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 +2),⇥ 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 2) = ⌅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 2), and ⌦x(t + 2 2) . 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. 2 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 2/3 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 +2),⇥ 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 2) = ⌅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 2), and ⌦x(t + 2 2) . 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. 2 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 2/3 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 ! = '.

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 ! = '.

• Very natural: reversible time evolution / (quantum) circuits should exhaust the state space. True in QT + CPT. • Brings in the power of group theory. • Enforces some symmetry in the state space:

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 ! = '.

• Very natural: reversible time evolution / (quantum) circuits should exhaust the state space. True in QT + CPT. • Brings in the power of group theory. • Enforces some symmetry in the state space:

1. The conjecture Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller Reversibility and tomographic locality

1. The conjecture Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller 4

release button outcomes x andx ¯ physical system

Reversibility and tomographic locality

! T x

Alice+Bob are given (many copies) of a state. Task: determine the state via measurements (tomography). True in QT+CPT: can be done via local measurements. Figure 1. General experimental setup. From left to right, there are the preparation, transformation and measurement devices. As soon as the release 1. The conjecture Compositionbutton and quantum theory: a conjecture, is and how it pressed, could fail the Markus preparationP. Müller device outputs a physical system in the state specified by the knobs. The next device performs the transformation specified by its knobs (which in particular can ‘do nothing’). The device on the right carries out the measurement specified by its knobs, and the outcome (x or x) is indicated by the corresponding light. ¯

2. Generalized probabilistic theories

In CPT there can always be a joint probability distribution for all random variables under consideration. The framework of generalized probabilistic theories (GPTs), also called the convex operational framework, generalizes this by allowing the possibility of random variables that cannot have a joint probability distribution or cannot be simultaneously measured (such as noncommuting observables in QT). This framework assumes that at some level there is a classical reality, where it makes sense to talk about experimentalists performing basic operations such as preparations, mixtures, measurements and counting the relative frequencies of outcomes. These are the primary concepts of this framework. It also provides a unified way for all GPTs to represent states, transformations and measurements. A particular GPT specifies which of these are allowed, but it does not tell their correspondence to actual experimental setups. On its own, a GPT can still make nontrivial predictions such as: the maximal violation of a Bell inequality [1], the complexity-theoretic computational power [2, 18] and, in general, all information-theoretic properties of the theory [6]. The framework of GPTs can be stated in different ways, but all lead to the same formalism [3–9]. This formalism is presented in this section at a very basic level, providing some elementary results without proofs.

2.1. States Definition of a system. We associate with a setup like figure 1 a system if, for each configuration of the preparation, transformation and measurement devices, the relative frequencies of the outcomes tend to a unique probability distribution (in the large sample limit). The probability of a measurement outcome x is denoted by p(x). This outcome can be associated with a binary measurement that tells whether x happens or not (this second event x has probability p(x) 1 p(x)). The above definition of a system allows one to associate with¯ each preparation¯ procedure= a list of probabilities of the outcomes of all the measurements that can be carried out on a system. As we show in section 4.3, our requirements imply that all these probabilities p(x) are determined by a finite set of them; the smallest such set is used to

New Journal of Physics 13 (2011) 063001 (http://www.njp.org/) 4

release button outcomes x andx ¯ physical system

Reversibility and tomographic locality

! T x

Alice+Bob are given (many copies) of a state. Task: determine the state via measurements (tomography). True in QT+CPT: can be done via local measurements. Figure 1. General experimental setup. From left to right, there are the preparation, transformation and measurement devices. As soon as the release 1. The conjecture Compositionbutton and quantum theory: a conjecture, is and how it pressed, could fail the Markus preparationP. Müller device outputs a physical system in the state specified by the knobs. The next device performs the transformation specified by its knobs (which in particular can ‘do nothing’). The device on the right carries out the measurement specified by its knobs, and the outcome (x or x) is indicated by the corresponding light. ¯

2. Generalized probabilistic theories

In CPT there can always be a joint probability distribution for all random variables under consideration. The framework of generalized probabilistic theories (GPTs), also called the convex operational framework, generalizes this by allowing the possibility of random variables that cannot have a joint probability distribution or cannot be simultaneously measured (such as noncommuting observables in QT). This framework assumes that at some level there is a classical reality, where it makes sense to talk about experimentalists performing basic operations such as preparations, mixtures, measurements and counting the relative frequencies of outcomes. These are the primary concepts of this framework. It also provides a unified way for all GPTs to represent states, transformations and measurements. A particular GPT specifies which of these are allowed, but it does not tell their correspondence to actual experimental setups. On its own, a GPT can still make nontrivial predictions such as: the maximal violation of a Bell inequality [1], the complexity-theoretic computational power [2, 18] and, in general, all information-theoretic properties of the theory [6]. The framework of GPTs can be stated in different ways, but all lead to the same formalism [3–9]. This formalism is presented in this section at a very basic level, providing some elementary results without proofs.

2.1. States Definition of a system. We associate with a setup like figure 1 a system if, for each configuration of the preparation, transformation and measurement devices, the relative frequencies of the outcomes tend to a unique probability distribution (in the large sample limit). The probability of a measurement outcome x is denoted by p(x). This outcome can be associated with a binary measurement that tells whether x happens or not (this second event x has probability p(x) 1 p(x)). The above definition of a system allows one to associate with¯ each preparation¯ procedure= a list of probabilities of the outcomes of all the measurements that can be carried out on a system. As we show in section 4.3, our requirements imply that all these probabilities p(x) are determined by a finite set of them; the smallest such set is used to

New Journal of Physics 13 (2011) 063001 (http://www.njp.org/) 4

release button outcomes x andx ¯ physical system

Reversibility and tomographic locality

! T x

Alice+Bob are given (many copies) of a state. Task: determine the state via measurements (tomography). True in QT+CPT: can be done via local measurements.

Tomographic locality: Every state of a composite system is completely Figurecharacterized by the 1.correlationsGeneral of measurements experimental setup. From left to right, there are the preparation,on the individual components. transformation and measurement devices. As soon as the release 1. The conjecture Compositionbutton and quantum theory: a conjecture, is and how it pressed, could fail the Markus preparationP. Müller device outputs a physical system in the state specified by the knobs. The next device performs the transformation specified by its knobs (which in particular can ‘do nothing’). The device on the right carries out the measurement specified by its knobs, and the outcome (x or x) is indicated by the corresponding light. ¯

2. Generalized probabilistic theories

In CPT there can always be a joint probability distribution for all random variables under consideration. The framework of generalized probabilistic theories (GPTs), also called the convex operational framework, generalizes this by allowing the possibility of random variables that cannot have a joint probability distribution or cannot be simultaneously measured (such as noncommuting observables in QT). This framework assumes that at some level there is a classical reality, where it makes sense to talk about experimentalists performing basic operations such as preparations, mixtures, measurements and counting the relative frequencies of outcomes. These are the primary concepts of this framework. It also provides a unified way for all GPTs to represent states, transformations and measurements. A particular GPT specifies which of these are allowed, but it does not tell their correspondence to actual experimental setups. On its own, a GPT can still make nontrivial predictions such as: the maximal violation of a Bell inequality [1], the complexity-theoretic computational power [2, 18] and, in general, all information-theoretic properties of the theory [6]. The framework of GPTs can be stated in different ways, but all lead to the same formalism [3–9]. This formalism is presented in this section at a very basic level, providing some elementary results without proofs.

