In this paper, we study the influence of nonabelian superselection rules on the transfer of quantum information with the help of qubits on the base of an algebraic model and formulate quantum protocols. We pay the main attention to the superselection structure of the algebra of observables [Formula: see text] defined by the Cuntz algebra [Formula: see text] (a field algebra) that contains [Formula: see text] as a pointwise fixed subalgebra with respect to the action of the gauge group [Formula: see text]. We prove that it is possible to code information only with the help of states such that projectors on them belong to the algebra of observables. These projectors commute with the elements of the representation of the group [Formula: see text], and therefore allow the recipient to restore the obtained information.
In this paper, we study the influence of nonabelian superselection rules on the transfer of quantum information with the help of qubits on the base of an algebraic model and formulate quantum protocols. We pay the main attention to the superselection structure of the algebra of observables [Formula: see text] defined by the Cuntz algebra [Formula: see text] (a field algebra) that contains [Formula: see text] as a pointwise fixed subalgebra with respect to the action of the gauge group [Formula: see text]. We prove that it is possible to code information only with the help of states such that projectors on them belong to the algebra of observables. These projectors commute with the elements of the representation of the group [Formula: see text], and therefore allow the recipient to restore the obtained information.
In this paper, we consider an extension of the previously proposed algebraic model and study the constraints of non-Abelian superselection rules on the transfer quantum information, taking into account conjugate endomorphism. The procedure of averaging (over the group G=SU(3) ) projectors to the basic states of coherent orthogonal subspaces into which the space of two three-level systems decomposes is considered. Main attention is paid to the superselection structure of the algebra of observables 0OG defined by the Cuntz algebra 0Od=3 (field algebra) containing 0OG as a pointwise fixed subalgebra with respect to the action of the gauge group G . As an application of the model, we consider the encoding of information using a three-level system and show that information can be transmitted only by those states whose projectors belong to the algebra of observables. These projectors commute with the elements of the representation of the group G , and therefore, allow the recipient to restore the obtained information.
We show that superselection rules do not enhance the information-theoretic security of quantum cryptographic protocols. Our analysis employs two quite different methods. The first method uses the concept of a reference system---in a world subject to a superselection rule, unrestricted operations can be simulated by parties who share access to a reference system with suitable properties. By this method, we prove that if an n-party protocol is secure in a world subject to a superselection rule, then the security is maintained even if the superselection rule is relaxed. However, the proof applies only to a limited class of superselection rules, those in which the superselection sectors are labeled by unitary irreducible representations of a compact symmetry group. The second method uses the concept of the format of a message sent between parties---by verifying the format, the recipient of a message can check whether the message could have been sent by a party who performed charge-conserving operations. By this method, we prove that protocols subject to general superselection rules (including those pertaining to non-Abelian anyons in two dimensions) are no more secure than protocols in the unrestricted world. However, the proof applies only to two-party protocols. Our results show in particular that, if no assumptions are made about the computational power of the cheater, then secure quantum bit commitment and strong quantum coin flipping with arbitrarily small bias are impossible in a world subject to superselection rules.
From the logic approach to quantum and classical mechanics, the W*−algebraic approach is deduced in dependence of a suitable ’’prestate.’’ An algebraic representation of the logic description is in fact constructed in a framework in which continuous superselection rules can be present. Logic propositions, observables, and states are represented by decomposable projections, decomposable self−adjoint operators, and normal states in a direct integral of Hilbert spaces. In this representation each algebraic term becomes the representative of a homologous logic one and the expectation values as well as the superselection rules are conserved. When a principle of ’’undistinguishability’’ is taken into account, the representation is faithful. In the classical case, the representation results in Koopman’s formalism.
Quantization is defined as the act of assigning an appropriate C*-algebra [Formula: see text] to a given configuration space Q, along with a prescription mapping self-adjoint elements of [Formula: see text] into physically interpretable observables. This procedure is adopted to solve the problem of quantizing a particle moving on a homogeneous locally compact configuration space Q=G/H. Here [Formula: see text] is chosen to be the transformation group C*-algebra corresponding to the canonical action of G on Q. The structure of these algebras and their representations are examined in some detail. Inequivalent quantizations are identified with inequivalent irreducible representations of the C*-algebra corresponding to the system, hence with its superselection sectors. Introducing the concept of a pre-Hamiltonian, we construct a large class of G-invariant time-evolutions on these algebras, and find the Hamiltonians implementing these time-evolutions in each irreducible representation of [Formula: see text]. “Topological” terms in the Hamiltonian (or the corresponding action) turn out to be representation-dependent, and are automatically induced by the quantization procedure. Known “topological” charge quantization or periodicity conditions are then identically satisfied as a consequence of the representation theory of [Formula: see text].
Abstract The operator algebra of fermionic modes is isomorphic to that of qubits, the difference between them is twofold: the embedding of subalgebras corresponding to mode subsets and multiqubit subsystems on the one hand, and the parity superselection in the fermionic case on the other. We discuss these two fundamental differences extensively, and illustrate these through the Jordan–Wigner representation in a coherent, self-contained, pedagogical way, from the point of view of quantum information theory. Our perspective leads us to develop useful new tools for the treatment of fermionic systems, such as the fermionic (quasi-)tensor product, fermionic canonical embedding, fermionic partial trace, fermionic products of maps and fermionic embeddings of maps. We formulate these by direct, easily applicable formulas, without mode permutations, for arbitrary partitionings of the modes. It is also shown that fermionic reduced states can be calculated by the fermionic partial trace, containing the proper phase factors. We also consider variants of the notions of fermionic mode correlation and entanglement, which can be endowed with the usual, local operation based motivation, if the parity superselection rule is imposed. We also elucidate some other fundamental points, related to joint map extensions, which make the parity superselection inevitable in the description of fermionic systems.
On One Conjugate Object in the Symmetric Tensor C∗-Category and the Statistics of Superselection Sectors
The paper presents a model of symmetric tensor C∗ -category with conjugation for an object of dimension d=3 . It is proved that the constructed conjugate object of the category satisfies the conjugation equations, and different classes of morphisms between the objects in the modelled category are considered and studied. This category allows for the generalization of the C∗ -algebraic model of observables in the presence of superselection rules generated by non-Abelian conjugate charges. As an application of the model, the area of quantum information transfer is explored, where the constraints imposed by the superselection rules need to be taken into account.