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Decoherence free subspaces for two--access quantum channels

Maciej Demianowicz

Published Sep 1, 2012• 0 citations
OpenAlex

Abstract

In this paper we consider construction of decoherence free subspaces for two--access random unitary channels. First, we concentrate on hermitian unitary noise model $U$ for a bi--unitary channel and show that in this case a code exists if the space of Schmidt matrices of an eigensubspace of $U$ exhibits certain properties of decomposability. Then, we show that our technique is also applicable in a generic case and consider its application two tri- (quatro-, ...) unitary channels.

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In this paper we consider construction of decoherence free subspaces for two--access random unitary channels. First, we concentrate on hermitian unitary noise model UUU for a bi--unitary channel and show that in this case a code exists if the space of Schmidt matrices of an eigensubspace of UUU exhibits certain properties of decomposability. Then, we show that our technique is also applicable in a generic case and consider its application two tri- (quatro-, ...) unitary channels.
In this paper we consider decoherence-free communication over multiple access and k-user quantum channels. First, we concentrate on a hermitian unitary noise model U for a two-access bi-unitary channel and show that in this case a decoherence-free code exists if the space of Schmidt matrices of an eigensubspace of U exhibits certain properties of decomposability. Then, we show that our technique is also applicable for generic random unitary two-access channels. Finally, we consider the applicability of the result to the case of a larger number of senders and general Kraus operators.
Maps of the form Phi(X) =sum_{i=1}^s A_iXA^*, where A_1, . . . ,A_s are fixed complex n by n matrices and X is any complex n by n matrix are used in quantum information theory as representations of quantum channels. This article deals with computable conditions for the existence of decoherence--free subspaces for Phi. Since the definition of decoherence-free subspace for quantum channels relies only on the matrices A1, . . . ,As, the term of common reducing unitary subspace is used instead of the original one. Among the main results of the paper, there are computable conditions for the existence of common eigenvectors. These are related to common reducing unitary subspaces of dimension one. The new results on common eigenvectors provide new effective condition for the existence of common invariant subspaces of arbitrary dimensions.
In certain situations the state of a quantum system, after transmission through a quantum channel, can be perfectly restored. This can be done by 'coding' the state space of the system before transmission into a 'protected' part of a larger state space, and by applying a proper 'decoding' map afterwards. We investigate two kinds of protected subspaces: the error correction subspaces and the dark subspaces. In neither case information leaks out of the system, but in the second it is more difficult to restore the state. Recent results on higher rank numerical ranges of operators are used to construct examples. In particular, dark spaces are constructed for any map of rank 2, for a biased permutations channel and for certain separable maps acting on multipartite systems. Furthermore, error correction subspaces are provided for a class of three unitary noise models.
In this work, inspired by the study of semidefinite programming for block-diagonalizing matrix *-algebras, we propose an algorithm that can find the algebraic structure of decoherence-free subspaces (DFS's) for a given noisy quantum channel. We prove that this algorithm will work for all cases with probability 1, and it is more efficient than the algorithm proposed by J. A. Holbrook, D. W. Kribs, and R. Laflamme, [Quantum. Inf. Proc. 80, 381 (2003)]. In fact, our results reveal that this previous algorithm only works for special cases. As an application, we discuss how this method can be applied to increase the efficiency of an optimization procedure for finding an approximate DFS.
All free-space quantum communications require the use of a quantum channel, which transports quantum bits in such a way that the quantum mechanical states of the qubits remain preserved from one end of the channel to the other one. In quantum computing the classical error coding methods could not be used, however we can construct a classical channel with zero redundancy error correction for any unitary channel. In our basically new quantum error correction approach, the classical states are coded into the eigenvectors and unitary transformations. In this paper, we show that with our new algorithm it’s possible to create redundancy-free quantum error correction. We also consider the redundancy-free implementation of a unitary error correcting operator. Our protocol achieves the redundancy-free quantum communication using local unitary operations and unitary matrices. These solutions could be useful for the free-space quantum communication.
We introduce a new quantum noise deconvolution technique that requires neither complete knowledge of the noise nor partial noise tomography, and is applicable to any invertible quantum noise channel. In this new method, we construct a set of observables with completely correctable expectation values despite our incomplete knowledge of noise. This task is achieved just by classical post-processing without extra quantum resources. We show that the number of parameters in the subset of observables with correctable expectation values is the same for all unitary quantum channels. For mixed unitary channels and the assumption that the probability distribution of unitary errors is unknown, we instruct the construction of the set of observables with correctable expectation values. For a particular case where the mixed unitary channel is made of just two random unitary Kraus operators acting on ddd-dimensional Hilbert-space, we show that the observable with correctable expectation value belongs to a set with at least ddd parameters. We extend our method by considering observables for which the partial recovery of the expectation value is possible, at the cost of having partial knowledge about the noise-free initial state.