Noiseless Subsystems and the Structure of the Commutant in Quantum Error Correction

Abstract

The effect of noise on a quantum system can be described by a set of operators obtained from the interaction Hamiltonian. Recently it has been shown that generalized quantum error correcting codes can be derived by studying the algebra of this set of operators. This led to the discovery of noiseless subsystems. They are described by a set of operators obtained from the commutant of the noise generators. In this paper we derive a general method to compute the structure of this commutant in the case of unital noise.

Generalized multiplicative domains and quantum error correction

“…The equivalence of (1) and (3) is in the same flavour as the commutant characterization of noiseless subsystems for unital CPTP maps given in [15], where by "noiseless" we mean a correctable subsystem that is corrected by the map R ≡ id. In fact, by taking π = id, it can be thought of as a generalization of Theorem 1 of [10], which states that if C = A ⊗ B is a subspace of H, then B is noiseless for E if and only if aE i P C − E i a = P C E * i a − aE * i = 0 for all a ∈ A B and all i, where E ≡ {E i }.…”

“…On the one hand, these characterizations of correctable codes provide the motivation for some of the characterizations of generalized multiplicative domains that are derived. On the other hand, we feel that they are of independent interest, as they generalize several known results for noiseless subsystems and decoherence-free subspaces [12,16,18,20,24,26,33,34], including the multiplicative domain results of [9] and the commutant relationships of [10,15]. In particular, it was recently shown that the standard multiplicative domain of a completely positive trace-preserving (CPTP) map encodes a subclass of what are known as "unitarily correctable codes" [9,22,23,31] (UCC).…”

Asymptotic evolution of random unitary operations

“…In the case of random unitary operations the following powerful theorem can be proved which allows us to specify the space of attractors of the RUO Φ. In this context it should be also mentioned that for the more general case of arbitrary unital quantum operations interesting general results have been derived by Kribs [13,14] recently. …”

“…A simple evaluation of Eq. (64) leads to the relations X (11) = , X (33) = , X (13) = , and X (31) = ( ∈ C). The remaining matrix block X (22) has to commute with the irreducible set of 2 × 2 matrices C (2) ( ∈ {1 2}) and has to be equal to a multiple of the identity matrix X (22) = I ( ∈ C).…”

Generalized Kac lemma for recurrence time in iterated open quantum systems

“…We generalized [7] this result to open quantum-mechanical systems whose dynamics is given by a quantum channel [8][9][10][11][12]. We found that if the channel is unital, i.e., if the limit distribution is the completely mixed state in an effective Hilbert space that contains the initial state [13][14][15], then the expected return time is equal to the dimensionality of that effective Hilbert space.…”