[1] A., Acin. Statistical distinguishability between unitary operations. Phys. Rev. Lett., 87: 177 901, 2001.Google Scholar

[2] S.T., Ali, J.-P., Antoine and J.-P., Gazeau. Coherent states, Wavelets and Their Generalizations. Springer-Verlag, New York, 2000.Google Scholar

[3] S.T., Ali, C., Carmeli, T., Heinosaari and A., Toigo. Commutative povms and fuzzy observables. Found. Phys., 39: 593–612, 2009.Google Scholar

[4] S.T., Ali and G.G., Emch. Fuzzy observables in quantum mechanics. J. Math, Phys. 15: 176–182, 1974Google Scholar

[5] A., Ambainis, M., Mosca, A., Tapp and R., de Wolf. Private quantum channels. In FOCS, Proc. 41st Annual Symp. on Foundations of Computer Science, pp. 547–553. IEEE Computer Society, 2000.

[6] H., Araki and M.M., Yanase. Measurement of quantum mechanical operators. Phys. Rev. 2, 120: 622–626, 1960.Google Scholar

[7] A., Arias, A., Gheondea and S., Gudder. Fixed points of quantum operations. J. Math. Phys., 43: 5872–5881, 2002.Google Scholar

[8] K., Audenaert and S., Scheel. On random unitary channels. New J. Phys., 10: 023 011, 2008.Google Scholar

[9] A., Barenco, C.H., Bennett, R., Cleve et al. Elementary gates for quantum computation. Phys. Rev. A, 52: 3457–3467, 1995.Google Scholar

[10] V., Bargmann. Note on Wigner's theorem on symmetry operations. J. Math. Phys., 5: 862–868, 1964.Google Scholar

[11] J.S., Bell. On the Einstein Podolsky Rosen paradox. Physics, 1: 195–200, 1964.Google Scholar

[12] J.S., Bell. On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys., 38: 447–452, 1966.Google Scholar

[13] E., Beltrametti, G., Cassinelli and P., Lahti. Unitary measurements of discrete quantities in quantum mechanics. J. Math. Phys., 31: 91–98, 1990.Google Scholar

[14] I., Bengtsson and K., Zyczkowski. Geometry of quantum states. Cambridge University Press, 2006.Google Scholar

[15] C.H., Bennett. Quantum cryptography using any two nonorthogonal states. Phys. Rev. Lett., 68: 3121–3124, 1992.Google Scholar

[16] C.H., Bennett and G., Brassard. Quantum cryptography: public-key distribution and coin-tossing. In Proc. IEEE Int. Conf. on Computers, Systems and Signal Processing, pp. 175–179. IEEE, New York, 1984.

[17] C.H., Bennett and S.J., Wiesner. Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett., 69: 2881–2884, 1992.Google Scholar

[18] C.H., Bennett, G., Brassard, C., Crepeau, R., Jozsa, A., Peres, and W.K., Wootters. Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Phys. Rev. Lett., 70: 1895–1899, 1993.Google Scholar

[19] C.H., Bennett, D.P., DiVincenzo, C.A., Fuchs, et al. Quantum nonlocality without entanglement. Phys. Rev. A, 59: 1070–1091, 1999.Google Scholar

[20] C.H., Bennett, D.P., DiVincenzo, T., Mor, P.W., Shor, J.A., Smolin, and B.M., Terhal. Unextendible product bases and bound entanglement. Phys. Rev. Lett., 82: 5385, 1999.Google Scholar

[21] J.A., Bergou, U., Herzogand M., Hillery. Discrimination of quantum states. In Quantum State Estimation, Vol. 649 of Lecture Notes in Physics, pp. 417–465. Springer, Berlin, 2004.