2.1. States Definition of a system. We associate with a setup like figure 1 a system if, for each configuration of the preparation, transformation and measurement devices, the relative frequencies of the outcomes tend to a unique probability distribution (in the large sample limit). The probability of a measurement outcome x is denoted by p(x). This outcome can be associated with a binary measurement that tells whether x happens or not (this second event x has probability p(x) 1 p(x)). The above definition of a system allows one to associate with¯ each preparation¯ procedure= a list of probabilities of the outcomes of all the measurements that can be carried out on a system. As we show in section 4.3, our requirements imply that all these probabilities p(x) are determined by a finite set of them; the smallest such set is used to

New Journal of Physics 13 (2011) 063001 (http://www.njp.org/) Reversibility and tomographic locality

More mathematical perspective:

• Given two state spaces ⌦ A and ⌦ B there are always infinitely many possible composites ⌦AB. • Only constraints: there are notions of "product states" and "product measurements". • Tomographic locality equivalent to the following property of state-space-carrying vector spaces: V = V V . AB A ⌦ B Tomographic locality: Every state of a composite system is completely characterized by the correlations of measurements on the individual components.

1. The conjecture Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller The conjecture

Conjecture:

If some ⌦AB is a locally tomographic composite of some ⌦A and ⌦B, and all three state spaces satisfy reversibility, and there is at least one reversible transformation T = T T , AB 6 A ⌦ B then ⌦AB is a (subspace of a) quantum state space.

1. The conjecture Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller The conjecture

Conjecture:

If some ⌦AB is a locally tomographic composite of some ⌦A and ⌦B, and all three state spaces satisfy reversibility, and there is at least one reversible transformation T = T T , AB 6 A ⌦ B then ⌦AB is a (subspace of a) quantum state space.

• If true: Gives very clear idea of "why the quantum?". • If wrong (which I actually hope): Physically interesting: counterexamples describe possible alternative/new physics. Mathematically interesting: interplay convex geometry/ group theory/ multilinear algebra. Computersciency interesting: contrast that new theory to quantum computation!

1. The conjecture 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?

1. The conjecture 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?

2. Evidence Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller 2. Evidence: composing bit balls

For many reasons, it's natural to expect that bits (binary alternatives) are described by Euclidean ball state spaces:

2. Evidence Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller 2. Evidence: composing bit balls

For many reasons, it's natural to expect that bits (binary alternatives) are described by Euclidean ball state spaces:

...

d =1 d =2 d =3 classical quantum bit bit

d =2, 5, 9 are bits in quantum theory over R, H, O.

2. Evidence Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller 2. Evidence: composing bit balls

Ll. Masanes, MM, D. Pérez-García, R. Augusiak, Entanglement and the three- dimensionality of the Bloch ball, J. Math. Phys. 55, 122203 (2014).

2. Evidence Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller 2. Evidence: composing bit balls

Ll. Masanes, MM, D. Pérez-García, R. Augusiak, Entanglement and the three- dimensionality of the Bloch ball, J. Math. Phys. 55, 122203 (2014). Consider two d-dimensional "Bloch" balls:

Bd Bd

A B

2. Evidence Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller 2. Evidence: composing bit balls

Ll. Masanes, MM, D. Pérez-García, R. Augusiak, Entanglement and the three- dimensionality of the Bloch ball, J. Math. Phys. 55, 122203 (2014). Consider two d-dimensional "Bloch" balls:

Bd Bd

A B

Assume tomographic locality, and reversibility for A, B, AB. group of reversible ) transformations A = B G G d d 1 must be transitive on @B = S .

2. Evidence Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller 2. Evidence: composing bit balls

Ll. Masanes, MM, D. Pérez-García, R. Augusiak, Entanglement and the three- dimensionality of the Bloch ball, J. Math. Phys. 55, 122203 (2014). Consider two d-dimensional "Bloch" balls: abstract groups d H d d SO(d) 3, 4, 5 ... B B V SU(d/2) 4, 6, 8 ... V V⇤ U(d/2) 2, 4, 6, 8 ... V V⇤ A B Sp(d/4) 8, 12, 16 ... V V⇤ Sp(d/4) U(1) 8, 12, 16 ... Assume tomographic locality, ⇥ V V⇤ Sp(d/4) SU(2) 4, 8, 12 ... irreducible and reversibility for A, B, AB. ⇥ G 7 group of reversible 2 V ) Spin(7) 8 transformations A = B V G G d d 1 must be transitive on @B = S . Spin(9) 16 V 2. Evidence Composition and quantum theory: a conjecture,TABLE and how I.it could The fail first column is the list of abstract groups Markus (or P. Müller families of groups parametrized by d)

that are transitive on the unit sphere within Rd. The second column contains the values of d for which this holds. The third column schematically specifies which representation of each abstract group corresponds to the matrix group ,where is the fundamental representation and H V V⇤ its dual (both irreducible). In cases where describing the representation is complicated we just mention whether it is irreducible.

where J =(i ) 1 and 1 is the n n identity matrix. For the definition of G see35, 2 ⌦ n n ⇥ 2 for the definition of Spin(n)see29. The fundamental representation is the defining one V (12-15). According to Table IV A, the representation for SO(d), denoted ,isthe H HSO(d) fundamental , hence =SO(d). The representation ⇤ makes use of a standard V HSO(d) V V trick to generate a real representation for a group of complex matrices. The particular map is:

n n 2n 2n C ⇥ R ⇥ ! Q 1 reQ +(i ) imQ. (16) 7! 2 ⌦ 2 ⌦

To see that this is a homomorphism, note that the real matrix (i2)behavesastheimag- inary unity (i )2 = 1 . This specifies the representation for the abstract groups 2 2 H SU(d/2), U(d/2), Sp(d/4), denoted , , . The group SO(d)withd =2 HSU(d/2) HU(d/2) HSp(d/4) is not in Table IV A because SO(2) = , and we choose to include it in the U(d/2) HU(1) family because SO(2) is reducible, while SO(d)ford 3 not. Another coincidence is 17 2. Evidence: composing bit balls

Ll. Masanes, MM, D. Pérez-García, R. Augusiak, Entanglement and the three- dimensionality of the Bloch ball, J. Math. Phys. 55, 122203 (2014). Consider two d-dimensional "Bloch" balls: abstract groups d H d d SO(d) 3, 4, 5 ... B B V SU(d/2) 4, 6, 8 ... V V⇤ U(d/2) 2, 4, 6, 8 ... V V⇤ A B Sp(d/4) 8, 12, 16 ... V V⇤ Sp(d/4) U(1) 8, 12, 16 ... Assume tomographic locality, ⇥ V V⇤ Sp(d/4) SU(2) 4, 8, 12 ... irreducible and reversibility for A, B, AB. ⇥ G 7 Additional assumption: 2 V Spin(7) 8 AB is a connected group. G V Spin(9) 16 V 2. Evidence Composition and quantum theory: a conjecture,TABLE and how I.it could The fail first column is the list of abstract groups Markus (or P. Müller families of groups parametrized by d)

that are transitive on the unit sphere within Rd. The second column contains the values of d for which this holds. The third column schematically specifies which representation of each abstract group corresponds to the matrix group ,where is the fundamental representation and H V V⇤ its dual (both irreducible). In cases where describing the representation is complicated we just mention whether it is irreducible.

