Abstract
From the perspective of quantum information theory, a system so simple as one restricted to just two nonorthogonal states can be surprisingly rich in physics. In this paper, we explore the extent of this statement through a review of three topics: (1) “nonlocality without entanglement” as exhibited in binary quantum communication channels, (2) the tradeoff between information gain and state disturbance for two prescribed states, and (3) the quantitative clonability of those states. Each topic in its own way quantifies the extent to which two states are “quantum” with respect to each other, i.e., the extent to which the two together violate some classical precept. It is suggested that even toy examples such as these hold some promise for shedding light on the foundations of quantum theory.
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References
K. Kraus, “States, Effects, and Operations,” Springer-Verlag, Berlin (1983).
For a very basic result in this respect, see Theorem 6 and Section V of I. Pitowsky, Infinite and finite Gleason’s theorems and the logic of indeterminacy, J. Math, Phys. 39:218 (1998).
W. K. Wootters and W. H. Zurek, A single quantum cannot be cloned, Nature 299:802 (1982); D. Dieks, Communication by EPR devices, Phys. Lett. A 92:271 (1982).
H. P. Yuen, Amplification of quantum states and noiseless photon amplifiers, Phys. Lett. A 113:405 (1986); H. Barnum, C. M. Caves, C. A. Fuchs, R. Jozsa, and B. Schumacher, Noncommuting mixed states cannot be broadcast, Phys. Rev. Lett. 76:2818 (1996).
C. H. Bennett, G. Brassard, and N. D. Mermin, Quantum cryptography without Bell’s theorem, Phys. Rev. Lett. 68:557 (1992).
C. A. Fuchs, Information gain vs. state disturbance in quantum theory, Fort. der Phys. 46:535 (1998); C. A. Fuchs and A. Peres, Quantum state disturbance vs. information gain: Uncertainty relations for quantum information, Phys. Rev. A 53:2038 (1996).
M. Sasaki, T. S. Usuda, O. Hirota, and A. S. Holevo, Applications of the Jaynes-Cummings model for the detection of nonorthogonal quantum states, Phys. Rev. A 53:1273 (1996).
C. A. Fuchs, Nonorthogonal quantum states maximize classical information capacity, Phys. Rev. Lett. 79:1163 (1997).
C. W. Helstrom, “Quantum Detection and Estimation Theory,” Academic Press, NY (1976).
C. M. Caves and C. A. Fuchs, Quantum information: How much information in a state vector?, in: “The Dilemma of Einstein, Podolsky and Rosen — 60 Years Later,” A. Mann and M. Revzen, eds., Israel Physical Society (1996).
C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W. Shor, J. A. Smolin, and W. K. Wootters, Quantum nonlocality without entanglement, quant-ph/9804053.
T. M. Cover and J. A. Thomas, “Elements of Information Theory,” Wiley, NY (1991), Sect. 8.9.
A. S. Holevo, Capacity of a quantum communication channel, Prob. Info. Trans. 15:247 (1979).
A. Fujiwara and H. Nagaoka, Operational capacity and pseudoclassicality of a quantum channel, IEEE Trans. Inf. Theory 44:1071 (1998).
L. B. Levitin, Optimal quantum measurements for two pure and mixed states, in: “Quantum Communications and Measurement,” V. P. Belavkin, O. Hirota, and R. L. Hudson, eds., Plenum Press, NY (1995); C. A. Fuchs and C. M. Caves, Ensemble-dependent bounds for accessible information in quantum mechanics, Phys. Rev. Lett. 73:3047 (1994).
P. Hausladen, R. Jozsa, B. Schumacher, M. Westmoreland, and W. K. Wootters, Classical information capacity of a quantum channel, Phys. Rev. A 54:1869 (1996); A. S. Holevo, The capacity of the quantum channel with general signal states, IEEE Trans. Inf. Theory 44:269 (1998); B. Schumacher and M. D. Westmoreland, Sending classical information via noisy quantum channels, Phys. Rev. A 56:131 (1997).
C. A. Fuchs, “Distinguishability and Accessible Information in Quantum Theory,” Ph.D. thesis, University of New Mexico, 1996. LANL archivequant-ph/9601020.
A. Peres and W. K. Wootters, Optimal detection of quantum information, Phys. Rev. Lett. 66:1119(1991).
W. Pauli, “Writings on Philosophy and Physics,” Springer-Verlag, Berlin (1995).
A. Peres, “Quantum Theory: Concepts and Methods,” Kluwer, Dordrecht (1993).
C. H. Bennett, Quantum cryptography using any two nonorthogonal states, Phys. Rev. Lett. 68:3121 (1992).
D. Bruß D. P. DiVincenzo, A. Ekert, C. A. Fuchs, C. Macchiavello, and J. A. Smolin, Optimal universal and state-dependent quantum cloning, Phys. Rev. A 57:2368 (1998).
N. D. Mermin, What is quantum mechanics trying to tell us?, Am. J. Phys. 66:753 (1998).
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Fuchs, C.A. (2002). Just Two Nonorthogonal Quantum States. In: Kumar, P., D’Ariano, G.M., Hirota, O. (eds) Quantum Communication, Computing, and Measurement 2. Springer, Boston, MA. https://doi.org/10.1007/0-306-47097-7_2
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DOI: https://doi.org/10.1007/0-306-47097-7_2
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