quantum channel.
We demonstrate that for an ensemble of qudits, subjected to collective decoherence in the form of perfectly correlated random SU(ddd) unitaries, quantum superpositions stored in the decoherence-free subspace are fully immune against the removal of one particle. This provides a feasible scheme to protect quantum information encoded in the polarization state of a sequence of photons against both collective depolarization and one-photon loss and can be demonstrated with photon quadruplets using currently available technology.
Decoherence-free subspaces (DFS) are to utilize the symmetric properties of the interaction between the system and environment so that they can be tolerant against the effect of decoherence. In this paper, we propose multiple-input multiple-output (MIMO) scheme to drive a finite-dimensional quantum system into DFS. For quantum system, the scheme is not only immune to dephasing but also feasible with currently available technology. Motivated by Shannon mutual information, we derive the capacity of MIMO quantum depolarizing channels.
We introduce a new quantum communication protocol for the transmission of quantum information under collective noise. Our protocol utilizes a decoherence-free subspace in such a way that an optimal asymptotic transmission rate is achieved, while at the same time encoding and decoding operations can be efficiently implemented. The encoding and decoding circuit requires a number of elementary gates that scale linearly with the number of transmitted qudits, m. The logical depth of our encoding and decoding operations is constant and depends only on the channel in question. For channels described by an arbitrary discrete group G, i.e. with a discrete number, |G|, of possible noise operators, perfect transmission at a rate m/(m+r) is achieved with an overhead that scales at most as O(dr)\mathcal{O}(d^r)O(dr) where the number of auxiliary qudits, r, depends solely on the group in question. Moreover, this overhead is independent of the number of transmitted qudits, m. For certain groups, e.g. cyclic groups, we find that the overhead scales only linearly with the number of group elements |G|.