where J =(i ) 1 and 1 is the n n identity matrix. For the definition of G see35, 2 ⌦ n n ⇥ 2 for the definition of Spin(n)see29. The fundamental representation is the defining one V (12-15). According to Table IV A, the representation for SO(d), denoted ,isthe H HSO(d) fundamental , hence =SO(d). The representation ⇤ makes use of a standard V HSO(d) V V trick to generate a real representation for a group of complex matrices. The particular map is:

n n 2n 2n C ⇥ R ⇥ ! Q 1 reQ +(i ) imQ. (16) 7! 2 ⌦ 2 ⌦

To see that this is a homomorphism, note that the real matrix (i2)behavesastheimag- inary unity (i )2 = 1 . This specifies the representation for the abstract groups 2 2 H SU(d/2), U(d/2), Sp(d/4), denoted , , . The group SO(d)withd =2 HSU(d/2) HU(d/2) HSp(d/4) is not in Table IV A because SO(2) = , and we choose to include it in the U(d/2) HU(1) family because SO(2) is reducible, while SO(d)ford 3 not. Another coincidence is 17 2. Evidence: composing bit balls

Ll. Masanes, MM, D. Pérez-García, R. Augusiak, Entanglement and the three- dimensionality of the Bloch ball, J. Math. Phys. 55, 122203 (2014).

d d B B Assumptions: Tomographic locality and reversibility for A, B, AB; ⌦ AB is connected (thus a Lie group). A B G

2. Evidence Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller 2. Evidence: composing bit balls

Ll. Masanes, MM, D. Pérez-García, R. Augusiak, Entanglement and the three- dimensionality of the Bloch ball, J. Math. Phys. 55, 122203 (2014).

d d B B Assumptions: Tomographic locality and reversibility for A, B, AB; ⌦ AB is connected (thus a Lie group). A B G

Theorem. Among all dimensions d and all groups A, there are only the following possibilities: G The trivial solution: = . • GAB GA ⌦ GB d = 3, = SO(3) (i.e. the quantum bit), PU(4), and • GA GAB ' ⌦AB is equivalent to the two-qubit quantum state space. In particular, continuous reversible interaction is only possible for d = 3, in standard complex two-qubit quantum theory.

2. Evidence Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller 2. Evidence: composing bit balls

Proof sketch: • Use Lie algebra properties to get generators that look "simple": 1 1 X gAB X0 := (A B)X(A B )dA dB gAB. 2 ) GA⌦GB ⌦ ⌦ 2 • Positivity of probabilitiesR imposes constraints on those generators.

Theorem. Among all dimensions d and all groups A, there are only the following possibilities: G The trivial solution: = . • GAB GA ⌦ GB d = 3, = SO(3) (i.e. the quantum bit), PU(4), and • GA GAB ' ⌦AB is equivalent to the two-qubit quantum state space. In particular, continuous reversible interaction is only possible for d = 3, in standard complex two-qubit quantum theory.

2. Evidence Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller 2. Evidence: composing bit balls

Proof sketch: 11 • Use Lie algebra properties to get generators that look "simple": for all x, y d with x = y = 1. By considering other 1 form1⇢ U⇢U must be contained in the global trans- X RgAB X0 := (A B)X(A B )dA dB † gAB. 2 | | | | A B 7! circuits of this2 kind,) we obtain aG long⌦G list of⌦ constraint ⌦formation group2 AB AB. H ✓ G equalities• Positivity and inequalities of probabilities whichR must imposes be satisfied constraints by all onThe those orbit generators. of this group on pure product states gen- global Lie algebra elements X. erates all pure quantum states, and one can show [55] that there cannot be any additional non-quantum states. Thus, we have recovered the state space of quantum the- ory on two qubits. Due to positivity, all e↵ects must be quantum e↵ects; in the noisy case (i.e. c =1/2 or a<1), not all quantum e↵ects may actually be implementable6 – that is, we might have a restricted set of measurements. We have thus proven: FIG. 7: Circuit model which yields constraints for the global Lie algebra elements X. We prepare a pure product state Theorem 3. From Postulates 1–4, it follows that A B !x !y , apply the transformation exp(tX), and perform a the state space of two direction bits is two-qubit product measurement A B . Since this gives probability 1 Mx My quantum state space (i.e. the set of 4 4 density for t = 0, and probabilities cannot be larger than 1 for other matrices), and time evolution is given⇥ by a one- (small) t, this implies that the derivative2. Evidence at t =0mustvanish, parameter group of unitaries, ⇢ U(t)⇢U(t)†. andComposition the second and quantum derivative theory: must a conjecture, be non-positive. and how it could fail Markus P. Müller 7!

As a simple consequence, there exists some 4 4 Hermi- Surprisingly, as shown in Appendix A and in [54], if tian matrix H such that U(t)=exp( iHt), i.e.⇥ a Hamil- d =3, then the only matrices X which satisfy all con- tonian which generates time evolution according to the straints6 are those of the form X = XA + XB. And Schr¨odinger equation. these elements generate non-interacting time evolution of product form: exp(tX)=exp(tXA)exp(tXB). Thus, if d = 3, AB contains only product transformations, and Postulate6 H 4 cannot be satisfied. VII. CONCLUSIONS Theorem 2. From Postulates 1–4 it follows that the spatial dimension must be d = 3. We have derived two facts about physics from information-theoretic postulates: the three- dimensionality of space [57], and the fact that proba- The main reason why d = 3 is special becomes visible bilities of measurement outcomes for some systems are by inspection of the proof in [54]. It boils down to the described by quantum theory. In order to do this, we group-theoretic fact that (at least for d 3) the subgroup assumed that there exist “reasonable” physical systems of SO(d) which fixes a given vector (that is, SO(d 1)) which, in a certain sense, carry minimal amounts of is Abelian only if d = 3. In other words, the fact that directional information. rotations commute in two dimensions, but not in higher Our result supports and clarifies the point of view that dimensions is the main reason why d =3survives.The the geometric structure of spacetime and the probabilis- cases d = 1 and d = 2 are special as well, but are ruled tic structure of quantum theory are closely intertwined, out in the proof for other reasons. similar in spirit to [1–4, 58–60]. As one can see in Fig. 3, It remains to show that we actually get quantum the- this conclusion becomes particularly obvious in the con- ory for two direction bits if d = 3. We already know that text of convex state spaces. This interrelation is not only the dimension of the global state space is dim ⌦AB = axiomatic, but also operational: as we have shown in (d + 1)(d + 1) 1 = 15, which agrees with the number of Sec. V, observers can measure – or even define – physical real parameters in a complex 4 4 density matrix. Thus, angles by measuring probabilities. ⇥ we can embed ⌦AB in the real space of Hermitian 4 4- Furthermore, these findings suggest exploring possible matrices of unit trace. Now we have global Lie algebra⇥ generalizations: the approach to construct state spaces elements X hAB that are not just sums of local genera- from physical symmetry properties [70], together with tors, i.e. X =2XA +XB. However, as shown in [55], these minimality assumptions, might reproduce quantum sys- elements are6 still highly restricted: they generate unitary tems of higher spin, or even physically interesting non- conjugations, i.e. transformations of the form ⇢ U⇢U †. quantum state spaces that have so far remained unex- By Postulate 4, at least one of these unitaries7! must be plored. entangling. Moreover, all local unitary transformations In summary, there seem to be two possible interpreta- are possible (in the ball representation, these are the ro- tions of the results in this paper. First, the results might tations in SO(3)). It is a well-known fact from quantum simply be mathematical coincidence, without any deep computation [56] that a set of unitaries of this kind gen- physical reason underlying them. This is perfectly con- erates the set of all unitaries – that is, every map of the ceivable; in this case, the main contribution of this paper 2. Evidence: composing bit balls