[22] P., Busch. Unsharp reality and joint measurements for spin observables. Phys. Rev. D, 33: 2253–2261, 1986.Google Scholar

[23] P., Busch. Quantum states and generalized observables: a simple proof of Gleason"s theorem. Phys. Rev. Lett., 91: 120403/1–4, 2003.Google Scholar

[24] P., Busch. The role of entanglement in quantum measurement and information processing. Int. J. Theor. Phys., 42: 937–941, 2003.Google Scholar

[25] P., Busch. ‘No information without disturbance’: quantum limitations of measurement. In J. Christian and W. Myrvold, eds., Quantum reality, relativistic causality, and closing the epistemic circle. Springer-Verlag, Berlin, 2008.Google Scholar

[26] P., Busch and P., Lahti. The determination of the past and the future of a physical system in quantum mechanics. Found. Phys., 19: 633–678, 1989.Google Scholar

[27] P., Busch and P, Lahti. Some remarks on unsharp quantum measurements, quantum nondemolition, and all that. Ann. Physik 7, 47: 369–382, 1990.

[28] P., Busch and P., Lahti. The complementarity of quantum observables: Theory and experiments. Riv. Nuovo Cimento, 18: 1–27, 1995.Google Scholar

[29] P., Busch and H.-J., Schmidt. Coexistence of qubit effects. Quantum Inf. Process., 9: 143,–169, 2010.Google Scholar

[30] P., Busch and C., Shilladay. Complementarity and uncertainty in Mach–Zehnder interferometry and beyond. Phys. Rep., 435: 1–31, 2006.Google Scholar

[31] P., Busch and J., Singh. Lüders theorem for unsharp quantum measurements. Phys. Lett. A, 249: 10–12, 1998.Google Scholar

[32] P., Busch, G., Cassinelli and P., Lahti. Probability structures for quantum state spaces. Rev. Math. Phys., 7: 1105–1121, 1995.Google Scholar

[33] P., Busch, M., Grabowski and P., Lahti.Repeatable measurements in quantum theory:their role and feasibility. Found. Phys., 25: 1239–1266, 1995.Google Scholar

[34] P., Busch, M., Grabowski and P., Lahti. Operational Quantum Physics. Springer-Verlag, Berlin, 1997; second, corrected, printing.Google Scholar

[35] P., Busch, P., Lahti and P., Mittelstaedt. The Quantum Theory of Measurement. Springer-Verlag, Berlin, second edition, 1996.Google Scholar

[36] A., Aspuru-Guzik. C.A., Rodríguez Rosario and K., Modi. Linearassignment maps for correlated system–environment states. Phys. Rev. A, 81: 012 313, 2010.Google Scholar

[37] C., Carmeli, T., Heinonen and A., Toigo. Intrinsic unsharpness and approximate repeatability of quantum measurements. J. Phys. A, 40: 1303–1323, 2007.Google Scholar

[38] G., Cassinelli, E., De Vito, P., Lahti and A., Levrero. The Theory of Symmetry Actions in Quantum Mechanics. Springer, Berlin, 2004.Google Scholar

[39] G., Cassinelli, E., De Vito and A., Levrero. On the decompositions of a quantum state. J. Math. Anal. Appl., 210: 472–483, 1997.Google Scholar

[40] C.M., Caves, C. A., Fuchs and R., Schack. Unknown quantum states: the quantum de Finetti representation. J. Math. Phys., 43: 4537–4559, 2002.Google Scholar

[41] N.J., Cerf and J., Fiurášek. Optimal quantum cloning – a review. In Progress in Optics, Vol. 49. Elsevier, 2006.Google Scholar

[42] A., Chefles. Quantum state discrimination. Contemp. Physics, 41: 401–424, 2000.Google Scholar

[43] M., Choi. Completely positive linear maps on complex matrices. Linear Alg. Appl., 10: 285–290, 1975.Google Scholar

[44] J.F., Clauser, M.A., Horne, A., Shimony and R.A., Holt. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett., 23: 880–884, 1969.Google Scholar

[45] J.B., Conway. A Course in Functional Analysis, second edition. Springer, 1990.Google Scholar

[46] J.B., Conway. A Course in Operator Theory. American Mathematical Society, Providence, Rhode Island, 2000.Google Scholar

[47] R., Cooke, M., Keane and W., Moran. An elementary proof of Gleason"s theorem. Math. Proc. Cambridge Philos. Soc., 98: 117–128, 1985.Google Scholar

[48] J., Corbett. The Pauli problem, state reconstruction and quantum-real numbers. Rep. Math. Phys., 57: 53–68, 2006.Google Scholar

