ISSN 0032-9460, Problems of Information Transmission, 2013, Vol. 49, No. 3, pp. 224–231.
Pleiades Publishing, Inc., 2013.
Original Russian Text
G.G. Amosov, 2013, published in Problemy Peredachi Informatsii, 2013, Vol. 49, No. 3, pp. 32–39.
On Estimating the Output Entropy of the Tensor
Product of a Phase-Damping Channel
and an Arbitrary Channel
G. G. Amosov
Steklov Mathematical Institute, Russian Academy of Sciences, Moscow
Received January 25, 2013; in ﬁnal form, March 6, 2013
Abstract—We obtain a lower estimate for the output entropy of a tensor product of the quan-
tum phase-damping channel and an arbitrary channel. We show that this estimate immediately
implies that strong superadditivity of the output entropy holds for this channel as well as for
the quantum depolarizing channel.
Let S(H) denote the set of all states, i.e., positive unit-trace operators, in a Hilbert space H,
dim H<+∞. By a quantum channel we mean a completely positive trace-preserving linear map
Φ: S(H) → S(K). A quantum channel Φ is said to be unital if Φ
and in what follows we denote by I
the identity operator in a Hilbert space L.
(Φ) = min
where S(ρ)=− Tr(ρ log ρ) is the von Neumann entropy of a state ρ.Inthequantity
was introduced. We shall say that strong superadditivity of the output entropy of the channel
Φ: S(H) → S(H)holdsif
for any quantum channel Ω: S(K) → S(K) and any states ρ ∈ S(H ⊗ K). In particular, if the
strong superadditivity holds for the channel Φ, then the minimal output entropy is additive with
respect to tensor product of channels , i.e.,
(Φ ⊗ Ω) = S
(Φ) + S
is satisﬁed for all quantum channels Ω. Unfortunately, additivity of the minimal output entropy (2)
does not hold in general . Nevertheless, it was proved for many signiﬁcant cases [3–6]. The strong
superadditivity holds for noiseless channels and for entanglement-breaking channels .
Supported in part by the Presidium of the Russian Academy of Sciences, Fundamental Research Program
“Dynamical Systems and Control Theory,” 2013, and the Russian Foundation for Basic Research, project
nos. 12-01-00319-a and 11-02-00456-a.