ISSN 0032-9460, Problems of Information Transmission, 2007, Vol. 43, No. 1, pp. 1–11.
Pleiades Publishing, Inc., 2007.
Original Russian Text
A.S. Holevo, 2007, published in Problemy Peredachi Informatsii, 2007, Vol. 43, No. 1, pp. 3–14.
One-Mode Quantum Gaussian Channels:
Structure and Quantum Capacity
A. S. Holevo
Steklov Mathematical Institute, RAS, Moscow
Received October 18, 2006
Abstract—A complete classiﬁcation of one-mode Gaussian channels is given up to canonical
unitary equivalence. We also comment on the quantum capacity of these channels. A channel
complementary to the quantum channel with additive classical Gaussian noise is described,
providing an example of a one-mode Gaussian channel which is neither degradable nor an-
Electromagnetic ﬁeld is a fundamental physical information carrier. Mathematically, radiation
ﬁeld is known to be equivalent to an ensemble of oscillators. In quantum optics one considers
quantized ﬁelds and hence quantum oscillators. This is a typical “continuous-variable” Bosonic
quantum system, whose basic observables (oscillator amplitudes) satisfy the canonical commutation
relations (CCR). Many of the current experimental realizations of quantum information processing
are carried out in such systems.
A mathematical description of linear Bosonic channels was given in [1, 2]. In the present paper,
we consider one-mode Gaussian channels. In the case of “continuous-variable” systems, one-mode
channels play a role similar to qubit channels for ﬁnite systems. Therefore it is interesting and
important to have their full classiﬁcation; this problem was motivated by [3, 4] and is solved in
the present paper. We also comment on the quantum capacity of these channels, following the
discussion of degradable and antidegradable channels in [3, 5]. A channel complementary to the
quantum channel with additive classical Gaussian noise is described, providing an example of a
one-mode Gaussian channel which is neither degradable nor antidegradable, thus giving a negative
answer to a conjecture in .
2. NORMAL FORM OF THE CHANNEL
For a mathematical description of linear Bosonic systems used in this paper, we refer the reader
to [1,2]. A Bosonic system with s degrees of freedom (modes) is described by canonical observables
] in a Hilbert space H satisfying the Heisenberg CCR. Introduce the unitary
Weyl operators V (z)=expiR · z,wherez =[x
] ∈ R
= Z; then these relations are
formally equivalent to the Weyl–Segal CCR
V (z)V (z
)V (z + z
Supported in part by the Russian Foundation for Basic Research, project no. 06-01-00164-a, and the Scien-
tiﬁc Program of the Branch of Mathematics of the RAS “Modern Problems of Theoretical Mathematics.”