[49] J., Corbett and C., Hurst. Are wave functions uniquely determined by their position and momentum distributions?J. Austral. Math. Soc. Ser. B, 20: 182–201, 1978.Google Scholar

[50] G.M., D'Ariano, P., Lo Presti and P., Perinotti. Classical randomness in quantum measurements. J. Phys. A, 38: 5979–5991, 2005.Google Scholar

[51] E.B., Davies. Quantum Theory of Open Systems. Academic Press,, London, 1976.Google Scholar

[52] E.B., Davies and J.T., Lewis. An operational approach to quantum probability. Comm. Math. Phys., 17: 239–260, 1970.Google Scholar

[53] W.M., de Muynck. Foundations of Quantum Mechanics, An Empiricist Approach. Kluwer, Dordrecht, 2002.

[54] P.A.M., Dirac. The Principles of Quantum Mechanics. Oxford University Press, 1930.Google Scholar

[55] W., Dür, G., Vidal and J.I., Cirac. Three qubits can be entangled in two inequivalent ways. Phys. Rev. A, 62: 062 314, 2000.Google Scholar

[56] T., Durt, B.-G., Englert, I., Bengtsson and K., Zyczkowski. On mutually unbiased bases. Int. J. Quant. Inf., 8: 535–640, 2010.Google Scholar

[57] A., Dvurečenskij. Gleason"s Theorem and Its Applications. Kluwer, Dordrecht, 1993.

[58] A., Einstein, B., Podolsky and N., Rosen. Can quantum-mechanical description of physical reality be considered complete?Phys. Rev., 47: 777–780, 1935.Google Scholar

[59] Y., Feng, R., Duan and M., Ying. Unambiguous discrimination between mixed quantum states. Phys. Rev. A, 70: 012 308, 2004.Google Scholar

[60] G.B., Folland. A Course in Abstract Harmonic Analysis. CRC Press, Boca Raton, Florida, 1995.Google Scholar

[61] G.B., Folland. Real Analysis, second edition. John Wiley & Sons, New York, 1999.Google Scholar

[62] A., Friedman. Foundations of Modern Analysis. Dover, New York<, 1982. Reprint of the 1970 original.Google Scholar

[63] C., Friedman. Some remarks on Pauli data in quantum mechanics. J. Austral. Math. Soc. Ser. B, 30: 298–303, 1989.Google Scholar

[64] C., Gerry and P., Knight. Introductory Quantum Optics. Cambridge University Press, 2005.Google Scholar

[65] N., Gisin, G., Ribordy, W., Tittel and H., Zbinden. Quantum cryptography. Rev. Mod. Phys., 74: 145–195, 2002.Google Scholar

[66] A.M., Gleason. Measures on the closed subspaces of a Hilbert space. J. Math. Mech., 6: 885–893, 1957.Google Scholar

[67] D.M., Greenberger, M.A., Horne and A., Zeilinger. Going beyond Bell"s theorem. In M. Kafatos, ed., Bell"s Theorem, Quantum Theory, and Conceptions of the Universe, pp. 69–72. Kluwer Academic Publishers, Dordrecht, 1989.Google Scholar

[68] S., Gudder. Lattice properties of quantum effects. J. Math. Phys., 37: 2637, 1996.Google Scholar

[69] S., Gudder and R., Greechie. Effect algebra counterexamples. Math. Slovaca, 46: 317–325, 1996.Google Scholar

[70] R., Haag and D., Kastler. An algebraic approach to quantum field theory. J. Math. Phys., 5: 848–861, 1964.Google Scholar

[71] N., Hadjisavvas. Properties of mixtures on non-orthogonal states. Lett. Math. Phys., 5: 327–332, 1981.Google Scholar

[72] M., Hayashi. Quantum Information. Springer-Verlag, Berlin, 2006. Translated from the 2003 Japanese original.Google Scholar

[73] T., Heinonen. Optimal measurements in quantum mechanics. Phys. Lett. A, 346: 77–86, 2005.Google Scholar

