Elements of quantum theory from limited information and complementarity Philipp H¨ ohn Perimeter Institute FFP ’14 Marseille July 16 th , 2014 July 16 th , 2014 P. H¨ ohn (PI) 1 / 14
What makes quantum theory special? which physical statements characterize QT within landscape of probabilistic theories? ⇒ wave of QT reconstructions within ‘generalized probabilistic theories’ framework [Hardy, Masanes, M¨ uller, Brukner, Dakic, D’Ariano, Chiribella, Perinotti......] complementary to this: understand quantum theory as framework for information inference [Rovelli, Zeilinger, Brukner, Fuchs, Spekkens,......] ⇒ Can one characterize/derive QT with primacy on information inference? Why useful? 1 gain novel perspective on QT within inference theory space 2 why or why not QT in its present form a fundamental theory July 16 th , 2014 P. H¨ ohn (PI) 2 / 14
Operational setup Observer O interrogating system S with binary questions Q i , i = 1 , . . . each Q i non-trivial 1-bit question (info measure later) O has tested identical S sufficiently often to have gained info about set of possible “states” Bayesian viewpoint: for specific S , O assigns probabilities p i to Q i accord. to his info about particular S , and about possible set of “states” p i encode all O can say about S ⇒ state of S (rel. to O ): collection of p i July 16 th , 2014 P. H¨ ohn (PI) 3 / 14
Question types require symmetry under relabeling ‘yes’ ↔ ‘no’ for any Q i ∃ special state of ‘no information’ p i = 1 2 ∀ i ⇒ call totally mixed state Q i , Q j are: independent if, relative to totally mixed state of S , answer to only Q i gives O no information about answer to Q j (and vice versa) ⇒ p ( Q i , Q j ) = p i · p j factorizes compatible if O may know answers to both simultaneously ⇒ p i , p j can be simultaneously 0 , 1 complementary if knowledge of Q i disallows O to know Q j at the same time (and vice versa) ⇒ p i = 0 , 1, then p j = 1 / 2 July 16 th , 2014 P. H¨ ohn (PI) 4 / 14
Basic postulates (for N qubit systems) for now, forget about QT, postulate: LI (limited information): “ O can acquire maximally N ∈ N independent bits of information about S at the same time.” ∃ Q i , i = 1 , . . . , N (mutually) independent compatible C (complementarity): “ O can always get up to N new (independent) bits of information about S . Whenever O asks a new question he experiences no net loss of information.” ∃ Q ′ i , i = 1 , . . . , N independent compatible but Q i , Q ′ j = i complementary (Postulates motivated by Rovelli’s Relational Quantum Mechanics and work by Brukner/Zeilinger) ⇒ postulates imply elements of qubit QT July 16 th , 2014 P. H¨ ohn (PI) 5 / 14
Number of degrees of freedom How many independent Q i for N qubits? N = 1: only individual Q i , i = 1 , . . . , D 1 ⇒ D 1 =? (know D 1 ≥ 2) July 16 th , 2014 P. H¨ ohn (PI) 6 / 14
Number of degrees of freedom How many independent Q i for N qubits? N = 1: only individual Q i , i = 1 , . . . , D 1 ⇒ D 1 =? (know D 1 ≥ 2) N = 2: 2 D 1 individual Q i qubit 1 qubit 2 Q ′ Q 1 1 vertex: individual question Q i , Q ′ Q 2 Q ′ j 2 Q ′ Q 3 3 . . . . . . . . . Q D 1 Q ′ D 1 July 16 th , 2014 P. H¨ ohn (PI) 6 / 14
Number of degrees of freedom How many independent Q i for N qubits? N = 1: only individual Q i , i = 1 , . . . , D 1 ⇒ D 1 =? (know D 1 ≥ 2) N = 2: 2 D 1 individual Q i + D 2 1 composite questions: Q ij := Q i ↔ Q ′ j “Are answers to Q i and Q ′ j the same?” + ??? qubit 1 qubit 2 Q 11 Q ′ Q 1 1 vertex: individual question Q i , Q ′ j Q ′ Q 2 2 edge: composite question Q ij Q ′ Q 3 3 . . . . . . . . . Q D 1 Q ′ D 1 July 16 th , 2014 P. H¨ ohn (PI) 6 / 14
Number of degrees of freedom How many independent Q i for N qubits? N = 1: only individual Q i , i = 1 , . . . , D 1 ⇒ D 1 =? (know D 1 ≥ 2) N = 2: 2 D 1 individual Q i + D 2 1 composite questions: Q ij := Q i ↔ Q ′ j “Are answers to Q i and Q ′ j the same?” + ??? qubit 1 qubit 2 Q 11 Q ′ Q 1 1 Q 31 vertex: individual question Q i , Q ′ Q 22 j Q ′ Q 2 2 edge: composite question Q ij Q 23 Q ′ Q 3 3 . . . . . . . . . Q D 1 D 1 Q D 1 Q ′ D 1 July 16 th , 2014 P. H¨ ohn (PI) 6 / 14
Number of degrees of freedom How many independent Q i for N qubits? N = 1: only individual Q i , i = 1 , . . . , D 1 ⇒ D 1 =? (know D 1 ≥ 2) N = 2: 2 D 1 individual Q i + D 2 1 composite questions: Q ij := Q i ↔ Q ′ j “Are answers to Q i and Q ′ j the same?” + ??? qubit 1 qubit 2 Q ′ Q 1 1 vertex: individual question Q i , Q ′ j Q ′ Q 2 2 edge: composite question Q ij Q ′ Q 3 3 . . . . . . . . . Q D 1 Q ′ D 1 July 16 th , 2014 P. H¨ ohn (PI) 6 / 14