Proof sketch: 11 • Use Lie algebra properties to get generators that look "simple": for all x, y d with x = y = 1. By considering other 1 form1⇢ U⇢U must be contained in the global trans- X RgAB X0 := (A B)X(A B )dA dB † gAB. 2 | | | | A B 7! circuits of this2 kind,) we obtain aG long⌦G list of⌦ constraint ⌦formation group2 AB AB. H ✓ G equalities• Positivity and inequalities of probabilities whichR must imposes be satisfied constraints by all onThe those orbit generators. of this group on pure product states gen- global Lie algebra elements X. erates all pure quantum states, and one can show [55] that there cannot be any additional non-quantum states. Thus, we have recovered the state space of quantum the- ory on two qubits. Due to positivity, all e↵ects must be quantum e↵ects; in the noisy case (i.e. c =1/2 or a<1), not all quantum e↵ects may actually be implementable6 – that is, we might have a restricted set of measurements. We have thus proven:

FIG. 7: CircuitA modelA which yields constraintsA B for thetX globalA B Lie algebra elementsx (!x )=0X. We preparex a pure producty e ( state!x !y ) Theoremt=0=0 3. From Postulates 1–4, it follows that A B M ) M ⌦ M ⌦ !x !y , apply the transformation exp(tX), and perform a the state space of two direction bits is two-qubit product measurement A B . Sinceis a this local gives minimum. probability 1 Mx My quantum state space (i.e. the set of 4 4 density for t = 0, and probabilities cannot be larger than 1 for other matrices), and time evolution is given⇥ by a one- (small) t, this implies that the derivative2. Evidence at t =0mustvanish, parameter group of unitaries, ⇢ U(t)⇢U(t)†. andComposition the second and quantum derivative theory: must a conjecture, be non-positive. and how it could fail Markus P. Müller 7!

As a simple consequence, there exists some 4 4 Hermi- Surprisingly, as shown in Appendix A and in [54], if tian matrix H such that U(t)=exp( iHt), i.e.⇥ a Hamil- d =3, then the only matrices X which satisfy all con- tonian which generates time evolution according to the straints6 are those of the form X = XA + XB. And Schr¨odinger equation. these elements generate non-interacting time evolution of product form: exp(tX)=exp(tXA)exp(tXB). Thus, if d = 3, AB contains only product transformations, and Postulate6 H 4 cannot be satisfied. VII. CONCLUSIONS Theorem 2. From Postulates 1–4 it follows that the spatial dimension must be d = 3. We have derived two facts about physics from information-theoretic postulates: the three- dimensionality of space [57], and the fact that proba- The main reason why d = 3 is special becomes visible bilities of measurement outcomes for some systems are by inspection of the proof in [54]. It boils down to the described by quantum theory. In order to do this, we group-theoretic fact that (at least for d 3) the subgroup assumed that there exist “reasonable” physical systems of SO(d) which fixes a given vector (that is, SO(d 1)) which, in a certain sense, carry minimal amounts of is Abelian only if d = 3. In other words, the fact that directional information. rotations commute in two dimensions, but not in higher Our result supports and clarifies the point of view that dimensions is the main reason why d =3survives.The the geometric structure of spacetime and the probabilis- cases d = 1 and d = 2 are special as well, but are ruled tic structure of quantum theory are closely intertwined, out in the proof for other reasons. similar in spirit to [1–4, 58–60]. As one can see in Fig. 3, It remains to show that we actually get quantum the- this conclusion becomes particularly obvious in the con- ory for two direction bits if d = 3. We already know that text of convex state spaces. This interrelation is not only the dimension of the global state space is dim ⌦AB = axiomatic, but also operational: as we have shown in (d + 1)(d + 1) 1 = 15, which agrees with the number of Sec. V, observers can measure – or even define – physical real parameters in a complex 4 4 density matrix. Thus, angles by measuring probabilities. ⇥ we can embed ⌦AB in the real space of Hermitian 4 4- Furthermore, these findings suggest exploring possible matrices of unit trace. Now we have global Lie algebra⇥ generalizations: the approach to construct state spaces elements X hAB that are not just sums of local genera- from physical symmetry properties [70], together with tors, i.e. X =2XA +XB. However, as shown in [55], these minimality assumptions, might reproduce quantum sys- elements are6 still highly restricted: they generate unitary tems of higher spin, or even physically interesting non- conjugations, i.e. transformations of the form ⇢ U⇢U †. quantum state spaces that have so far remained unex- By Postulate 4, at least one of these unitaries7! must be plored. entangling. Moreover, all local unitary transformations In summary, there seem to be two possible interpreta- are possible (in the ball representation, these are the ro- tions of the results in this paper. First, the results might tations in SO(3)). It is a well-known fact from quantum simply be mathematical coincidence, without any deep computation [56] that a set of unitaries of this kind gen- physical reason underlying them. This is perfectly con- erates the set of all unitaries – that is, every map of the ceivable; in this case, the main contribution of this paper 2. Evidence: composing bit balls

Proof sketch: 11 • Use Lie algebra properties to get generators that look "simple": for all x, y d with x = y = 1. By considering other 1 form1⇢ U⇢U must be contained in the global trans- X RgAB X0 := (A B)X(A B )dA dB † gAB. 2 | | | | A B 7! circuits of this2 kind,) we obtain aG long⌦G list of⌦ constraint ⌦formation group2 AB AB. H ✓ G equalities• Positivity and inequalities of probabilities whichR must imposes be satisfied constraints by all onThe those orbit generators. of this group on pure product states gen- global Lie algebra elements X. erates all pure quantum states, and one can show [55] that there cannot be any additional non-quantum states. Thus, we have recovered the state space of quantum the- • If d =3 then it follows 6 ory on two qubits. Due to positivity, all e↵ects must be X = XquantumA 1B e+↵ects;1A inX theB. noisy case (i.e. c =1/2 or a<1), not⌦ all quantum⌦ e↵ects may actually be implementable6 – that is, we might have a restricted set of measurements. We have thus proven:

FIG. 7: CircuitA modelA which yields constraintsA B for thetX globalA B Lie algebra elementsx (!x )=0X. We preparex a pure producty e ( state!x !y ) Theoremt=0=0 3. From Postulates 1–4, it follows that A B M ) M ⌦ M ⌦ !x !y , apply the transformation exp(tX), and perform a the state space of two direction bits is two-qubit product measurement A B . Sinceis a this local gives minimum. probability 1 Mx My quantum state space (i.e. the set of 4 4 density for t = 0, and probabilities cannot be larger than 1 for other matrices), and time evolution is given⇥ by a one- (small) t, this implies that the derivative2. Evidence at t =0mustvanish, parameter group of unitaries, ⇢ U(t)⇢U(t)†. andComposition the second and quantum derivative theory: must a conjecture, be non-positive. and how it could fail Markus P. Müller 7!

