Probabilistic theories: classical, quantum and beyond quantum

1. Probabilistic theories: classical, quantum andbeyond quantum byBorivoje Dakic (with Caslav Brukner) Speakable in quantum mechanics: atomic, nuclear and subnuclear physics tests, Trento 2011

2. Groucho Marx on quantum theory: “Very interesting theory – it makes no sense at all.”

3. Historyteachesus … ... that every theory that was established and broadly accepted at a certain time was later inevitably replaced by a deeper and more fundamental theoryof which the old one either remains a special case or is completely rejected. PtolemaicgeocentricsystembyPortuguesecosmographerandcartographerBartolomeu Velho, 1568 Copernicus' heliocentric system in his manuscript De revolutionibusorbiumcoelestium, 1543

4. Whataboutquantumtheory? • Non-contextualhidden-variable theories (Kochen-Speckertheorem) • Local hidden-variable theories (Bell’s theorem) • Crypto-nonlocal hidden-variable theories (Leggett) • Nonlinear Schrödinger equation (Biaynicki-Birula …) • Collapsmodels (Diosi, Pearle, Penrose …) All wantto save oneortheotherfeatureofclassicalphysics: determinism, localizability, macroscopicrealism, … „Ournexttheoryislikelytobecrazierthanquantumtheoryitself.“ (A. Zeilinger & D. Greenberger)

5. Underlying physicalprinciples Special relativity Quantum mechanics Mathematicalapparatus Lorentz transfomations States (Hilbert space) Unitaryevolution Mesurements Born rule Physicalprinciles Frame dependentspaceand time Causality in all frames Nosignaling ?

6. „Andyou‘requitesureit‘s just a hypotheticalcat? “ Toavoidmisunderstandings: „Quantum phenomena do not occur in a Hilbert space, they occur in a laboratory.“ (A. Peres)

7. Laboratory Laser, optical fibers, fiber couplers, mirrors, modulators, lenses, beamsplitters, half-waveplates, quarter-waveplates, down-conversioncrystals, polarizers, filters, photodiodes, etc. Never ending story… ??? Whatisthis?

8. Black box description Click! Click! 2 2 2 1 1 1 3 3 3 0 0 0 4 4 4 Analyzer Analyzer Analyzer Source Bob Click! Alice P(A,B,C|a,b,c)=? Charlie

9. State vector Transformation Measurement Preparation M1 M2 M3 …. Pure Mixtures: Mixed

10. Transformations T →T is linear T1 T2 T12 →T is a group

11. Measurement Measurement Born rule Measurement vector Pure states:

12. “State”: Whatdefines a „system“? L Dimension – Maximal numberofdistinguishableoutcomes in a singleshotexperiment Numberofdegreesoffreedom – Minimal numberofparametersdescribingthestatecompletely (log L – maximal numberofbitsencoded in thesystem) d Wootters (1986), Hardy (2001)

13. 1 Bit Systems (L=2) d=3 real numbers needed to specify quantum state but only 1 bit can be read out (Holevo Bound) The same Born rule 1bit encoded in thesystem(A. Zeilinger)

14. Can a theoryexibitentanglementif: Information Capacity Axiom: The statespaceof 1-bit systemis a „sphere“ LocalityAxiom: The stateof a compositesystemcanbedeterminedfrom (local) measurementsofsubsystems Subspace Axiom: All 1-bit systemsareequivalent ? Yes ... but only quantum theory! → Axioms

15. 1. Information Capacity Axiom x States Axiom 1: Any state (pure or mixed) of two-level system can be prepared by mixing at most two orthogonal states (i.e. distinguishable in a single-shot experiment) “Any state can be prepared as a mixture of classical bits 0 and 1” → Sphere

16. General Bit (G-bit) States, Measurement vectors = Sd-1 Probabilities Transformations reversible d independent measurements preserving purity orthogonal matrix (sub)group of O(d)

17. 2. Locality Axiom Axiom 2: The state of a composite system is completely determined by local measurements on its subsystems andtheircorrelations („LocalTomography“) Example: Local vectors Correlations d d Wootters (1986), Hardy (2001), Barrett (2005)

18. 2. Locality Axiom Axiom 2: The state of a composite system is completely determined by local measurements on its subsystems andtheircorrelations („LocalTomography“) Local vectors Correlations d d Real Quantum Mechanics: 2 + 2 + 2x2 < 9 Quaternionic QM: 5 + 5 + 5x5 > 27 Complex QM: 3 + 3 + 3x3 = 15 Wootters (1986), Hardy (2001), Barrett (2005)

19. Productstates: Probabilityrule: Product State General State Normalization: Composite System Correlations Local vectors Entanglementwitness: The stateisentangledifandonlyif||T||>1

20. 3. Subspace Axiom Axiom 3: All 1-bit systems are equivalent. (Every system constrained to belong to a two-dimensional subspace behavies like a genuine two-level system). Mathematical: Upon any two orthogonal state one can construct a two-dimensional state space that is isomorphic to d-1 sphere 1 11 TwoBits All statesexcept „00“ and „11“ areentangled! 0 00 Single Bit

21. No entanglement if d even Flipingallcoordinatesof a local (Bloch) vectorx: Repeat the same fory: Ey = -y • If d oddand SO(d) areallowedphysicaltransformations, Eisforbidden (detE=-1) • If d even, Ebelongstophysicaltranformations

22. Heuristic argument Locality Axiom Subspace Axiom 1 1 11 10 d-1 d d d-1 d-1 d-1 d-1 0 0 d-1 01 00 = 4-1 + 6(d-1) d + d2 + d d=1 Classical d=3 Quantum

23. Noentanglementif d ≠ 3 See arXiv:0911.0695 „Quantum TheoryandBeyond: Is Entanglement Special?“ B. Dakic & C. Brukner Conclusions: No probabilistic theory other than quantum theory can exhibit entanglement without contradicting one or more of the axioms. → The new theories are crazy, but not crazy enough

24. Thank you for your attention!