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Erschienen in:
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2015 | OriginalPaper | Buchkapitel

5. Quantum Decision Theory: Analysis and Optimization

verfasst von : Gianfranco Cariolaro

Erschienen in: Quantum Communications

Verlag: Springer International Publishing

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Abstract

Decision is the heart of any quantum digital communication system and is concerned with the formulation of how a measurement must be taken to argue on the transmitted message in the presence of uncertainties (due to the nature of any quantum measurement). The state of the quantum system is considered as assigned through a constellation of pure states or of density operators, whereas the measurement operators must be found with the goal of achieving the “best decision” (optimization), usually obtained by minimizing the error probability. The chapter, after a mathematical formulation of decision, moves on to optimization, where the guidelines are given by Holevo’s and Kennedy’s theorems. These theorems determine the conditions that must be fulfilled by an optimal system of measurement operators, but do not provide any clue on how to identify it. In any case, the problem of optimization is very difficult, and exact solutions (nonnumerical) are only known in few cases, mainly in the binary systems and in general when the state constellation has a symmetry. The specific symmetry that simplifies detection and optimization is called geometrically uniform symmetry (GUS).

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Vorheriges Kapitel Introduction to Part II: Quantum Communications
Nächstes Kapitel Quantum Decision Theory: Suboptimization
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Fußnoten
1
In Sect. 4.​2 we justified the advantage of dealing with a single symbol instead of a sequence of symbols.
 
2
In practice, the performance of a telecommunication system (classical or quantum) is often expressed in terms of the error probability, but in theoretical formulation it is more convenient to refer to the correct decision probability.
 
3
This point will be clarified in Sect. 5.11, Proposition 5.4. The eigenvectors \(|\eta _i\rangle \) are called measurement vectors because they form the measurement operators as \(Q_i=|\eta _i\rangle \langle \eta _i|\).
 
4
To the author’s knowledge the general expression of the eigenvectors (with \(X\) complex and not equally likely symbols) does not seem to be available in the literature.
 
5
The measurement vectors, previously obtained as eigenvectors and denoted by \(|\eta _i\rangle \), are hereafter denoted by \(|\mu _i\rangle \).
 
6
To express \(\cos \phi \) and \(\sin \phi \) from \(\tan 2\phi \) we use the trigonometric identities
$$ \sin \phi =2^{-1/2}\sqrt{1-1/\sqrt{1+\tan ^2 2\phi }},\qquad \cos \phi =2^{-1/2}\sqrt{1+1/\sqrt{1+\tan ^2 2\phi }} $$
which hold for \(0\le \phi \le \pi /4\). This range of \(\phi \) covers the cases of interest.
 
7
Another way to obtain a factorization is given by Choleski’s decomposition (see Sect. 2.​12.​5).
 
8
The term operator, in practice represented by a square matrix, is reserved to linear transformations from one space to the same space.
 
9
The interest of the GUS is confined to the case in which the a priori probabilities are equal (\(q_i=1/K\)).
 
10
In the literature [3] the set of the states that satisfy (5.120) is called cyclic state set, whereas the term geometrically uniform symmetry indicates the general case, which is obtained with a multiplicative group of unitary matrices.
 
11
An operator from one space to another space is called isometric if it preserves norms and inner products [17].
 
Literatur
1.
C.W. Helstrom, J.W.S. Liu, J.P. Gordon, Quantum-mechanical communication theory. Proc. IEEE 58(10), 1578–1598 (1970)CrossRefMathSciNet
2.
K. Kraus, States, Effect and Operations: Fundamental Notions of Quantum Theory, Lecture Notes in Physics, vol. 190 (Springer, New York, 1983)
3.
Y.C. Eldar, A. Megretski, G.C. Verghese, Optimal detection of symmetric mixed quantum states. IEEE Trans. Inf. Theory 50(6), 1198–1207 (2004)CrossRefMATHMathSciNet
4.
5.
H.P. Yuen, R. Kennedy, M. Lax, Optimum testing of multiple hypotheses in quantum detection theory. IEEE Trans. Inf. Theory 21(2), 125–134 (1975)CrossRefMATHMathSciNet
6.
7.
M. Grant, S. Boyd, Graph implementations for nonsmooth convex programs, in: Recent Advances in Learning and Control, ed. by V. Blondel, S. Boyd, H. Kimura, (eds). Lecture Notes in Control and Information Sciences. (Springer, 2008), pp. 95–110, http://​stanford.​edu/​~ boyd/​graph_​dcp.​html
8.
R.S. Kennedy, A near-optimum receiver for the binary coherent state quantum channel. Massachusetts Institute of Technology, Cambridge (MA), Technical Report, January 1973. MIT Research Laboratory of Electronics Quarterly Progress Report 108
9.
V. Vilnrotter and C.W. Lau, Quantum detection theory for the free-space channel. NASA, Technical Report, August 2001. Interplanetary Network Progress (IPN) Progress Report 42–146
10.
V. Vilnrotter, C.W. Lau, Quantum detection and channel capacity using state-space optimization. NASA, Technical Report, February 2002. Interplanetary Network Progress (IPN) Progress Report 42–148
11.
V. Vilnrotter, C.W. Lau, Binary quantum receiver concept demonstration. NASA, Technical Report, Interplanetary Network Progress (IPN) Progress Report 42–165, May 2006
12.
K. Kato, M. Osaki, M. Sasaki, O. Hirota, Quantum detection and mutual information for QAM and PSK signals. IEEE Trans. Commun. 47(2), 248–254 (1999)CrossRef
13.
R.A. Horn, C.R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1998)
14.
Y.C. Eldar, G.D. Forney, On quantum detection and the square-root measurement. IEEE Trans. Inf. Theory 47(3), 858–872 (2001)
15.
A. Assalini, G. Cariolaro, G. Pierobon, Efficient optimal minimum error discrimination of symmetric quantum states. Phys. Rev. A 81, 012315 (2010)CrossRef
16.
G. Cariolaro, R. Corvaja, G. Pierobon, Compression of pure and mixed states in quantum detection. in Global Telecommunications Conference, vol. 2011 (GLOBECOM, IEEE, 2011), pp. 1–5
17.
A.S. Holevo, V. Giovannetti, Quantum channels and their entropic characteristics. Rep. Prog. Phys. 75(4), 046001 (2012)CrossRefMathSciNet
Metadaten
Titel
Quantum Decision Theory: Analysis and Optimization
verfasst von
Gianfranco Cariolaro
Copyright-Jahr
2015
Buch
Quantum Communications

Print ISBN: 978-3-319-15599-9

Electronic ISBN: 978-3-319-15600-2

Copyright-Jahr: 2015

DOI
https://doi.org/10.1007/978-3-319-15600-2_5

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