The central issue in this article is to transmit a quantum state in such a way that after some decoherence occurs, most of the information can be restored by a suitable decoding operation. For this purpose, we incorporate redundancy by mapping a given initial quantum state to a messenger state on a larger-dimensional Hilbert space via a $C^*$-algebra embedding. Our noise model for the transmission is a phase damping channel which admits a noiseless or decoherence-free subspace or subsystem. More precisely, the transmission channel is obtained from convex combinations of a set of lowest rank yes/no measurements that leave a component of the messenger state unchanged. The objective of our encoding is to distribute quantum information optimally across the noise-susceptible component of the transmission when the noiseless component is not large enough to contain all the quantum information to be transmitted. We derive simple geometric conditions for optimal encoding and construct examples.
The central issue in this article is to transmit a quantum state in such a way that after some decoherence occurs, most of the information can be restored by a suitable decoding operation. For this purpose, we incorporate redundancy by mapping a given initial quantum state to a messenger state on a larger-dimensional Hilbert space via a C∗-algebra embedding. Our noise model for the transmission is a phase damping channel which admits a noiseless or decoherence-free subspace or subsystem. More precisely, the transmission channel is obtained from convex combinations of a set of lowest rank yes/no measurements that leave a component of the messenger state unchanged. The objective of our encoding is to distribute quantum information optimally across the noise-susceptible component of the transmission when the noiseless component is not large enough to contain all the quantum information to be transmitted. We derive simple geometric conditions for optimal encoding and construct examples.
Decoherence-Insensitive Quantum Communication by Optimal C∗-Encoding
The central issue in this paper is to transmit a quantum state in such a way that after some decoherence occurs, most of the information can be restored by a suitable decoding operation. For this purpose, we incorporate redundancy by mapping a given initial quantum state to a messenger state on a larger dimensional Hilbert space via a C* -algebra embedding. Our noise model for the transmission is a phase damping channel which admits a noiseless subsystem or decoherence-free subspace. More precisely, the transmission channel is obtained from convex combinations of a set of lowest rank yes/no measurements that leave a component of the messenger state unchanged. The objective of our encoding is to distribute quantum information optimally across the noise-susceptible component of the transmission when the noiseless component is not large enough to contain all the quantum information to be transmitted. We derive simple geometric conditions for optimal encoding and construct examples of such encodings.
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) 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|.
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) 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∣.
Quantum information transmission is subject to imperfections in communication processes and systems. These phenomena alter the original content due to decoherence and noise. However, suitable communication architectures incorporating quantum and classical redundancy can selectively remove these errors, boosting destructive interference. In this work, a selection of architectures based on path superposition or indefinite causal order were analyzed under appropriate configurations, alongside traditional methods such as classical redundancy, thus enhancing transmission. For that purpose, we examined a broad family of decoherent channels associated with the qubit chain transmission by passing through tailored arrangements or composite architectures of imperfect channels. The outcomes demonstrated that, when combined with traditional redundancy, these configurations could significantly improve the transmission across a substantial subset of the channels. For quantum key distribution purposes, two alternative bases were considered to encode the information chain. Because a control system must be introduced in the proposed architectures, two strategies for its disposal at the end of the communication process were compared: tracing and measurement. In addition, eavesdropping was also explored under a representative scenario, to quantify its impact on the most promising architecture analyzed. Thus, in terms of transmission quality and security, the analysis revealed significant advantages over direct transmission schemes.
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.
Near-term quantum communication protocols suffer inevitably from channel noises, whose alleviation has been mostly attempted with resources such as multiparty entanglement or sophisticated experimental techniques. Generation of multiparty higher dimensional entanglement is not easy. This calls for exploring realistic solutions which are implementable with current devices. Motivated particularly by the difficulty in generation of multiparty entangled states, in this paper, we have investigated error-free information transfer with minimal requirements. For this, we have proposed a new information encoding scheme for communication purposes. The encoding scheme is based on the fact that most noisy channels leave some quantities invariant. Armed with this fact, we encode information in these invariants. These invariants are functions of expectation values of operators. This information passes through the noisy channel unchanged. Pertinently, this approach is not in conflict with other existing error correction schemes. In fact, we have shown how standard quantum error-correcting codes emerge if suitable restrictions are imposed on the choices of logical basis states. As applications, for illustration, we propose a quantum key distribution protocol and an error-immune information transfer protocol.