Axioms of measurement theory

史郎 石川

Ｔｈｅ ｐｈｉｌｏｓｏｐｈｙ ｏｆ ｍｅａｓｕｒｅｍｅｎｔ ｔｈｅｏｒｙ Ｍｅａｓｕｒｅｍｅｎｔｓ（Ａｘｉｏｍ １） Ｔｈｅ ｒｅｌａｔｉｏｎ ａｍｏｎｇ ｓｙｓｔｅｍｓ（Ａｘｉｏｍ ２） Ｂｏｌｔｚｍａｎｎ’ｓ ｅｑｕｉｌｉｂｒｉｕｍ ｓｔａｔｉｓｔｉｃａｌ ｍｅｃｈａｎｉｃｓ Ｆｉｓｈｅｒ’ｓ ｓｔａｔｉｓｔｉｃｓ Ｉ（ｕｎｄｅｒ Ａｘｉｏｍ １） Ｆｉｓｈｅｒ’ｓ ｓｔａｔｉｓｔｉｃｓ ＩＩ（ｒｅｌａｔｅｄ ｔｏ Ａｘｉｏｍｓ １ ａｎｄ ２） Ｐｒａｃｔｉｃａｌ ｌｏｇｉｃ Ｓｔａｔｉｓｔｉｃａｌ ｍｅａｓｕｒｅｍｅｎｔｓ ｉｎ Ｃ＊‐ａｌｇｅｂｒａｉｃ ｆｏｒｍｕｌａｔｉｏｎ Ｓｔａｔｉｓｔｉｃａｌ ｍｅａｓｕｒｅｍｅｎｔｓ ｉｎ Ｗ＊‐ａｌｇｅｂｒａｉｃ ｆｏｒｍｕｌａｔｉｏｎ Ｎｅｗｔｏｎｉａｎ ｍｅｃｈａｎｉｃｓ ｉｎ ｍｅａｓｕｒｅｍｅｎｔ Ｔｈｅｏｒｙ Ｍｅａｓｕｒｅｍｅｎｔ ｅｒｒｏｒ Ｈｅｉｓｅｎｂｅｒｇ’ｓ ｕｎｃｅｒｔａｉｎｔｙ ｒｅｌａｔｉｏｎ

Henry Krips

Daniel I. Fivel

Conventional quantum mechanics with a complex Hilbert space and the Born Rule is derived from five axioms describing properties of probability distributions for the outcome of measurements. Axioms I,II,III are common to quantum mechanics and hidden variable theories. Axiom IV recognizes a phenomenon, first noted by Turing and von Neumann, in which the increase in entropy resulting from a measurement is reduced by a suitable intermediate measurement. This is shown to be impossible for local hidden variable theories. Axiom IV, together with the first three, almost suffice to deduce the conventional rules but allow some exotic, alternatives such as real or quaternionic quantum mechanics. Axiom V recognizes a property of the distribution of outcomes of random measurements on qubits which holds only in the complex Hilbert space model. It is then shown that the five axioms also imply the conventional rules for all dimensions. 1

Roger Balian

John F. Cyranski

Jiří Souček

We present the axiomatization of quantum mechanics which does not contain axioms concerning the measurement. Instead of the concept of measurement this axiomatization uses the concept of the observation of the individual state of the measuring system after the run of the experiment. It is proved that the resulting theory is empirically equivalent to the standard quantum mechanics but it is also shown that these two theories are (theoretically) different.