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).
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).
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).
Phase covariant qubit dynamics describes an evolution of a two-level system under simultaneous action of pure dephasing, energy dissipation, and energy gain with time-dependent rates γz(t) , γ−(t) , and γ+(t) , respectively. Non-negative rates correspond to completely positive divisible dynamics, which can still exhibit such peculiarities as non-monotonicity of populations for any initial state. We find a set of quantum channels attainable in the completely positive divisible phase covariant dynamics and show that this set coincides with the set of channels attainable in semigroup phase covariant dynamics. We also construct new examples of eternally indivisible dynamics with γz(t)<0 for all t>0 that is neither unital nor commutative. Using the quantum Sinkhorn theorem, we for the first time derive a restriction on the decoherence rates under which the dynamics is positive divisible, namely, γ±(t)≥0 , γ+(t)γ−(t)+2γz(t)>0 . Finally, we consider phase covariant convolution master equations and find a class of admissible memory kernels that guarantee complete positivity of the dynamical map.
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.
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.
The time evolution of an initially uncorrelated system is governed by a completely positive (CP) map. More generally, the system may contain initial (quantum) correlations with an environment, in which case the system evolves according to a not-completely positive (NCP) map. It is an interesting question what the relative measure is for these two types of maps within the set of positive maps. After indicating the scope of the full problem of computing the true volume for generic maps acting on a qubit, we study the case of Pauli channels in an abstract space whose elements represent an equivalence class of maps that are identical up to a non-Pauli unitary. In this space, we show that the volume of NCP maps is twice that of CP maps.
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.