As a simple consequence, there exists some 4 4 Hermi- Surprisingly, as shown in Appendix A and in [54], if tian matrix H such that U(t)=exp( iHt), i.e.⇥ a Hamil- d =3, then the only matrices X which satisfy all con- tonian which generates time evolution according to the straints6 are those of the form X = XA + XB. And Schr¨odinger equation. these elements generate non-interacting time evolution of product form: exp(tX)=exp(tXA)exp(tXB). Thus, if d = 3, AB contains only product transformations, and Postulate6 H 4 cannot be satisfied. VII. CONCLUSIONS Theorem 2. From Postulates 1–4 it follows that the spatial dimension must be d = 3. We have derived two facts about physics from information-theoretic postulates: the three- dimensionality of space [57], and the fact that proba- The main reason why d = 3 is special becomes visible bilities of measurement outcomes for some systems are by inspection of the proof in [54]. It boils down to the described by quantum theory. In order to do this, we group-theoretic fact that (at least for d 3) the subgroup assumed that there exist “reasonable” physical systems of SO(d) which fixes a given vector (that is, SO(d 1)) which, in a certain sense, carry minimal amounts of is Abelian only if d = 3. In other words, the fact that directional information. rotations commute in two dimensions, but not in higher Our result supports and clarifies the point of view that dimensions is the main reason why d =3survives.The the geometric structure of spacetime and the probabilis- cases d = 1 and d = 2 are special as well, but are ruled tic structure of quantum theory are closely intertwined, out in the proof for other reasons. similar in spirit to [1–4, 58–60]. As one can see in Fig. 3, It remains to show that we actually get quantum the- this conclusion becomes particularly obvious in the con- ory for two direction bits if d = 3. We already know that text of convex state spaces. This interrelation is not only the dimension of the global state space is dim ⌦AB = axiomatic, but also operational: as we have shown in (d + 1)(d + 1) 1 = 15, which agrees with the number of Sec. V, observers can measure – or even define – physical real parameters in a complex 4 4 density matrix. Thus, angles by measuring probabilities. ⇥ we can embed ⌦AB in the real space of Hermitian 4 4- Furthermore, these findings suggest exploring possible matrices of unit trace. Now we have global Lie algebra⇥ generalizations: the approach to construct state spaces elements X hAB that are not just sums of local genera- from physical symmetry properties [70], together with tors, i.e. X =2XA +XB. However, as shown in [55], these minimality assumptions, might reproduce quantum sys- elements are6 still highly restricted: they generate unitary tems of higher spin, or even physically interesting non- conjugations, i.e. transformations of the form ⇢ U⇢U †. quantum state spaces that have so far remained unex- By Postulate 4, at least one of these unitaries7! must be plored. entangling. Moreover, all local unitary transformations In summary, there seem to be two possible interpreta- are possible (in the ball representation, these are the ro- tions of the results in this paper. First, the results might tations in SO(3)). It is a well-known fact from quantum simply be mathematical coincidence, without any deep computation [56] that a set of unitaries of this kind gen- physical reason underlying them. This is perfectly con- erates the set of all unitaries – that is, every map of the ceivable; in this case, the main contribution of this paper Outline

1. The conjecture tomographic locality + reversibility QT )

d d 2. Evidence B B d=3 and QT )

3. Multipartite interaction beyond QT?

2. Evidence 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?

3. Multipartite interaction? Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller 3. Multipartite interaction?

In QT, multipartite interactions can be decomposed into pairs: =

3. Multipartite interaction? Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller 3. Multipartite interaction?

In QT, multipartite interactions can be decomposed into pairs: =

But maybe this is false in many "nice" theories beyond QT?

3. Multipartite interaction? Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller 3. Multipartite interaction?

In QT, multipartite interactions can be decomposed into pairs: The classical limit of a physical theory and the dimensionality of space

