Institute for Experimental Physics University of Vienna Institute for Quantum Optics and Quantum Information Austrian Academy of Sciences Undecidability in Quantum Physics Časlav Brukner Leeds, United Kingdom March 2007

What constraints on our physical theories are implied by the foundations of mathematics?

Tomek Paterek Peter Klimek Č.B. Johannes Kofler

: Foundational crisis of mathematics David Hilbert Can every mathematical proposition in principle be proved or disproved? Idea: There must be a formal system in which a proof can be evaluated by a recursive procedure involving only logical and arithmetical manipulations, i.e. there must be algorithm for testing the validity of proofs.

Undecidability and Gödel Kurt Gödel (1931) Incompleteness theorem: Every theory of a certain expressive strength is either inconsistent or incomplete. Complete: Every statement can be proven or disproven, i.e. is decidable. Consistent: It is not possible to prove both statement A and !A. The paradox of the liar: „This statement is false.“ is changed to: „This statement is unprovable.“

Halting problem and Turing Alan Turing (1936) Halting problem: Given a description of a program and a finite input, decide whether the program terminates or will run forever. A general algorithm to solve the halting problem for all possible program-input pairs cannot exist.

Information and Chaitin Gregory J. Chaitin (1982) Information-theoretical formulation of Gödel’s theorem: If a theorem contains more information than a given set of axioms, then it is impossible for the theorem to be derived from the axioms. In a formal system with n bits of axioms it is impossible to prove that a particular binary string is of complexity greater than n.

Logical Complementarity Axiom: “The value f(0) of the function is 0”. 1 bit Theorem: “The value f(0) of the function is 0 and the value f(1) of the function is 0”.Undecidable, 2 bits Theorem: “The value f(0) of the function is 0 and f(0) = f(1)”. Consequence of Gödel’s theorem: certain mathematical statements are “logically complementary”, i.e. one cannot associate to them definite truth values simultaneously: 1.“The value f(0) of the function is 0”. 2.“The value f(1) of the function is 0”. 3.“The function f is constant”.

Closer look … 1 bit of information available: three binary statements are logically complementary: „Local“ properties „Global“ property

Physics of finite informational resources Mathematics: Total information contained in mutually complementary “theorems” cannot exceed the information contained in the “axioms” (1 bit) Physics: If computational resources are constrained to 1 bit, no machine can be built that gives more than 1 bit of information about f(0), f(1) and f(0) = f(1). This is true for every physical theory underlying computation

Physical representation of mathematical functions Encoding bit values:... presence: "1"... absence: "0" (, )  00 (, )  01 (, )  10 (, )  11 black box x  {0,1}  f (x)  {, } (, )  (f (0),f (1)) 0, 1

Classical scenarios for black box computation Deterministic: always execute one specific strategy Probabilistic: execute a probabilistic mixture of deterministic strategies

What can we learn from the classical scenarios? Given 1 bit of resources, one can classically learn either f(0) or f(1),… … or probabilistically one or the other … … but absolutely nothing about „Is f(0) = f(1)?“ f (0)=? f (1)=? f (0)=f (1) ? local properties global propery

Quantum computation: Deutsch algorithm information gain even for f (0)=? f (1)=? f (0)=f (1) ? local properties global propery Quantum mechanics allows us to access information, which would be classically not available using the same resources Interference crucial!

Partial knowledge of all 3 statements Beam-splitters: f (0)=f (1) ? f (0)=? f (1)=?

Information measure Q 1 and Q 2 are two complement subsets of functions measure of distinguishabilty of the two subsets prior to computation the „updated“ information after obtaining the readout D i information gain averaged over all detection events.

Complementarity Relation Quantum physics allows us to obtain partial knowledge about the complete set of three logically complementary questions

Are logical and quantum complementarity equivalent? f (0)=? It did not take this path. Partial which- path information Complete which- path information It took this path.

Generalizations … QFT:  The global property is complementary to all the local properties. The local properties are all mutually complementary iff dimension is prime.

Conclusions & Future Directions Gödels theorem defines to certain extent the structure of physical theories with constrained informational resources Can we find other set-ups in which quantum complementarities can be derived from the concept of logical complementarity? What can we learn from this concept about the structure of quantum theory? What is the connection to MUBs? Is randomness in physics an expression of mathematical undecidability?

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Formalizing the scenarios {Q i } set of all possible configurations in the black box {S j } set of all possible strategies (= computational paths) {D k }set of all detectors (= readouts after computation) The mutual information I(Q;D,S) = H(Q)–H(Q|D,S) tells us what we learn about Q by obtaining the readout D for given strategy S.

What is black-box computation?