Surprisingly often decoherence is due to classical fluctuations of ambient fields and may thus be described in terms of random unitary (RU) dynamics. However, there are decoherence channels where such a representation cannot exist. Based on a simple and intuitive geometric measure for the distance of an extremal channel to the convex set of RU channels we are able to characterize the set of true quantum phase-damping channels. Remarkably, using the Caley-Menger determinant, our measure may be assessed directly from the matrix representation of the channel. We find that the channel of maximum quantumness is closely related to a symmetric, informationally complete positive operator-valued measure on the environment. Our findings are in line with numerical results based on the entanglement of assistance.
Surprisingly often decoherence is due to classical fluctuations of ambient fields and may thus be described in terms of random unitary (RU) dynamics. However, there are decoherence channels where such a representation cannot exist. Based on a simple and intuitive geometric measure for the distance of an extremal channel to the convex set of RU channels we are able to characterize the set of true quantum phase-damping channels. Remarkably, using the Caley-Menger determinant, our measure may be assessed directly from the matrix representation of the channel. We find that the channel of maximum quantumness is closely related to a symmetric, informationally complete positive operator-valued measure on the environment. Our findings are in line with numerical results based on the entanglement of assistance.
It is commonly stated that decoherence in open quantum systems is due to growing entanglement with an environment. In practice, however, surprisingly often decoherence may equally well be described by random unitary dynamics without invoking a quantum environment at all. For a single qubit, for instance, pure decoherence (or phase damping) is always of random unitary type. Here, we construct a simple example of true quantum decoherence of two qubits: we present a feasible phase damping channel of which we show that it cannot be understood in terms of random unitary dynamics. We give a very intuitive geometrical measure for the positive distance of our channel to the convex set of random unitary channels and find remarkable agreement with the so-called Birkhoff defect based on the norm of complete boundedness.
We study pure phase damping of two qubits due to fluctuating fields. As frequently employed, decoherence is thus described in terms of random unitary (RU) dynamics, that is, a convex mixture of unitary transformations. Based on a separation of the dynamics into an average Hamiltonian and a noise channel, we are able to analytically determine the evolution of both entanglement and purity. This enables us to characterize the dynamics in a concurrence-purity (CP) diagram: We find that RU phase-damping dynamics sets constraints on accessible regions in the CP plane. We show that initial state and dynamics contribute to final entanglement independently.
Decoherence, when viewed in the Heisenberg picture, can turn incompatible measurements into jointly measurable ones. This provides an observable-level description of emergent classicality in terms of the incompatibility destroyed by a noise channel. The corresponding loss of incompatibility can be captured by the channel's \emph{compatibility region}---the observables in a probe class that become jointly measurable with every noisy observable. The geometry and volume of this region provide a refined operational way to compare noise channels. We apply this framework to decoherence channels, each of which acts by Schur multiplication with extreme points of the convex set of unit-diagonal positive semidefinite matrices. Restricting to extremals eliminates convex mixing and reveals subtle phase-dependent decoherence effects thar are not captured by damping rates or conventional coherence quantifiers. Such channels have a rigid dilation structure described by a rank-one operator frame, which reduces joint measurability to a positivity test, making the compatibility region analytically tractable, and its volume computable from the frame's Gram matrix. If the frame can be normalised to a minimal informationally complete (MIC) POVM, the compatibility region acquires a geometric interpretation in the associated probability representation of the quantum state space, familiar with Qbism. Among maximal-rank extremals, those associated with symmetric informationally complete (SIC) POVMs maximise the compatibility volume and hence destroy the greatest amount of incompatibility, providing an operational characterisation of the special role of SICs in terms of joint measurability. This leads to a new formulation of the well-known SIC existence problem as a joint measurability question for noisy mutually unbiased bases.
Based on the superposition principle, any two states of a quantum system may be coherently superposed to yield a novel state. Such a simple construction is at the heart of genuinely quantum phenomena such as interference of massive particles or quantum entanglement. Yet, these superpositions are susceptible to environmental influences, eventually leading to a complete disappearance of the system's quantum character. In principle, two distinct mechanisms responsible for this process of decoherence may be identified. In a classical decoherence setting, on the one hand, stochastic fluctuations of classical, ambient fields are the relevant source. This approach leads to a formulation in terms of stochastic Hamiltonians; the dynamics is unitary, yet stochastic. In a quantum decoherence scenario, on the other hand, the system is described in the language of open quantum systems. Here, the environmental degrees of freedom are to be treated quantum mechanically, too. The loss of coherence is then a direct consequence of growing correlations between system and environment. The purpose of the present thesis is to clarify the distinction between classical and quantum decoherence. It is known that there exist decoherence processes that are not reconcilable with the classical approach. We deem it desirable to have a simple, feasible model at hand of which it is known that it cannot be understood in terms of fluctuating fields. Indeed, we find such an example of true quantum decoherence. The calculation of the norm distance to the convex set of classical dynamics allows for a quantitative assessment of the results. In order to incorporate genuine irreversibility, we extend the original toy model by an additional bath. Here, the fragility of the true quantum nature of the dynamics under increasing coupling strength is evident. The geometric character of our findings offers remarkable insights into the geometry of the set of non-classical decoherence maps. We give a very intuitive geometrical measure---a volume---for the quantumness of dynamics. This enables us to identify the decoherence process of maximum quantumness, that is, having maximal distance to the convex set of dynamics consistent with the stochastic, classical approach. In addition, we observe a distinct correlation between the decoherence potential of a given dynamics and its achievable quantumness. In a last step, we study the notion of quantum decoherence in the context of a bipartite system which couples locally to the subsystems' respective environments. A simple argument shows that in the case of a separable environment the resulting dynamics is of classical nature. Based on a realistic experiment, we analyze the impact of entanglement between the local environments on the nature of the dynamics. Interestingly, despite the variety of entangled environmental states scrutinized, no single instance of true quantum decoherence is encountered. In part, the identification of the classical nature relies on numerical schemes. However, for a large class of dynamics, we are able to exclude analytically the true quantum nature.
The loss of the coherence happens due to the interaction between the desired system and its surroundings.Addressing decoherence is one of the main concepts in the study of quantum channels to access their potential for various scale.In this process the information of the system is penetrated into the environment.Therefore, the measurement of the environment could be used as the error correction method.Phase damping channels principally belong to random unitary channels and can be corrected by classical information.This paper presents a method to generate non-random unitary phase damping channels based on the Bloch vectors in two qubits systems.A phase damping channel which consisted of a two qubits system and a single qubit environment was investigated.The results demonstrated that the phase damping channels belong to random unitary dynamics if the three-dimensional tetrahedron volume spanned by the Bloch vectors in R 3 is not zero, or, the same hyperplane in R 3 was not pointed by the Bloch vectors.It is found that the Bell state belongs to random unitary class and can be corrected based on classical information obtained from environmental measurements.
This paper studies the class of stochastic maps, or channels, for which (I⊗Φ)(Γ) is always separable (even for entangled Γ). Such maps are called entanglement breaking, and can always be written in the form Φ(ρ)=∑ k R k Tr F k ρ where each R k is a density matrix and F k >0. If, in addition, Φ is trace-preserving, the {F k } must form a positive operator valued measure (POVM). Some special classes of these maps are considered and other characterizations given. Since the set of entanglement-breaking trace-preserving maps is convex, it can be characterized by its extreme points. The only extreme points of the set of completely positive trace preserving maps which are also entanglement breaking are those known as classical-quantum or CQ. However, for d≥3, the set of entanglement breaking maps has additional extreme points which are not extreme CQ maps.