Compatibility and independence of composite questions assume Specker’s principle: if n Q i are pairwise compatible then they are also mutually compatible Specker’s principle + Complementarity ⇒ { Q ij } D 1 i , j =1 pairwise independent July 16 th , 2014 P. H¨ ohn (PI) 7 / 14
Compatibility and independence of composite questions assume Specker’s principle: if n Q i are pairwise compatible then they are also mutually compatible Specker’s principle + Complementarity ⇒ { Q ij } D 1 i , j =1 pairwise independent Q ij complementary if corresponding edges intersect . . . July 16 th , 2014 P. H¨ ohn (PI) 7 / 14
Compatibility and independence of composite questions assume Specker’s principle: if n Q i are pairwise compatible then they are also mutually compatible Specker’s principle + Complementarity ⇒ { Q ij } D 1 i , j =1 pairwise independent Q ij complementary if corresponding edges intersect Specker + C ⇒ Q ij compatible if corresponding edges non-intersecting (otherwise net loss of information possible) . ⇒ entanglement: > 1 bit in Q ij [Brukner, Zeilinger] . . July 16 th , 2014 P. H¨ ohn (PI) 7 / 14
Compatibility and independence of composite questions assume Specker’s principle: if n Q i are pairwise compatible then they are also mutually compatible Specker’s principle + Complementarity ⇒ { Q ij } D 1 i , j =1 pairwise independent Q ij complementary if corresponding edges intersect Specker + C ⇒ Q ij compatible if corresponding edges non-intersecting (otherwise net loss of information possible) . ⇒ entanglement: > 1 bit in Q ij [Brukner, Zeilinger] . . July 16 th , 2014 P. H¨ ohn (PI) 7 / 14
Compatibility and independence of composite questions assume Specker’s principle: if n Q i are pairwise compatible then they are also mutually compatible Specker’s principle + Complementarity ⇒ { Q ij } D 1 i , j =1 pairwise independent Q ij complementary if corresponding edges intersect Specker + C ⇒ Q ij compatible if corresponding edges non-intersecting (otherwise net loss of information possible) . ⇒ entanglement: > 1 bit in Q ij [Brukner, Zeilinger] . . how many independent Q s for N = 2? ⇒ depends on D 1 July 16 th , 2014 P. H¨ ohn (PI) 7 / 14
Why is the Bloch sphere 3-dim.? [see also M¨ uller, Masanes, v. Weizsaecker,....] Q 11 Logical argument from N = 2 case: Q ii , i = , . . . , D 1 pairwise independent, Q 22 compatible O can acquire answers to all D 1 composites Q ii simultaneously (Specker) Q 33 . . . Q D 1 D 1 July 16 th , 2014 P. H¨ ohn (PI) 8 / 14
Why is the Bloch sphere 3-dim.? [see also M¨ uller, Masanes, v. Weizsaecker,....] Q 11 Logical argument from N = 2 case: Q ii , i = , . . . , D 1 pairwise independent, Q 22 compatible O can acquire answers to all D 1 composites Q ii simultaneously (Specker) Q 33 LI: O cannot know more than N = 2 . . independent bits about S . Q D 1 D 1 July 16 th , 2014 P. H¨ ohn (PI) 8 / 14
Why is the Bloch sphere 3-dim.? [see also M¨ uller, Masanes, v. Weizsaecker,....] Q 11 Logical argument from N = 2 case: Q ii , i = , . . . , D 1 pairwise independent, Q 22 compatible O can acquire answers to all D 1 composites Q ii simultaneously (Specker) Q 33 LI: O cannot know more than N = 2 . . independent bits about S . Q D 1 D 1 ⇒ answers to any two Q ii determine answers to all other Q jj July 16 th , 2014 P. H¨ ohn (PI) 8 / 14
Why is the Bloch sphere 3-dim.? [see also M¨ uller, Masanes, v. Weizsaecker,....] Q 11 Logical argument from N = 2 case: Q ii , i = , . . . , D 1 pairwise independent, Q 22 compatible O can acquire answers to all D 1 composites Q ii simultaneously (Specker) Q 33 LI: O cannot know more than N = 2 . . independent bits about S . Q D 1 D 1 ⇒ answers to any two Q ii determine answers to all other Q jj e.g., truth table for any three Q ii ( a � = b ): Q 11 Q 22 Q 33 ⇒ Q 33 = Q 11 ↔ Q 22 or ¬ ( Q 11 ↔ Q 22 ) 0 1 a 1 0 a 1 1 b 0 0 b July 16 th , 2014 P. H¨ ohn (PI) 8 / 14
Why is the Bloch sphere 3-dim.? [see also M¨ uller, Masanes, v. Weizsaecker,....] Q 11 Logical argument from N = 2 case: Q ii , i = , . . . , D 1 pairwise independent, Q 22 compatible O can acquire answers to all D 1 composites Q ii simultaneously (Specker) Q 33 LI: O cannot know more than N = 2 . . independent bits about S . Q D 1 D 1 ⇒ answers to any two Q ii determine answers to all other Q jj e.g., truth table for any three Q ii ( a � = b ): Q 11 Q 22 Q 33 ⇒ Q 33 = Q 11 ↔ Q 22 or ¬ ( Q 11 ↔ Q 22 ) 0 1 a ⇒ holds for all compatible sets of Q ij : 1 0 a 2 ≤ D 1 ≤ 3 1 1 b 0 0 b July 16 th , 2014 P. H¨ ohn (PI) 8 / 14
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