Axioms and Measurements

John Earman, Abner Shimony

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

Bernard d’Espagnat

Tomasz Paterek, Časlav Brukner, Robert Prevedel +4 more

We propose a new link between mathematical undecidability and quantum physics. We demonstrate that the states of elementary quantum systems are capable of encoding mathematical axioms and show that quantum measurements are capable of revealing whether a given proposition is decidable or not within the axiomatic system. Whenever a mathematical proposition is undecidable within the axioms encoded in the state, the measurement associated with the proposition gives random outcomes. Our results support the view that quantum randomness is irreducible and a manifestation of mathematical undecidability. Whenever a proposition and a given set of axioms together contain more information than the axioms themselves, the proposition can neither be proved nor disproved from the axioms { it is mathematically undecidable [1, 2]. Here we propose a novel link between mathematical undecidability and quantum physics. We demonstrate that the states of elementary quantum systems are capable of encoding mathematical axioms. Quantum mechanics imposes an upper limit on how much information can be encoded in a quantum state [3, 4], thus limiting the information content of the set of axioms. We show that quantum measurements are capable of revealing whether a given proposition is decidable or not. Whenever a mathematical proposition is undecidable within the system of axioms encoded in the state, the measurement associated with the proposition gives random outcomes. This allows for an experimental test of mathematical undecidability by realizing in the laboratory both the actual quantum states and the required quantum measurements. (To illustrate these ideas, we conducted experiments using the polarization of photons.) Our results support the view that quantum randomness is irreducible [5, 6] and a manifestation of mathematical undecidability. Any formal system is based on axioms, which are propositions that are dened to be true. A proposition is logically independent from a given set of axioms if it can neither be proved nor disproved from the axioms. If a proposition is independent from the axioms, neither the proposition itself nor its negation creates an inconsistency together with the axiomatic system. ^

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

Philip R. Wallace