Borivoje Daki´c1, 2 and Caslavˇ Brukner1, 3 1Vienna Center for Quantum Science= and Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria 2Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543 3Institute of Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria InThe the operational classical approach limit of to general a physical probabilistic theory theor andies one the distinguishes dimensional two spaces,ity theof spacestate space of the “elementary systems” and the physical space in which “laboratory devices” are embedded. Each of those spaces has its own dimension– the minimal number1, 2 of realˇ parameters (coordinates)1, 3 needed to specify the stateBut maybe this is false of system or a point within theBorivoje physical space.Daki´c Withinand anCaslav operational Brukner framework to a physical theory, the two dimensions1 coincideVienna Center in a natural for Quantum way under Science the following and Technology “closeness” (VCQ), requirement: Faculty of the Physics, dynamics of a single elementary system canUniversity be generated of Vienna, by the invariant Boltzmanngasse interaction 5, between A-1090 theVienna, system Aus andtria the “macroscopic trans-in many "nice" theories 2Centreformation for Quantumdevice” that Technologies, itself is described National from University within the of theory Singapore, in the macroscopic 3 Science Drive (classical) 2, Singapore limit. Quantum 117543 mechanics fulfils this3Institute requirement of Quantum since an Optics arbitrary and unita Quantumry transformation Information of an (IQOQI elementary), system (spin-1/2beyond QT? or qubit) can beAustrian generated Academy by the pairwise of Sciences, invariant Boltzmanngasse interaction between 3, A-1090 the spin Vienna, and the Austria constituents of a large coherent state (“classical magnetic field”). Both the spin state space and the “classical field” are then embedded Inin the Euclidean operational three-dimensional approach to general space. probabilistic Can we have theor a geneies oneral probabilistic distinguishes theory, two spaces, other than the state quantum space the- of theory, “elementary in which the systems” elementary and the system physical (“generalized space in spin”) which an “ladthe“classicalfields”generatingitsdynamicsareboratory devices” are embedded. Each of those spacesembedded has its in own a higher-dimensional dimension– the minimal physical number space? of We real show para thmetersat as long (coordinates) as the interaction needed to is specify pairwise, the this state is ofimpossible, system or a and point quantum within mechanics the physical and space. the three-dimension Within an operalational space remain framework the only to a solution. physical However, theory, the having two dimensionsmulti-particle coincide interactions in a natural and a way generalized under the notion following of “classical “closeness” field” may requirement: open up such the adynamics possibility. of a single elementary system can be generated by the invariant interaction between the system and the “macroscopic trans- formation device” that itself is described from within the theory in the macroscopic (classical) limit. Quantum mechanicsI. fulfils INTRODUCTION this requirement since an arbitrary unitarynot transformation further analyzed. of an elementary The “position” system (spin-1 of the/2 switch at the or qubit) can be generated by the pairwise invariant interactiontransformation between the device spin and or the the constituents record on theof a observation large screen coherent state (“classical magnetic field”). Both the spin statehave space only and an the abstract “classical meaning field” are and then are embedded not linked to the con- “Physicalin space the Euclidean is not a three-dimensional space of states” writesspace. Can Bengtsson we have a general probabilistic theory, other than quantum the- in his article entitled “Why is space three dimensional?”1.In- cepts of position, time, direction, or energy of “traditional” ory, in which the elementary system (“generalized spin”) anphysicsdthe“classicalfields”generatingitsdynamicsare (or to use Barnum’s words “the full, meaty physical deed, althoughembedded the state in a space higher-dimensional dimension for physical a macroscopic space? We show that as long as the interaction31 is pairwise, this is object is exponentially large (in the number of object’s con- theory” is still missing ). As3. aMultipartite result of the interaction? reconstructions impossible, and quantum mechanics and the three-dimensionofal quantum space remain theory, the only one derivessolution. a However, finite-dimensional, having or count- stituents), wemulti-particleComposition still find ourselves interactions and quantum organizing and a generalized theory: data into notiona a conjecture, three- of “classical and field” how may it opencould up suchfail a possibility. Markus P. Müller dimensional manifold called “space”. Why is this discrep- able infinite-dimensional, Hilbert space as an operationally ancy? Can there be more dimensions? In past different ap- testable, abstract formalism concerned with predictions offre- proaches have beenI. taken INTRODUCTION to show that the three-dimensional notquency further counts analyzed. in future The experiments “position” with of no the appointment switch at the of space is special, such as bio-topological argument2,stabil- transformationconcrete physical device labels or the to record physical on states the observation or measurement screen ity of planet orbits2,stabilityofatoms3 or elementary parti- haveoutcomes. only an In abstract standard meaning textbook and approach are not linked to quantum to the con-me- “Physical space4 is not a space of states” writes Bengtsson chanics this appointment is “inherited” from classical me- cle properties .Theexistenceofextradimensionshasbeen1 cepts of position, time, direction, or energy of “traditional” in his article entitled “Why is space three dimensional?” .In- chanics and is formalized through the first quantization – the deed,proposed although as thea possibility state space for dimension physics beyond for a macroscopic the standard physics (or to use Barnum’s words “the full, meaty physical model5–10. theory”set of explicit is still missing rules that31). relate As a classical result of phase the reconstructions variables with object is exponentially large (in the number of object’s con- quantum-mechanical operators. However, these rules lack an of quantum theory, one derives a finite-dimensional, or count- stituents),In this we work still find we willourselves address organizing the questions data into given a th aboveree- immediate operational justification. This calls for a “comple- able infinite-dimensional, Hilbert space as an operationally dimensionalwithin the manifoldoperational called approach “space”. to general Why probabilistic is this discrep- theo- tion” of operational approaches to quantum mechanics with 11,12,16 testable, abstract formalism concerned with predictions offre- ancy?ries Can there.Therethebasicingredientsofthetheoryareprimi- be more dimensions? In past different ap- the “meaty physics”. Our work can be understood as a step in arXiv:1307.3984v1 [quant-ph] 15 Jul 2013 quency counts in future experiments with no appointment of proachestive laboratory have been procedures taken to by show which that physical the three-dimensional systems are pre- this direction. spacepared, is special, transformed such and as measuredbio-topological by laboratory argument devices,2,stabil- but concrete physical labels to physical states or measurement itythe of planetsystems orbits are not2,stabilityofatomsnecessarily described3 or by elementary quantum theory.parti- outcomes.In an operational In standard approach textbook one approach interprets to parameters quantum that me- General probabilistic theories are shown to share many fea- cle properties4.Theexistenceofextradimensionshasbeen chanicsdescribe this physical appointment states, transformations, is “inherited” from and measureme classicalnts, me- tures that one previously have expected to be uniquely quan- proposed as a possibility for physics beyond the standard chanicsas the parameters and is formalized that specify through the the configurations first quantization of macro- – the tum, such as probabilistic predictions for individual outcomes, model5–10. setscopic of explicit instruments rules in that physical relate space classical by which phase the variables state is prwithe- the impossibility of copying unknown states (no cloning)14,or quantum-mechanicalpared, transformed, andoperators. measured. However, Within these this rules approach lack a itn In this work we will address13,58 the questions given above violation of Bell’s inequalities .Whythennatureobeys immediateis natural operational to assume the justification. state space This and calls the physical for a “comp spacele- withinquantum the operational mechanics approach rather than to general other probabilistic probabilistic theory? theo- 11,12,16 tion”to be of isomorphic operational to approaches each other. to quantumThe isomorphism mechanics of with the riesRecently,.Therethebasicingredientsofthetheoryareprimi- there have been several approaches, answering this thetwo “meaty spaces physics”. is realized Our in work quantum can be mechanics understood for as athe step ele- in arXiv:1307.3984v1 [quant-ph] 15 Jul 2013 tivequestion laboratory by proceduresreconstructing by which quantum physical theory systems from a areplausible pre- thismentary direction. directional degree of freedom (spin-1/2). The state pared,set of transformed axioms that and demarcate measured phenomena by laboratory that are devices, exclusively but space of the spin is a three-dimensional unit ball (the Bloch thequantum systems from are not thosenecessarily that are common described to by more quantum general theory. proba- ball)In an and operational its dimension approach and the one symmetry interprets coincide parameters with those that Generalbilistic probabilistic theories15–30. theories are shown to share many fea- describeof the Euclidian physical (non-relativistic) states, transformations, three-dimensional and measureme spacents, in turesIn that probabilistic one previously theories have the expected macroscopic to be laboratory uniquely devi quan-ces aswhich the parameters classical macroscopic that specify instruments the configurations are embedded. of macro- This tum,are such standardly as probabilistic assumed predictions to be classically for individual describable, outcomes, but are scopicwas first instruments pointed out in physical by von space Weizs¨acker by which who the writes state32 is:“It pre- the impossibility of copying unknown states (no cloning)14,or pared, transformed, and measured. Within this approach it violation of Bell’s inequalities13,58.Whythennatureobeys is natural to assume the state space and the physical space quantum mechanics rather than other probabilistic theory? to be isomorphic to each other. The isomorphism of the Recently, there have been several approaches, answering this two spaces is realized in quantum mechanics for the ele- question by reconstructing quantum theory from a plausible mentary directional degree of freedom (spin-1/2). The state set of axioms that demarcate phenomena that are exclusively space of the spin is a three-dimensional unit ball (the Bloch quantum from those that are common to more general proba- ball) and its dimension and the symmetry coincide with those bilistic theories15–30. of the Euclidian (non-relativistic) three-dimensional space in In probabilistic theories the macroscopic laboratory devices which classical macroscopic instruments are embedded. This are standardly assumed to be classically describable, but are was first pointed out by von Weizs¨acker who writes32:“It 3. Multipartite interaction?

In QT, multipartite interactions can be decomposed into pairs: The classical limit of a physical theory and the dimensionality of space

