M. Keyl. Fundamentals of quantum information theory. to appear in Phys. Rep., quant-ph/0202122 (2001).  Home/Search   Document Details and Download   Summary   Related Articles   Check

....measurement, filtering, and quantum feedback control (V.P. Belavkin, organizer) at the International Conference on Physics and Control (PhysCon 2003) August 20 22, St. Petersburg, Russia. Electronic Mail: maxim ece.northwestern.edu 1 Introduction and background In quantum information theory [1] all admissible devices are described mathematically by means of the so called quantum operations (or quantum channels) 2, 3] Given a complex Hilbert space H, the # algebra of all bounded operators on H. In this paper we will work primarily with finite dimensional Hilbert spaces, so ....

M. Keyl, "Fundamentals of quantum information theory," Phys. Rep. 369, 431--548 (2002).

....46L60, 47L07 PACS Classi cation (2003) 02.30.Tb, 03.67. a Keywords: completely positive maps, quantum operations, quantum channels, noncommutative RadonNikodym theorem Electronic Mail: maxim ece.northwestern.edu 1 Introduction In the mathematical framework of quantum information theory [18], all admissible devices are modelled by the so called quantum operations [9, 20] that is, completely positive linear contractions on the algebra of observables of the physical system under consideration. Thus it is of paramount importance to have at one s disposal a good analysis toolkit for ....

....one says that the transformation 7 T ( succeeds with probability Tr T ( this probability is equal to unity for all density operators if and only if T is unital, i.e. T (1) 1, so that T is trace preserving. Unital quantum operations are also referred to as quantum channels [18]. The Kraus theorem implies that we can write any quantum operation T as a sum of pure operations [9, Sect. 2.3] i.e. maps of the form A 7 X AX with X X 1 (this is equivalent to X being a contraction, kXk 1 where k k is the usual operator norm, kXk = sup 2H kX k=k k) The quali ....

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M. Keyl, \Fundamentals of quantum information theory," Phys. Rep. 369(5), 431 (2002).

....we have Q(Id) log 2 (d) for the ideal channel. A precise proof of this statement is, however, not so easy as it looks like and we skip the details here. Maybe the most easy approach is to use the quantity log 2 (k Tk cb ) where denotes the transposition) which is an upper bound on Q(T ) cf. [9] or [22] The same idea can be used to show that the quantum capacity of a classical channel, or more generally a channel T which uses classical information at an intermediate step, is zero. This is a reformulation of the no classical teleportation theorem (cf. again [22] Another useful ....

....1 ) Q(T 2 )g: 10) Alternatively we can use the two channels in parallel, i.e. we consider the tensor product T T 2 . In this case the capacity of the resulting channel is at least as big as the sum of Q(T 1 ) and Q(T 2 ) i.e. Q is superadditive: Q(T T 2 ) Q(T 1 ) Q(T 2 ) 11) cf. [9] for a proof of both statements) To decide whether Q is even additive, i.e. whether equality holds in (11) is another big open question about channel capacities. 4 Quantum error correction The de nition of capacity requires that we correct errors in a collection of n parallel channels . ....

M. Keyl. Fundamentals of quantum information theory. quant-ph/0202122 (2001).

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M. Keyl. Fundamentals of quantum information theory. to appear in Phys. Rep., quant-ph/0202122 (2001).

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