Dividing Quantum Channels

By Michael M. Wolf, J. I. Cirac. 2006

297 citations recorded by the source corpus.

We investigate the possibility of dividing quantum channels into concatenations of other channels, thereby studying the semigroup structure of the set of completely-positive trace-preserving maps. We show the existence of ‘indivisible’ channels which can not be written as non-trivial products of other channels and study the set of ‘infinitesimal divisible’ channels which are elements of continuous completely positive evolutions. For qubit channels we obtain a complete characterization of the sets of indivisible and infinitesimal divisible channels. Moreover, we identify those channels which are solutions of time-dependent master equations for both positive and completely positive evolutions. For arbitrary finite dimension we prove a representation theorem for elements of continuous completely positive evolutions based on new results on determinants of quantum channels and Markovian approximations.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/0bd4e105fb4e18ab74c88a52b58bdad786127a30

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Li Li
2. Norbert Schuch
3. Thomas Schulte-Herbrüggen
4. Michael J. W. Hall
5. David Pérez-García
6. Howard M. Wiseman
7. I. Cirac
8. Frank Verstraete

In Atomic and Molecular Physics, and Optics

1. Norbert Schuch
2. Frank Verstraete
3. I. Cirac
4. David Pérez-García
5. Simone Montangero
6. Thomas Schulte-Herbrüggen
7. K. Kawabata
8. Michael J. W. Hall

Citations in the corpus

• A Postquantum Theory of Classical Gravity? (2018, unknown): “…for quantum systems, a linear map E ð ˆ σ Þ which is completely positive trace preserving at every time t (we say it is infinitesimally divisible [122]) is always generated by the GKSL or Lindblad equation [108,109] with ˆ H ð t Þ the Hamiltonian, the set f ˆ L α g a traceless basis of operators…”
• Matrix product states and projected entangled pair states: Concepts, symmetries, theorems (2020, background): “However, this is subtle as there exist quantum channels T that cannot be divided (Wolf and Cirac, 2008), in the sense that there does not exist any other quantum channel R so that R2 = T .”
• Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe (2022, background): “corresponding decision problem (encompassing both the time-independent and timedependent notion via infinitesimal cp-divisibility [622] on all time scales) is np hard [152, 153].”
• The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement (2012, background): “Excluding such uninteresting compositions we can define the concept of channel divisibility, originally introduced and studied by Wolf and Cirac [145].”
• Concepts of quantum non-Markovianity: A hierarchy (2017, background): “Wolf and Cirac originally considered only a particular subclass of divisible dynamical maps to be ‘Markovian’ [110] (see also Section 3.”
• Topological Phases of Non-Hermitian Systems (2018, background): “With the TP property imposed, the eigenvalues of E are enforced to be on or inside the unit circle in the complex plane [155].”
• Quantum channels and memory effects (2012, background): “According to (Wolf and Cirac, 2008) a quantum channel is called Markovian if it is an element of a completely positive continuous one-parameter semigroup.”