Borivoje Daki´c1, 2 and Caslavˇ Brukner1, 3 1Vienna Center for Quantum Science= and Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria 2Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543 3Institute of Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria InThe the operational classical approach limit of to general a physical probabilistic theory theor andies one the distinguishes dimensional two spaces,ity theof spacestate space of the “elementary systems” and the physical space in which “laboratory devices” are embedded. Each of those spaces has its own dimension– the minimal number1, 2 of realˇ parameters (coordinates)1, 3 needed to specify the stateBut maybe this is false of system or a point within theBorivoje physical space.Daki´c Withinand anCaslav operational Brukner framework to a physical theory, the two dimensions1 coincideVienna Center in a natural for Quantum way under Science the following and Technology “closeness” (VCQ), requirement: Faculty of the Physics, dynamics of a single elementary system canUniversity be generated of Vienna, by the invariant Boltzmanngasse interaction 5, between A-1090 theVienna, system Aus andtria the “macroscopic trans-in many "nice" theories 2Centreformation for Quantumdevice” that Technologies, itself is described National from University within the of theory Singapore, in the macroscopic 3 Science Drive (classical) 2, Singapore limit. Quantum 117543 mechanics fulfils this3Institute requirement of Quantum since an Optics arbitrary and unita Quantumry transformation Information of an (IQOQI elementary), system (spin-1/2beyond QT? or qubit) can beAustrian generated Academy by the pairwise of Sciences, invariant Boltzmanngasse interaction between 3, A-1090 the spin Vienna, and the Austria constituents of a large coherent state (“classical magnetic field”). Both the spin state space and the “classical field” are then embedded Inin the Euclidean operational three-dimensional approach to general space. probabilistic Can we have theor a geneies oneral probabilistic distinguishes theory, two spaces, other than the state quantum space the- of theory, “elementary in which the systems” elementary and the system physical (“generalized space in spin”) which an “ladthe“classicalfields”generatingitsdynamicsareboratory devices” are embedded. Each of those spacesembedded has its in own a higher-dimensional dimension– the minimal physical number space? of We real show para thmetersat as long (coordinates) as the interaction needed to is specify pairwise, the this state is ofimpossible, system or a and point quantum within mechanics the physical and space. the three-dimension Within an operalational space remain framework the only to a solution. physical However, theory, the having two dimensionsmulti-particle coincide interactions in a natural and a way generalized under the notion following of “classical “closeness” field” may requirement: open up such the adynamics possibility. of a single elementary system can be generated by the invariant interaction between the system and the “macroscopic trans- formation device” that itself is described from within the theory in the macroscopic (classical) limit. Quantum mechanicsI. fulfils INTRODUCTION this requirement since an arbitrary unitarynot transformation further analyzed. of an elementary The “position” system (spin-1 of the/2 switch at the or qubit) can be generated by the pairwise invariant interactiontransformation between the device spin and or the the constituents record on theof a observation large screen coherentInteraction state (“classical magnetic among field”). Both the(d spin-1) statehave spacemany only and an the abstract “classicald-balls? meaning field” are and then are embedded not linked to the con- “Physicalin space the Euclidean is not a three-dimensional space of states” writesspace. Can Bengtsson we have a general probabilistic theory, other than quantum the- in his article entitled “Why is space three dimensional?”1.In- cepts of position, time, direction, or energy of “traditional” ory, in which the elementary system (“generalized spin”) anphysicsdthe“classicalfields”generatingitsdynamicsare (or to use Barnum’s words “the full, meaty physical deed, althoughembedded the state in a space higher-dimensional dimension for physical a macroscopic space? We show that as long as the interaction31 is pairwise, this is object is exponentially large (in the number of object’s con- theory” is still missing ). As3. aMultipartite result of the interaction? reconstructions impossible, and quantum mechanics and the three-dimensionofal quantum space remain theory, the only one derivessolution. a However, finite-dimensional, having or count- stituents), wemulti-particleComposition still find ourselves interactions and quantum organizing and a generalized theory: data into notiona a conjecture, three- of “classical and field” how may it opencould up suchfail a possibility. Markus P. Müller dimensional manifold called “space”. Why is this discrep- able infinite-dimensional, Hilbert space as an operationally ancy? Can there be more dimensions? In past different ap- testable, abstract formalism concerned with predictions offre- proaches have beenI. taken INTRODUCTION to show that the three-dimensional notquency further counts analyzed. in future The experiments “position” with of no the appointment switch at the of space is special, such as bio-topological argument2,stabil- transformationconcrete physical device labels or the to record physical on states the observation or measurement screen ity of planet orbits2,stabilityofatoms3 or elementary parti- haveoutcomes. only an In abstract standard meaning textbook and approach are not linked to quantum to the con-me- “Physical space4 is not a space of states” writes Bengtsson chanics this appointment is “inherited” from classical me- cle properties .Theexistenceofextradimensionshasbeen1 cepts of position, time, direction, or energy of “traditional” in his article entitled “Why is space three dimensional?” .In- chanics and is formalized through the first quantization – the deed,proposed although as thea possibility state space for dimension physics beyond for a macroscopic the standard physics (or to use Barnum’s words “the full, meaty physical model5–10. theory”set of explicit is still missing rules that31). relate As a classical result of phase the reconstructions variables with object is exponentially large (in the number of object’s con- quantum-mechanical operators. However, these rules lack an of quantum theory, one derives a finite-dimensional, or count- stituents),In this we work still find we willourselves address organizing the questions data into given a th aboveree- immediate operational justification. This calls for a “comple- able infinite-dimensional, Hilbert space as an operationally dimensionalwithin the manifoldoperational called approach “space”. to general Why probabilistic is this discrep- theo- tion” of operational approaches to quantum mechanics with 11,12,16 testable, abstract formalism concerned with predictions offre- ancy?ries Can there.Therethebasicingredientsofthetheoryareprimi- be more dimensions? In past different ap- the “meaty physics”. Our work can be understood as a step in arXiv:1307.3984v1 [quant-ph] 15 Jul 2013 quency counts in future experiments with no appointment of proachestive laboratory have been procedures taken to by show which that physical the three-dimensional systems are pre- this direction. spacepared, is special, transformed such and as measuredbio-topological by laboratory argument devices,2,stabil- but concrete physical labels to physical states or measurement itythe of planetsystems orbits are not2,stabilityofatomsnecessarily described3 or by elementary quantum theory.parti- outcomes.In an operational In standard approach textbook one approach interprets to parameters quantum that me- General probabilistic theories are shown to share many fea- cle properties4.Theexistenceofextradimensionshasbeen chanicsdescribe this physical appointment states, transformations, is “inherited” from and measureme classicalnts, me- tures that one previously have expected to be uniquely quan- proposed as a possibility for physics beyond the standard chanicsas the parameters and is formalized that specify through the the configurations first quantization of macro- – the tum, such as probabilistic predictions for individual outcomes, model5–10. setscopic of explicit instruments rules in that physical relate space classical by which phase the variables state is prwithe- the impossibility of copying unknown states (no cloning)14,or quantum-mechanicalpared, transformed, andoperators. measured. However, Within these this rules approach lack a itn In this work we will address13,58 the questions given above violation of Bell’s inequalities .Whythennatureobeys immediateis natural operational to assume the justification. state space This and calls the physical for a “comp spacele- withinquantum the operational mechanics approach rather than to general other probabilistic probabilistic theory? theo- 11,12,16 tion”to be of isomorphic operational to approaches each other. to quantumThe isomorphism mechanics of with the riesRecently,.Therethebasicingredientsofthetheoryareprimi- there have been several approaches, answering this thetwo “meaty spaces physics”. is realized Our in work quantum can be mechanics understood for as athe step ele- in arXiv:1307.3984v1 [quant-ph] 15 Jul 2013 tivequestion laboratory by proceduresreconstructing by which quantum physical theory systems from a areplausible pre- thismentary direction. directional degree of freedom (spin-1/2). The state pared,set of transformed axioms that and demarcate measured phenomena by laboratory that are devices, exclusively but space of the spin is a three-dimensional unit ball (the Bloch thequantum systems from are not thosenecessarily that are common described to by more quantum general theory. proba- ball)In an and operational its dimension approach and the one symmetry interprets coincide parameters with those that Generalbilistic probabilistic theories15–30. theories are shown to share many fea- describeof the Euclidian physical (non-relativistic) states, transformations, three-dimensional and measureme spacents, in turesIn that probabilistic one previously theories have the expected macroscopic to be laboratory uniquely devi quan-ces aswhich the parameters classical macroscopic that specify instruments the configurations are embedded. of macro- This tum,are such standardly as probabilistic assumed predictions to be classically for individual describable, outcomes, but are scopicwas first instruments pointed out in physical by von space Weizs¨acker by which who the writes state32 is:“It pre- the impossibility of copying unknown states (no cloning)14,or pared, transformed, and measured. Within this approach it violation of Bell’s inequalities13,58.Whythennatureobeys is natural to assume the state space and the physical space quantum mechanics rather than other probabilistic theory? to be isomorphic to each other. The isomorphism of the Recently, there have been several approaches, answering this two spaces is realized in quantum mechanics for the ele- question by reconstructing quantum theory from a plausible mentary directional degree of freedom (spin-1/2). The state set of axioms that demarcate phenomena that are exclusively space of the spin is a three-dimensional unit ball (the Bloch quantum from those that are common to more general proba- ball) and its dimension and the symmetry coincide with those bilistic theories15–30. of the Euclidian (non-relativistic) three-dimensional space in In probabilistic theories the macroscopic laboratory devices which classical macroscopic instruments are embedded. This are standardly assumed to be classically describable, but are was first pointed out by von Weizs¨acker who writes32:“It 3. Multipartite interaction?