[74] T., Heinosaari and M.M., Wolf. Nondisturbing quantum measurements. J. Math. Phys., 51: 092201, 2010.Google Scholar

[75] C.W., Helstrom. Quantum Detection and Estimation Theory. Academic Pressm New York, 1976.Google Scholar

[76] A.S., Holevo. Probabilistic and Statistical Aspects of Quantum Theory. North-Holland, Amsterdam, 1982.Google Scholar

[77] M., Horodecki, P., Horodecki and R., Horodecki. Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A, 223: 1–8, 1996.Google Scholar

[78] M., Horodecki, P.W., Shor and M.B., Ruskai. Entanglement breaking channels. Rev. Math. Phys., 15: 629–641, 2003.Google Scholar

[79] P., Horodecki. Separability criterion and inseparable mixed states with positive partial transposition. Phys. Lett. A, 232, 1997.Google Scholar

[80] R., Horodecki, P., Horodecki, M., Horodecki and K., Horodecki. Quantum entanglement. Rev. Mod. Phys. 81: 865–942, 209.

[81] L.P., Hughston, R., Jozsa and W.K., Wootters. A complete classification of quantum ensembles having a given density matrix. Phys. Lett. A, 183: 14–18, 1993.Google Scholar

[82] I.D., Ivanović. Geometrical description of quantal state determination. J. Phys. A: Math. Gen., 14: 3241–3245, 1981.Google Scholar

[83] I.D., Ivanović. How to differentiate between nonorthogonal states. Phys. Lett. A, 123: 257–259, 1987.Google Scholar

[84] E.T., Jaynes. Information theory and statistical mechanics. II. Phys. Rev., 108: 171–190, 1957.Google Scholar

[85] A., Jenčová and S., Pulmannová. Characterizations of commutative POV measures. Found. Phys., 39: 613–624, 2009.Google Scholar

[86] D., Jonathan and M.B., Plenio. Entanglement-assisted local manipulation of pure quantum states. Phys. Rev. Lett., 83: 3566–3569, 1999.Google Scholar

[87] Y., Kinoshita, R., Namiki, T., Yamamoto, M., Koashi and N., Imoto. Selective entanglement breaking. Phys. Rev. A, 75: 032 307, 2007.Google Scholar

[88] K., Kraus. General state changes in quantum theory. Ann. Physics, 64: 311–335, 1971.Google Scholar

[89] K., Kraus. States, Effects, and Operations. Springer-Verlag, Berlin, 1983.Google Scholar

[90] S., Kullback and R.A., Leibler. On information and sufficiency. Ann. Math. Statistics, 22: 79–86, 1951.Google Scholar

[91] P.J., Lahti and M., Maczynski. Partial order of quantum effects. J. Math. Phys., 36: 1673–1680, 1995.Google Scholar

[92] L.J., Landau and R.F., Streater. On Birkhoff"s theorem for doubly stochastic completely positive maps of matrix algebras. Linear Algebra Appl., 193: 107–127, 1993.Google Scholar

[93] G., Lindblad. Completely positive maps and entropy inequalities. Comm. Math. Phys., 40: 147–151, 1975.Google Scholar

[94] L., Loveridge and P., Busch. ‘Measurement of quantum mechanical operators’ revisited. Eur. Phys. J. D, 2011.Google Scholar

[95] G., Lüders. Über die Zustandsänderung durch den Messprozess. Ann. Physik, 6 8: 322–328, 1951.Google Scholar

[96] G., Ludwig. Foundations of Quantum Mechanics I. Springer-Verlag, New York, 1983.Google Scholar

[97] D., Mayers. Unconditional security in quantum cryptography. J. Assoc. Comp. Mach., 48: 351, 2001.Google Scholar

[98] C.B., Mendl and M.M., Wolf. Unital quantum channels – convex structure and revivals of Birkhoff"s theorem. Comm. Math. Phys., 289: 1057–1086, 2009.Google Scholar

[99] D.N., Mermin. Quantum Computer Science. Cambridge University Press, 2007.Google Scholar

[100] M., Mičuda, M., Ježek, M., Dušek and J., Fiurášek. Experimental realization of programmable quantum gate. Phys. Rev. A, 78: 062 311, 2008.Google Scholar

