The Holevo bound is a bound on the mutual information for a given quantum encoding. In 1996 Schumacher, Westmoreland, and Wootters [Phys. Rev. Lett.76, 3452 (1996)] derived a bound that reduces to the Holevo bound for complete measurements, but that is tighter for incomplete measurements. The most general quantum operations may be both incomplete and inefficient. Here we show that the bound derived by SWW can be further extended to obtain one that is yet again tighter for inefficient measurements. This allows us, in addition, to obtain a generalization of a bound derived by Hall, and to show that the average reduction in the von Neumann entropy during a quantum operation is concave in the initial state, for all quantum operations. This is a quantum version of the concavity of the mutual information. We also show that both this average entropy reduction and the mutual information for pure state ensembles, are Schur concave for unitarily covariant measurements; that is, for these measurements, information gain increases with initial uncertainty.

1.
J. P.
Gordon
in
Quantum Electronics and Coherent Light
,
Proceedings of the International School of Physics
“Enrico Fermi” XXX1, edited by
P. A.
Miles
(
Academic
, New York,
1964
).
2.
L. B.
Levitin
, in
Proceedings of the All-Union Conference on Information Complexity and Control in Quantum Physics
, (
Mockva-Tashkent
, Tashkent,
1969
), Sec. II (in Russian);
L. B.
Levitin
, in
Information Complexity and Control in Quantum Physics
, edited by
A.
Blaquieve
,
S.
Diner
, and
G.
Lochak
(
Springer-Verlag
, New York,
1987
), pp.
15
47
.
3.
A. S.
Holevo
,
Probl. Peredachi Inf.
9
,
3
(
1973
)
A. S.
Holevo
, [
Probl. Inf. Transm.
(USSR)
9
,
177
(
1973
)].
4.
H. P.
Yuen
and
M.
Ozawa
,
Phys. Rev. Lett.
70
,
363
(
1993
).
5.
C. A.
Fuchs
and
C. M.
Caves
,
Phys. Rev. Lett.
73
,
3047
(
1994
).
6.
B.
Schumacher
,
M.
Westmoreland
, and
W. K.
Wootters
,
Phys. Rev. Lett.
76
,
3452
(
1996
).
7.

While it is an abuse of notation to denote the ensemble probabilities by P(i), and the (in general, unrelated) outcome probabilities by P(j), we use it systematically throughout, since we feel it keeps the notation simpler, and thus ultimately clearer.

8.
H.
Carmichael
,
An Open Systems Approach to Quantum Optics
(
Springer-Verlag
, Berlin,
1993
);
H. M.
Wiseman
and
G. J.
Milburn
,
Phys. Rev. A
47
,
642
(
1993
).
[PubMed]
9.

If the observer has only partial information about the outcome of a measurement, then if we label the outcomes by n (with associated measurement operators Bn), the most general situation is one in which the observer knows instead the value of a second variable m, where m is related to n by an arbitrary conditional probability P(mn). This general case is encompassed by the two-index formulation we use in the text. To see this, one sets k=n, j=m and chooses Anm(Akj)=αnmBn. Then by giving the observer complete knowledge of j, and no knowledge of k, we reproduce precisely the general case described above by choosing αnm so that αnm2=P(mn).

10.

The von Neumann entropy is not the only quantity that can be used to measure the achieved level of control. The von Neumann entropy specifically gives the minimum possible entropy of the results of a measurement on the system. It therefore measures the maximum information (strictly, the minimum information deficit) that the user who is performing the control has about of the future behavior of the system under measurement. An example of another measure of control is the maximum eigenvalue of the density matrix. Under the assumption that all unitary operations are available to the controller, this measures the probability that the controlled system will be found in the desired state.

11.
A.
Doherty
,
K.
Jacobs
, and
G.
Jungman
,
Phys. Rev. A
63
,
062306
(
2001
).
12.
M. J. W.
Hall
,
Phys. Rev. A
55
,
100
(
1997
).
13.
K.
Jacobs
,
Phys. Rev. A
68
,
054302
(BR) (
2003
).
14.
15.
K.
Kraus
,
States, Effects and Operations: Fundamental Notions of Quantum Theory
,
Lecture Notes in Physics
Vol.
190
(
Springer-Verlag
, Berlin,
1983
).
E. H.
Lieb
and
M. B.
Ruskai
,
Phys. Rev. Lett.
30
,
434
(
1973
);
E. H.
Lieb
and
M. B.
Ruskai
,
J. Math. Phys.
14
,
1938
(
1973
);
In addition, a much simpler proof of strong subadditivity has been obtained by Petz [
Rep. Math. Phys.
23
,
57
(
1986
)] and is described in
M. A.
Nielsen
and
D.
Petz
, e-print: quant-ph/0408130.
17.
M.
Ozawa
,
J. Math. Phys.
27
,
759
(
1986
).
18.
M. A.
Nielsen
,
Phys. Rev. A
63
,
022114
(
2001
).
19.
C. A.
Fuchs
and
K.
Jacobs
,
Phys. Rev. A
63
,
062305
(
2001
).
20.
A discussion of this point may be found in
K.
Jacobs
,
Quantum Inf. Process.
1
,
73
(
2002
).
21.
H.
Barnum
, “
Information-disturbance tradeoff in quantum measurement on the uniform ensemble and on the mutually unbiased bases
,” e-print: quant-ph/0205155.
22.
G.
Cassinelli
,
E.
De Vito
,
A.
Toigo
, “
Positive operator valued measures covariant with respect to an irreducible representation
,” e-print: quant-ph/0302187.
23.
A. W.
Marshall
and
I.
Olkin
,
Inequalities: Theory of Majorization and Its Applications
(
Academic
, New York,
1979
).
24.
R.
Bhatia
,
Matrix Inequalities
(
Springer-Verlag
, Berlin,
1997
).
25.
M. A.
Nielsen
,
Phys. Rev. Lett.
83
,
436
(
1999
).
26.
D.
Jonathan
and
M. B.
Plenio
,
Phys. Rev. Lett.
83
,
1455
(
1999
);
D.
Jonathan
and
M. B.
Plenio
,
Phys. Rev. Lett.
83
,
3566
(
1999
);
27.
M. A.
Nielsen
and
J.
Kempe
,
Phys. Rev. Lett.
86
,
5184
(
2001
).
28.
A.
Chefles
,
Phys. Rev. A
65
,
052314
(
2002
).
29.
30.
G. M.
D'Ariano
,
P.
Lo Presti
, and
P.
Perinotti
, “
Classical randomness in quantum measurements
,” e-print: quant-ph/0408115.
You do not currently have access to this content.