Marius Ernst HagenJoint Krumm work (in progress, unpublished) Resident PhD Student, Perimeter Institute for Theoretical Physics, Waterloo ON, Canada, and PhD Candidate, Departmentwith ofMarius Applied Mathematics, Krumm University of Western Ontario, London ON, Canada [email protected] Born 13.10.1990 in Worms, Germany

Education

Perimeter Institute for Theoretical Physics Waterloo ON, Canada • Resident PhD Student Since October 2016 University of Western Ontario London ON, Canada • PhD Candidate, Department of Applied Mathematics Since September 2015 Heidelberg University Heidelberg, Germany • M.Sc. Physics 2012 - 2015 – Master Thesis: Thermodynamics and the Structure of Quantum Theory as a Generalized Probabilistic Theory – Final grade 1.0 (Very good, best possible grade in German university system) Heidelberg University Heidelberg, Germany • B.Sc. Physics 2009 - 2012 – Bachelor Thesis: Flussgleichungs-Methode fur¨ tridiagonale Hamilton-Operatoren – Final grade 1.0 (Very good) 3. Multipartite interaction? High School: Saarlouiser Gymnasium am Stadtgarten Saarlouis, Germany Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller • High School Diploma: Abitur 2001 - 2009 – Final grade 1.1 (Very good)

Awards & Honors

Otto-Haxel Prize 2015 of Heidelberg University for the best Master thesis in theoretical physics • DPG-Buchpreis 2009 - Book Award of the German Physical Society for high school achievements in physics • Mathematik-Preis 2009 - Mathematics award of Saarland University for high school achievements in • mathematics DMV-Abiturpreis 2009 - Award of the German Mathematical Society for high school achievements in • mathematics

Publications

M. Krumm, H. Barnum, J. Barrett, M. P. Muller,¨ Thermodynamics and the structure of quantum theory, • submitted preprint: http://arxiv.org/abs/1608.04461

H. Barnum, J. Barrett, M. Krumm, M. P. Muller,¨ Entropy, majorization and thermodynamics in general • probabilistic theories, Proceedings QPL 2015, EPTCS 195, 2015, pp. 43-58, http://arxiv.org/abs/1508.03107 3. Multipartite interaction?

Marius Ernst HagenJoint Krumm work (in progress, unpublished) Resident PhD Student, Perimeter Institute for Theoretical Physics, Waterloo ON, Canada, and PhD Candidate, Departmentwith ofMarius Applied Mathematics, Krumm University of Western Ontario, London ON, Canada [email protected] Born 13.10.1990 in Worms, Germany Theorem: There is no tomographically local interaction among Education n 2 many d-ball state spaces, if d 5, 7, 9, 11, 13,... 2 { } andPerimeter if Institute the group for Theoretical of Physics local reversible transformationsWaterloo ON, Canada is SO(d). • Resident PhD Student Since October 2016 University of Western Ontario London ON, Canada • PhD Candidate, Department of Applied Mathematics Since September 2015 Heidelberg University Heidelberg, Germany • M.Sc. Physics 2012 - 2015 – Master Thesis: Thermodynamics and the Structure of Quantum Theory as a Generalized Probabilistic Theory – Final grade 1.0 (Very good, best possible grade in German university system) Heidelberg University Heidelberg, Germany • B.Sc. Physics 2009 - 2012 – Bachelor Thesis: Flussgleichungs-Methode fur¨ tridiagonale Hamilton-Operatoren – Final grade 1.0 (Very good) 3. Multipartite interaction? High School: Saarlouiser Gymnasium am Stadtgarten Saarlouis, Germany Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller • High School Diploma: Abitur 2001 - 2009 – Final grade 1.1 (Very good)

Awards & Honors

Otto-Haxel Prize 2015 of Heidelberg University for the best Master thesis in theoretical physics • DPG-Buchpreis 2009 - Book Award of the German Physical Society for high school achievements in physics • Mathematik-Preis 2009 - Mathematics award of Saarland University for high school achievements in • mathematics DMV-Abiturpreis 2009 - Award of the German Mathematical Society for high school achievements in • mathematics

Publications

M. Krumm, H. Barnum, J. Barrett, M. P. Muller,¨ Thermodynamics and the structure of quantum theory, • submitted preprint: http://arxiv.org/abs/1608.04461

H. Barnum, J. Barrett, M. Krumm, M. P. Muller,¨ Entropy, majorization and thermodynamics in general • probabilistic theories, Proceedings QPL 2015, EPTCS 195, 2015, pp. 43-58, http://arxiv.org/abs/1508.03107 3. Multipartite interaction?

Marius Ernst HagenJoint Krumm work (in progress, unpublished) Resident PhD Student, Perimeter Institute for Theoretical Physics, Waterloo ON, Canada, and PhD Candidate, Departmentwith ofMarius Applied Mathematics, Krumm University of Western Ontario, London ON, Canada [email protected] Born 13.10.1990 in Worms, Germany Theorem: There is no tomographically local interaction among Education n 2 many d-ball state spaces, if d 5, 7, 9, 11, 13,... 2 { } andPerimeter if Institute the group for Theoretical of Physics local reversible transformationsWaterloo ON, Canada is SO(d). • Resident PhD Student Since October 2016 University of Western Ontario London ON, Canada • PhD Candidate, Department of Applied Mathematics Since September 2015 Heidelberg University Heidelberg, Germany • DisprovesM.Sc. Physics part of Brukner's and Dakic's conjecture. 2012 - 2015 – Master Thesis: Thermodynamics and the Structure of Quantum Theory as a Generalized Probabilistic Theory – Final grade 1.0 (Very good, best possible grade in German university system) Heidelberg University Heidelberg, Germany • B.Sc. Physics 2009 - 2012 – Bachelor Thesis: Flussgleichungs-Methode fur¨ tridiagonale Hamilton-Operatoren – Final grade 1.0 (Very good) 3. Multipartite interaction? High School: Saarlouiser Gymnasium am Stadtgarten Saarlouis, Germany Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller • High School Diploma: Abitur 2001 - 2009 – Final grade 1.1 (Very good)

Awards & Honors

Otto-Haxel Prize 2015 of Heidelberg University for the best Master thesis in theoretical physics • DPG-Buchpreis 2009 - Book Award of the German Physical Society for high school achievements in physics • Mathematik-Preis 2009 - Mathematics award of Saarland University for high school achievements in • mathematics DMV-Abiturpreis 2009 - Award of the German Mathematical Society for high school achievements in • mathematics

Publications

M. Krumm, H. Barnum, J. Barrett, M. P. Muller,¨ Thermodynamics and the structure of quantum theory, • submitted preprint: http://arxiv.org/abs/1608.04461

H. Barnum, J. Barrett, M. Krumm, M. P. Muller,¨ Entropy, majorization and thermodynamics in general • probabilistic theories, Proceedings QPL 2015, EPTCS 195, 2015, pp. 43-58, http://arxiv.org/abs/1508.03107 Summary

Conjecture:

If some ⌦AB is a locally tomographic composite of some ⌦A and ⌦B, and all three state spaces satisfy reversibility, and there is at least one reversible transformation T = T T , AB 6 A ⌦ B then ⌦AB is a (subspace of a) quantum state space.

• Counterexamples would be extremely interesting for physics, mathematics and computer science. • Evidence for conjecture: only pairs of quantum Bloch balls can interact reversibly (singling out d=3 and the quantum state space). Ll. Masanes, MM, D. Pérez-García, R. Augusiak, Entanglement and the three- dimensionality of the Bloch ball, J. Math. Phys. 55, 122203 (2014).

• Hope for "counterex".: multipartite interaction. Being killed now. :(

Summary Composition and quantum theory: a conjecture, and how it could fail Markus P. Müller