[101] L., Molnár. Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces, Vol. 1895 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2007.Google Scholar

[102] M.A., Nielsen. Conditions for a class of entanglement transformations. Phys. Rev. Lett., 83: 436–439, 1999.Google Scholar

[103] M.A., Nielsen and I.L., Chuang. Programmable quantum gate arrays. Phys. Rev. Lett., 79: 321–324, 1997.Google Scholar

[104] M.A., Nielsen and I.L., Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2000.Google Scholar

[105] M., Ohya and D., Petz. Quantum Entropy and Its Use. Springer-Verlag, Berlin. Second, corrected printing, 2004.

[106] M., Ozawa. Conditional expectation and repeated measurements of continuous quantum observables. In Probability Theory and Mathematical Statistics Tbilisi, 1982, Vol. 1021 of Lecture Notes in Mathematics, pp. 518–525. Springer, Berlin, 1983.Google Scholar

[107] M., Ozawa. Quantum measuring processes of continuous observables. J. Math. Phys., 25: 79–87, 1984.Google Scholar

[108] M., Ozawa. Conditional probability and a posteriori states in quantum mechanics. Publ. RIMS, Kyoto Univ., 21: 279–295, 1985.Google Scholar

[109] M., Ozawa. Operations, disturbance, and simultaneous measurability. Phys. Rev. A, 63: 032 109, 2001.Google Scholar

[110] M., Paris and J., Řeháček, eds. Quantum State Estimation, Vol. 649 of Lecture Notes in Physics. Springer-Verlag, Berlin, 2004.Google Scholar

[111] K.R., Parthasarathy. Extremal decision rules in quantum hypothesis testing. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 24: 557–568, 1999.Google Scholar

[112] V., Paulsen. Completely Bounded Maps and Operator Algebras. Cambridge University Press, Cambridge, 2003.Google Scholar

[113] G., Pedersen. Analysis Now. Springer-Verlag, New York, 1989.Google Scholar

[114] J.-P., Pellonpää. Complete characterization of extreme quantum observables in infinite dimensions. J. Phys. A: Math. Theor., 44: 085 304, 2011.Google Scholar

[115] A., Peres. Separability criterion for density matrices. Phys. Rev. Lett., 77: 1413– 1415, 1996.Google Scholar

[116] D., Petz. Quantum Information Theory and Quantum Statistics. Springer, Berlin, 2008.Google Scholar

[117] D., Petz. Algebraic complementarity in quantum theory. J. Math. Phys., 51: 015 215, 2010.Google Scholar

[118] M.B., Plenio and S., Virmani. An introduction to entanglement measures. Quant. Inf. Comp., 7: 1, 2007.Google Scholar

[119] E., Prugovečki. Information-theoretical aspects of quantum measurements. Int. J. Theor. Phys., 16: 321–331, 1977.Google Scholar

[120] M., Raginsky. Strictly contractive quantum channels and physically realizable quantum computers. Phys. Rev. A, 65: 032 306, 2002.Google Scholar

[121] M., Reed and B., Simon. Methods of Modern Mathematical Physics, Vol. I: Functional Analysis, revised and enlarged ed. Academic Press, London, 1980.Google Scholar

[122] W., Rudin. Real and Complex Analysis, third ed. McGraw-Hill, New York, 1987.Google Scholar

[123] W., Rudin. Functional Analysis, second ed., McGraw-Hill, New York, 1991.Google Scholar

[124] M.B., Ruskai, S., Szarek and E., Werner. An analysis of completely positive racepreserving maps on M2. Linear Algebra Appl., 347: 1593–187, 2002.Google Scholar

[125] B., Russo and H.A., Dye. A note on unitary operators in C?-algebras. Duke Math. J., 33: 413–416, 1966.Google Scholar

[126] V., Scarani, S., Iblisdir, N., Gisin and A., Acin. Quantum cloning. Rev. Mod. Phys., 77: 1225–1256, 2005.Google Scholar

[127] E., Schrödinger. Discussion of probability relations between separated systems. Math. Proc. Camb. Phil. Soc., 31: 555–563, 1935.Google Scholar

