Every completely positive map G that commutes which the Hamiltonian time evolution is an integral or sum over CP-maps G_\\sigma where \\sigma is the energy that is transferred to or taken from the environment. If the spectrum is non-degenerated each G_\\sigma is an energy shift followed by a dephasing channel. The Kraus operator of the energy shift is a partial isometry which defines a translation on R with respect to a non-translation-invariant measure. The dephasing is given by the Hadamard product of the density operator with a positive operator. As an example, I calculate this decomposition explicitly for the gaussian channel on a single mode. For a special type of channels, a lower bound on the quantum capacity is derived using the Fourier transform of the CP-map-valued measure (G_\\sigma).
Every completely positive map G that commutes which the Hamiltonian time evolution is an integral or sum over CP-maps G_\sigma where \sigma is the energy that is transferred to or taken from the environment. If the spectrum is non-degenerated each G_\sigma is an energy shift followed by a dephasing channel. The Kraus operator of the energy shift is a partial isometry which defines a translation on R with respect to a non-translation-invariant measure. The dephasing is given by the Hadamard product of the density operator with a positive operator. As an example, I calculate this decomposition explicitly for the gaussian channel on a single mode. For a special type of channels, a lower bound on the quantum capacity is derived using the Fourier transform of the CP-map-valued measure (G_\sigma).
Every completely positive map G that commutes with the Hamiltonian time evolution is an integral or sum over (densely defined) CP-maps Gσ where σ is the energy that is transferred to or taken from the environment. If the spectrum is non-degenerate each Gσ is a dephasing channel followed by an energy shift. The dephasing is given by the Hadamard product of the density operator with a (formally defined) positive operator. The Kraus operator of the energy shift is a partial isometry which defines a translation on R with respect to a non-translation-invariant measure. As an example, this decomposition is explicitly calculated for the rotation invariant Gaussian channel on a single mode. The question of under what conditions a covariant channel destroys superpositions between mutually orthogonal states on the same orbit is addressed. For channels which allow mutually orthogonal output states on the same orbit, a lower bound on the quantum capacity is derived using the Fourier transform of the CP-map-valued measure (Gσ).
We study the reduced time-evolution of general open quantum systems by combining insights from quantum-information and statistical field theory. Inspired by prior work [Eur. Phys. Lett.~102, 60001 (2013) and Phys. Rev. Lett.~111, 050402 (2013)] we establish the explicit structure guaranteeing the complete positivity (CP) and trace-preservation (TP) of the real-time evolution expansion in terms of the microscopic system-environment coupling. This reveals a fundamental two-stage structure of the coupling expansion: Whereas the first stage naturally defines the dissipative timescales of the system -before having integrated out the environment completely- the second stage sums up elementary physical processes, each described by a CP superoperator. This allows us to establish the highly nontrivial functional relation between the (Nakajima-Zwanzig) memory-kernel superoperator for the reduced density operator and novel memory-kernel operators that generate the Kraus operators of an operator-sum. We illustrate the physically different roles of the two emerging coupling-expansion parameters for a simple solvable model. Importantly, this operational approach can be implemented in the existing Keldysh real-time technique and allows approximations for general time-nonlocal quantum master equations to be systematically compared and developed while keeping the CP and TP structure explicit. Our considerations build on the result that a Kraus operator for a physical measurement process on the environment can be obtained by `cutting' a group of Keldysh real-time diagrams `in half'. This naturally leads to Kraus operators lifted to the system plus environment which have a diagrammatic expansion in terms of time-nonlocal memory-kernel operators. These lifted Kraus operators obey coupled time-evolution equations which constitute an unraveling of the original Schroedinger equation for system plus environment. Whereas both equations lead to the same reduced dynamics, only the former explicitly encodes the operator-sum structure of the coupling expansion.
Quantum systems which interact with an unknown environment cannot be described in terms of a unitary evolution on the system alone. For such evolution one can use a map from one density operator to another and use any other known information to model the system. Such maps are required to be positive (at least on their domain) — they take positive density operators to positive density operators — so as to be physically reasonable. A map [Formula: see text] which is positive but not completely positive (CP) is one which takes positive operators to positive operators, but when extended by the identity [Formula: see text], i.e. [Formula: see text] does not give a positive map for some [Formula: see text]. The map is CP if and only if the extension is positive for all [Formula: see text]. Recently some effort has been put forth to try to understand if, and/or, under what circumstances one might utilize a non-CP map. Here, after a tutorial-type introduction to maps from one density operator to another, some examples which do not fit the standard prescription (SP) for deriving a CP map are given. In cases where the physical system is constrained by some knowledge of the interaction, it is possible that the SP cannot be used directly. Our example is robust to changes in the initial and final conditions.
Abstract In this chapter we visit the ‘church of the larger Hilbert space’: we take a closer look at the relationship of pure and mixed quantum states via purification, and discuss a similar connection between completely positive and trace-preserving (CPTP) maps and unitary dynamics on a larger Hilbert space. We show that such CPTP maps, which represent quantum channels, are equivalent to operations admitting a Kraus decomposition, as well as to unitary dynamics on a larger Hilbert space with factoring initial condition, the famous Stinespring dilation of quantum operations. In this context we introduce the Choi-Jamiołkowski isomorphism, which expresses a duality between maps between two Hilbert spaces and states on tensor products of these Hilbert spaces, and we review several important examples of quantum channels such as the dephasing channel, amplitude-damping channel, and depolarizing channel. Finally, we examine operations that are not possible in quantum theory, focusing on the no-cloning theorem.
Completely positive maps are useful in modeling the discrete evolution of quantum systems. Spectral properties of operators associated with such maps are relevant for determining the asymptotic dynamics of quantum systems subjected to multiple interactions described by the same quantum channel. We discuss a connection between the properties of the peripheral spectrum of completely positive and trace preserving map and the algebra generated by its Kraus operators A(A1,…,AK). By applying the Shemesh and Amitsur-Levitzki theorems to analyse the structure of the algebra A(A1,…,AK), one can predict the asymptotic dynamics for a class of operations.
This paper develops the open-system companion to the unitary BV-clock quantum dynamics paper. We consider density matrices evolving over a finite-horizon sigma clock with bounded Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) continuous generators and locally finite completely positive trace-preserving (CPTP) atom maps. Under bounded measurable coefficients with integrable generator norm, the sigma-chronological product exists as a CPTP channel evolution, composes through admissible cuts, preserves trace, positivity, and complete positivity, and inserts atom maps in clock order. We also prove that averaging over a probability distribution of admissible sigma clocks preserves CPTP structure by convexity. This random-clock result is deliberately not stated as a semigroup or divisibility theorem: mixtures of admissible channels need not inherit Markovian divisibility without additional hypotheses. A finite open-qubit benchmark checks a GKSL flow with amplitude damping, excitation, dephasing, reset/depolarizing/bit-flip atoms, and random timing averages using trace, Hermiticity, state eigenvalue, Choi eigenvalue, and trace-preservation ledgers.