Transition behavior in the channel capacity of two-quibit channels with memory

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 69, Issue 1, 2004, Pages 103031-103034

  Macchiavello, Chiara ^a   Palma, G Massimo ^b   Virmani, S ^c,d  

^a INFM   (Italy)

^b UNIVERSITY OF MILAN   (Italy)

^c IMPERIAL COLLEGE LONDON   (United Kingdom)

^d UNIVERSITY OF HERTFORDSHIRE   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; ENTROPY; INFORMATION THEORY; MATHEMATICAL OPERATORS; MATRIX ALGEBRA; OPTIMIZATION; PROBABILITY DISTRIBUTIONS; QUANTUM THEORY; ROTATION; SET THEORY; SPURIOUS SIGNAL NOISE; TENSORS;

PAULI CHANNELS; QUANTUM CHANNELS;

CHANNEL CAPACITY;

EID: 18444394082     PISSN: 10502947     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (19)

1. C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, New York, 1976).
2. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).
3. C. Macchiavello, G. M. Palma, and A. Zeilinger, Quantum Computation and Quantum Information Theory (World Scientific, Singapore, 2001).
4. D. Bruß, L. Faoro, C. Macchiavello, and G.M. Palma, J. Mod. Opt. 47, 325 (2000).
5. C. King and M.B. Ruskai, IEEE Trans. Inf. Theory 47, 192 (2001); C. King, e-print quant-ph/0103156.
6. C. King and M.B. Ruskai, IEEE Trans. Inf. Theory 47, 192 (2001); C. King, e-print quant-ph/0103156.
7. K. Matsumoto, T. Shimono, and A. Winter, e-print quant-ph/0206148; K.M.R. Audenaert and S.L. Braunstein, e-print quant-ph/0303045; P.W. Shor, e-print quant-ph/0305035.
8. K. Matsumoto, T. Shimono, and A. Winter, e-print quant-ph/0206148; K.M.R. Audenaert and S.L. Braunstein, e-print quant-ph/0303045; P.W. Shor, e-print quant-ph/0305035.
9. K. Matsumoto, T. Shimono, and A. Winter, e-print quant-ph/0206148; K.M.R. Audenaert and S.L. Braunstein, e-print quant-ph/0303045; P.W. Shor, e-print quant-ph/0305035.
10. C. Macchiavello and G.M. Palma, Phys. Rev. A 65, 050301 (R) (2002).
11. G. Bowen and S. Mancini, e-print quant-ph/0305010.
12. B. Schumacher and M.D. Westmoreland, Phys. Rev. A 56, 131 (1997); A.S. Holevo, IEEE Trans. Inf. Theory 44, 269 (1998).
13. B. Schumacher and M.D. Westmoreland, Phys. Rev. A 56, 131 (1997); A.S. Holevo, IEEE Trans. Inf. Theory 44, 269 (1998).
14. J. Cortese, e-print quant-ph/0211093; A.S. Holevo, e-print quant-ph/0212025.
15. J. Cortese, e-print quant-ph/0211093; A.S. Holevo, e-print quant-ph/0212025.
16. This is a consequence of Schur's Lemma; see, for example, C. J. Isham, Lectures on Groups and Vector Spaces for Physicists (World Scientific, Singapore, 1989).
17. 1). To ensure this we have picked the probabilities such that any elements drawn from the same left coset of the subgroup have identical weight. This is why the channel that we consider is invariant under preoperating with the "averaging over the subgroup" operation, and the same reasoning also applies to any subgroup/channel combination such that the weights are constant over each left coset.
18. It might be argued that our derivation does not strictly imply that entanglement is necessary to achieve the capacity above the threshold, as our solution is not necessarily unique, and so it might be possible to achieve capacity using alternative ensembles of separable states. To show that this is not possible, we must show that there is no input separable state that gives minimal output entropy. As the preoperation F is not entangling, it is sufficient to consider input separable states of the form (16). Moreover, we have already seen that product pure states of this form, such as |01〉, etc., do not have minimal output entropy, so we should only be concerned with mixed separable states with the structure (16). It is not difficult (although slightly tedious) to verify that each of optimal input Bell states derived for memories above the threshold is taken to a distinct output state by the channel. Therefore, by the strict concavity of the entropy, no mixed separable state of the form (16) can have minimal output entropy either.
19. C. Macchiavello, G. M. Palma, and S. Virmani (unpublished).