[128] A.J., Scott and M., Grassl. Symmetric informationally complete positive-operatorvalued measures: a new computer study. J. Math. Phys., 51: 042 203, 2010.Google Scholar

[129] P., Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput., 26: 1484–1509, 1997.Google Scholar

[130] P.W., Shor and J., Preskill. Simple proof of security of the BB84 quantum key distribution protocol. Phys. Rev. Lett., 85: 441–444, 2000.Google Scholar

[131] M., Singer and W., Stulpe. Phase-space representations of general statistical physical theories. J. Math. Phys., 33: 131–142, 1992.Google Scholar

[132] P., Stano, D., Reitzner and T., Heinosaari. Coexistence of qubit effects. Phys. Rev. A, 78: 012 315, 2008.Google Scholar

[133] W.F., Stinespring. Positive functions on C?-algebras. Proc. Amer. Math. Soc., 6: 211–216, 1955.Google Scholar

[134] E., Størmer. Positive linear maps of operator algebras. Acta Math., 110: 233–278, 1963.Google Scholar

[135] W., Thirring. Atoms, molecules and large systems, in Quantum Mathematical Physics, second ed. Springer-Verlag, Berlin, 2002. Translated from the 1979 and 1980 German originals by Evans M. Harrell II.Google Scholar

[136] L., Vaidman, Y., Aharonov and D., Albert. How to ascertain the values of σx, σy, and σz of a spin-1/2 particle. Phys. Rev. Lett., 58: 1385–1387, 1987.Google Scholar

[137] G., Vernam. Cipher printing telegraph system for secret wire and radio telegraphic communications. J. Am. Inst. Electr. Eng., 45: 109–115, 1926.Google Scholar

[138] J., von Neumann. Mathematical Foundations of Quantum Mechanics. Princeton University Press, 1955. Translated by R.T. Beyer from Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, 1932.Google Scholar

[139] D.F., Walls and G.J., Milburn. Quantum Optics, second ed. Springer-Verlag, 2008.Google Scholar

[140] T.-C., Wei, K., Nemoto, P.M., Goldbart, P.G., Kwiat, W.J., Munro and F., Verstraete. Maximal entanglement versus entropy for mixed quantum states. Phys. Rev. A, 67:022 110, 2003.Google Scholar

[141] R.F., Werner. Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model. Phys. Rev. A, 40: 4277–4281, 1989.Google Scholar

[142] R.F., Werner. All teleportation and dense coding schemes. J. Phys. A, 34: 7081, 2001.Google Scholar

[143] E.P., Wigner. Die Messung quantenmechanischer Operatoren. Z. Physik, 133: 101–108, 1952.Google Scholar

[144] E.P., Wigner. Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra. Expanded and improved edition, translated from the German by J. J. Griffin. Academic Press, New York, 1959.Google Scholar

[145] M.M., Wolf and J.I., Cirac. Dividing quantum channels. Commun. Math. Phys., 279: 147–168, 2008.Google Scholar

[146] M.M., Wolf, D., Perez-Garcia and C., Fernandez. Measurements incompatible in quantum theory cannot be measured jointly in any other no-signaling theory. Phys. Rev. Lett., 103: 230 402, 2009.Google Scholar

[147] W.K., Wootters and B.D., Fields. Optimal state-determination by mutually unbiased measurements. Ann. Physic, 191: 363–381, 1989.Google Scholar

[148] S. L., Woronowicz. Positive maps of low dimensional matrix algebras. Rep. Math. Phys, 10: 165–183, 1976.Google Scholar

[149] S., Yu, N., Liu, L., Li and C.H., Oh. Joint measurement of two unsharp observables of a qubit. Phys. Rev. A 81: 062 116, 2010.Google Scholar

[150] M., Ziman and V., Bužek. Entanglement-induced state ordering under local operations. Phys. Rev. A, 73: 012 312, 2006.Google Scholar

[151] M., Ziman and V., Bužek. Entanglement measures: state ordering vs. local operations. In M. Zukowski, ed., Quantum Communication and Security, pp. 196–204. IOS Press, 2007.Google Scholar