Protected subspaces in quantum information

Abstract

Please be advised that this information was generated on 2018-05-13 and may be subject to change.arXiv:0902.4126v2 [quant-ph] 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. By a version of the Heisenberg Principle, which we prove, such a protected space must be "dark" in the sense that no information leaks out during the transmission. We explain the role of the Knill-Laflamme condition in relation to protection and darkness, and we analyze several degrees of protection, whether related to error correction, or to state restauration after a measurement. 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 tri-unitary noise models. P r o te c te d S u b sp a ces in Q u an tu m In form ation I. IN T R O D U C T IO NWe consider a quantum channel of finite dimension through which a quantum system in some state is sent. The output consists of another quantum state, and possibly some classical information. We are interested in the question to what extent the original quantum state can be recovered from that state and that information. In particular, we investigate if there are subspaces of the Hilbert space of the original system , on which the state can be perfectly restored.In the literature a hierarchy of such spaces, which we shall call protected subspaces here, has been described. The strongest protection possible is provided in the case of a "decoherence free subspace" [1-4]. In this case the channel acts on the subspace as a isometric transformation. All we have to do in order to recover the state, is to rotate it back.The next strongest form of protection occurs when the channel acts on the subspace as a random choice between isometries, whose image spaces are m utually orthogonal. Then by measuring along a suitable partition of the output Hilbert space, it can be inferred from the output state which isometry has occurred, so that it can be rotated back. This situation is characterized by the well-known Knill-Laflamme criterion, [5, 6] and the protected subspace in this case is usually called an error correction subspace.The weakest form of protection is provided in yet a third situation, which was encountered in the context of quantum trajectories and the purification tendency of states along these paths [7]. In this case the deformation of the state is not caused by some given external device, but by the experimenter himself, who is performing a Kraus measurement [8]. Also in this case the "channel" acts as a random isometry, but the image spaces need not be orthogonal. It is now the m eas...

Restricted numerical range: A versatile tool in the theory of quantum information

“…where X m = Y † m Y m and λ m ∈ C no information goes outside of this subspace [53], so P l is called a dark subspace [54].…”

Quantifying the Performance of Quantum Codes

“…For instance, in [24] it is suggested that the entanglement fidelity [25] is the important quantity to maximize in schemes for quantum error correction. In [26], it is proposed that the code entropy is an important quantitative measure of the quantum correction schemes by determining a hierarchical structure in the set of protected spaces [27].…”

Global Estimates of Errors in Quantum Computation by the Feynman–Vernon Formalism

“…If some quantum dynamics Φ on some space H admits quantum error correction therefore reduces to the question if there exists a code space C. In general this is not trivial to decide, see e.g. [42] for general rank-2 super-operators and [40] for general 2-qubit maps.…”