Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller* and Marius Krumm Departments of Applied Mathematics and Philosophy, UWO Perimeter Institute for Theoretical Physics, Waterloo Outline 1. The conjecture tomographic locality + reversibility QT ) d d 2. Evidence B B d=3 and QT ) 3. Multipartite interaction beyond QT? Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller Outline 1. The conjecture tomographic locality + reversibility QT ) d d 2. Evidence B B d=3 and QT ) 3. Multipartite interaction beyond QT? Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller 1. The conjecture Starting point: convex-operational theory Preparation, transformation, measurement. 1. The conjecture Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller 1. The conjecture Starting point: convex-operational theory Preparation, transformation, measurement. State ! = equivalence class of preparation procedures 1. The conjecture Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller 1. The conjecture Starting point: convex-operational theory Preparation, transformation, measurement. State ! = equivalence class of preparation procedures State space ⌦ = set of all possible states of a given system 1. The conjecture Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller 1. The conjecture Starting point: convex-operational theory Preparation, transformation, measurement. State ! = equivalence class of preparation procedures State space ⌦ = set of all possible states of a given system QT: ΩN = set of N×N density matrices CPT: ΩN = set of prob. distributions (p1,...,pN ). 1. The conjecture Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller 1. The conjecture Any compact convex set (in a real space) is a possible state space: 6 classical quantum "gbit" bit bit FIG. 6: The set of pure states in 3 is connected, but for the cylinder the pure states form twoQ circles. Arbitrary convex Classical "trit" Quantum "trit". FIG. 8: a) The space curve ⌅x(t) modelling pure quantum state space (3-level-system) states is obtained by rotating an equilateral triangle according to Eq.Complicated, (16) —three positions of the triangle 8D! are shown); b) The convex hull C of the curve models the set of all quantum 1. The conjecture states. Composition and quantum theory:FIG. 7: a Thisconjecture, is now the and convex how hull ofit acould single spacefail curve, Markus P. Müller but one cannot inscribe copies of the classical set ∆2 in it. planes. Since the set of pure states is connected this is our best model so far of the set of quantum pure states, we consider the space curve although the likeness is not perfect. It is interesting to think a bit more about the boundary T ⌦x(t)= cos(t) cos(3t), cos(t)sin(3t), sin(t) . (16) of C. There are three flat faces, two triangular ones and − one rectangular. The remaining part of the boundary Note that the curve is closed, ⌦x(t)=⌦x(t +2π),⇥ and be- consists of ruled surfaces: they are curved, but contain longs to the unit sphere, ⌦x(t) = 1. Moreover || || one dimensional faces (straight lines). The boundary of the set shown in Fig. 7 has similar properties. The ruled ⌦x(t) ⌦x(t + 1 2π) = ⌅3 (17) || − 3 || surfaces of C have an analogue in the boundary of the set of quantum states , we have already noted that a for every value of t. Hence every point ⌦x(t) belongs to Q3 generic point in the boundary of 3 belongs to a copy of an equilateral triangle with vertices at (the Bloch ball), arising as theQ intersection of with Q2 Q3 ⌦x(t), ⌦x(t + 1 2π), and ⌦x(t + 2 2π) . a hyperplane. The flat pieces of C have no analogues in 3 3 the boundary of , apart from Bloch balls (rank two) Q3 They span a plane including the z-axis for all times t. and pure states (rank one) no other faces exist. 2π During the time ∆t = 3 this plane makes a full turn Still this model is not perfect: Its set of pure states about the z-axis, while the triangle rotates by the angle has self-intersections. Although it is created by rotating 2π/3 within the plane—so the triangle has returned to a a triangle, the triangles are not cross-sections of C.It congruent position. The curve ⌦x(t) is shown in Fig. 8 a) is not true that every point on the boundary belongs together with exemplary positions of the rotating trian- to a face that touches the largest inscribed sphere, as gle, and Fig. 8 b) shows its convex hull C. This convex it happens for the set of quantum states [17]. Indeed its hull is symmetric under reflections in the (x-y) and (x-z) boundary is not quite what we want it to be, in particular 1. The conjecture Any compact convex set (in a real space) is a possible state space: 6 More formal definition: Set of unnormalized classicalstates is a closed,quantum convex, pointed, generating"gbit" bitcone in a (finite-dim.bit here) real vector space. Transformations, measurements: analogous def's. FIG. 6: The set of pure states in 3 is connected, but for the All herecylinder theis pure math. states form two Qrigoros circles. , but I keep this talk simple. Arbitrary convex Classical "trit" Quantum "trit". FIG. 8: a) The space curve ⌅x(t) modelling pure quantum state space (3-level-system) states is obtained by rotating an equilateral triangle according to Eq.Complicated, (16) —three positions of the triangle 8D! are shown); b) The convex hull C of the curve models the set of all quantum 1. The conjecture states. Composition and quantum theory:FIG. 7: a Thisconjecture, is now the and convex how hull ofit acould single spacefail curve, Markus P. Müller but one cannot inscribe copies of the classical set ∆2 in it. planes. Since the set of pure states is connected this is our best model so far of the set of quantum pure states, we consider the space curve although the likeness is not perfect. It is interesting to think a bit more about the boundary T ⌦x(t)= cos(t) cos(3t), cos(t)sin(3t), sin(t) . (16) of C. There are three flat faces, two triangular ones and − one rectangular. The remaining part of the boundary Note that the curve is closed, ⌦x(t)=⌦x(t +2π),⇥ and be- consists of ruled surfaces: they are curved, but contain longs to the unit sphere, ⌦x(t) = 1. Moreover || || one dimensional faces (straight lines). The boundary of the set shown in Fig. 7 has similar properties. The ruled ⌦x(t) ⌦x(t + 1 2π) = ⌅3 (17) || − 3 || surfaces of C have an analogue in the boundary of the set of quantum states , we have already noted that a for every value of t. Hence every point ⌦x(t) belongs to Q3 generic point in the boundary of 3 belongs to a copy of an equilateral triangle with vertices at (the Bloch ball), arising as theQ intersection of with Q2 Q3 ⌦x(t), ⌦x(t + 1 2π), and ⌦x(t + 2 2π) . a hyperplane. The flat pieces of C have no analogues in 3 3 the boundary of , apart from Bloch balls (rank two) Q3 They span a plane including the z-axis for all times t. and pure states (rank one) no other faces exist. 2π During the time ∆t = 3 this plane makes a full turn Still this model is not perfect: Its set of pure states about the z-axis, while the triangle rotates by the angle has self-intersections. Although it is created by rotating 2π/3 within the plane—so the triangle has returned to a a triangle, the triangles are not cross-sections of C.It congruent position. The curve ⌦x(t) is shown in Fig. 8 a) is not true that every point on the boundary belongs together with exemplary positions of the rotating trian- to a face that touches the largest inscribed sphere, as gle, and Fig. 8 b) shows its convex hull C. This convex it happens for the set of quantum states [17]. Indeed its hull is symmetric under reflections in the (x-y) and (x-z) boundary is not quite what we want it to be, in particular Reversibility and tomographic locality Transformations map states to states and are linear. A transformation T is reversible if T-1 exists and is a transformation, too. 1. The conjecture Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller Reversibility and tomographic locality Transformations map states to states and are linear. A transformation T is reversible if T-1 exists and is a transformation, too. QM: reversible transformations = unitaries, ⇢ U⇢U †. 7! CPT: permutations, (p ,...,p ) (p ,...,p ) 1 n 7! ⇡(1) ⇡(n) 1. The conjecture Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller Reversibility and tomographic locality Transformations map states to states and are linear. A transformation T is reversible if T-1 exists and is a transformation, too. QM: reversible transformations = unitaries, ⇢ U⇢U †. 7! CPT: permutations, (p ,...,p ) (p ,...,p ) 1 n 7! ⇡(1) ⇡(n) Reversible transformations are linear symmetries of the state space. They map pure states to pure states (pure state = extremal point of convex set). 1. The conjecture Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller Reversibility and tomographic locality Reversibility postulate: For every pair of pure states !, ', there is a reversible transformation T such that T ! = '.