Quantum entanglement and contextuality with complexifications of E 8 root system

The gist: This work introduces a new configuration with 120 quantum states derived from the complexification of 240 minimal vectors of an E8 real lattice, providing an analysis of nonlocality and contextuality in quantum mechanics through analogies with the Witting configuration.

Derivation from E8 Root System

The configurations discussed are both derived from the complexification of 240 minimal vectors corresponding to the root system of Lie algebra E8 (Page 1). The paper notes that despite having a different number of states, both configurations can be derived from this common source (Page 1). The symmetry group of the configuration with 5040 transformations is a 4D unitary representation of a double cover of alternating group A7 (Page 2). This representation, denoted as A˜7, is the symmetry group for the configuration Λ˜A7 with 240 complex vectors (Page 2).

Symmetry and State Representation

The symmetry group A˜7 has 5040 transformations (Page 2). The symmetries of Witting configuration correspond to a subgroup VI from the list, while the investigated subgroup has index IV (Page 2). The representation A˜7 is generated by matrices P1 and P2, along with a matrix S of seventh order (Page 3). The group A7 itself represents symmetries of 4D complex configuration Λ˜A7 with 240 vectors (Page 3). Each vector in this configuration corresponds to one state, and each state is included together with its opposite vector (Page 3).

Bases and Contextuality Analysis

The configuration has some analogy with the Witting polytope because natural maps into a configuration of 240 8D real vectors correspond to the root system of Lie algebra E8 after some orthogonal transformation (Page 2). The Witting configuration includes 40 rays or quantum states with 40 different bases (orthogonal tetrads) and Λ˜A7 corresponds to 120 states and 210 bases (Page 2). The number of bases for the 120 states is found to be 210 (Page 7). A diagram with links corresponding to orthogonal states is known as the Kochen-Specker graph, and all these bases can be found using search for maximal 4-cliques in this graph (Page 7).

Entanglement Properties

The paper considers an entangled state omegai = (Page 10). For a general entangled state, it can be described by a matrix J as omegaJ i = Tr(JJ∗) X jk Jjkjiki (Page 10). A natural example of measurement bases for both systems is obtained by the same unitary transformation A with property (A ⊗ A)omegaJ i = ±omegaJ i =⇒ AJ¯ = ±JA (Page 10). Subgroups like A˜5 and S˜5 are relevant for analyzing entanglement, as the orbit of S˜5 includes all 120 vectors, whereas A˜5 has two orbits with 60 vectors (Page 11). The J-opposite state is defined by an antiunitary transformation up to unsignificant phase ψJ i ≃ Jψ¯i (Page 17).

Measurement Schemes and Bases

The analysis explores measurement schemes similar to those discussed in Ref. [2] by applying transformations from A˜5 or S˜5 (Page 10). The number of bases between the 210 available bases is limited to 30 when using only a subset of the symmetry group (Page 11). Alternative entangled states can be constructed using matrices J C = CJC¯−1 = CJCT, yielding different cosets CS˜5 (Page 12). The comparison between two configurations involves checking property Eq. (14) for generators of some subgroup, as products also respect the same relation (Page 12). Maximal size of nonorthogonal cliques found for the ΛA7 configuration is 24, and maximal size of nonorthogonal states required for a noncontextual classical model is 30 (Page 9).

Conclusion

The new configuration with 120 quantum states provides analogies with Witting configuration and geometry of dodecahedron (Page 16). The polynomial z squared + z + 1 = 0 and z squared + z + 2 = 0 may be used for constructing coordinates in the configurations (Page 16). Maximal size of nonorthogonal cliques is size 7, while minimal size is size 4 (Page 16). The equal bases respected by an appropriate subgroup suggest possibility to apply for entangled configurations the same transformation to both systems in a simpler measurement scheme (Page 16). The analysis suggests that the second measurement scheme can be used instead of the first one for contextuality analysis (Page 9).

--- Page 1 ---

Quantum entanglement and contextuality with complexifications of E8 root system Alexander Yu. Vlasov Abstract The Witting configuration with 40 complex rays was suggested as a possible reformulation of Penrose model with two spin-3/2 systems based on geometry of dodecahedron and used for analysis of nonlocality and contextuality in quantum mechanics. Yet another configuration with 120 quantum states is considered in presented work. Despite of different number of states both configurations can be derived from complexification of 240 minimal vectors of 8D real lattice corresponding to root system of Lie algebra E8. An analysis of properties of suggested configuration of quantum states is provided using many analogies with properties of Witting configuration.

--- Page 2 ---

Symmetry group VI from the list and the subgroup investigated in presented work has index IV. Such a group with 5040 transformations is a 4D unitary representation of a double cover of alternating group A7 [6], i.e., group of 2520 even permutations of seven elements. The unitary equivalent representation of the group with different choice of generators is denoted A˜7 in Section 2. For such representation A˜7 is the symmetry group of a configuration Λ˜A7 with 240 complex vectors introduced in Section 3. The configuration has some analogy with Witting polytope, because in both cases natural maps into configuration of 240 8D real vectors correspond to the root system of Lie algebra E8 after some orthogonal transformation. Despite of the equivalence in 8D real space the complex configurations provide quite different models of system with quantum states in 4D Hilbert space. The Witting configuration includes 40 rays or quantum states with 40 different bases (orthogonal tetrads) and Λ˜A7 corresponds to 120 states and 210 bases.

--- Page 3 ---

The matrices P1 = (Page 3) P2 = (Page 3). The matrices P1 and P2 generate group of 24 transformations isomorphic with double cover A˜4 of group A4 of even permutation of four elements in quite natural way as permutations of four basic states. An isomorphism of the 4D unitary representation of group A˜7 with different choice of generators is less trivial, but it can be generated by P1 and P2 together with matrix S = (Page 3). where ι = i√7, ι' = 1 + ι/2, ι'' = 1 + 3ι/2. Let us note ι2 = 1/7, ι'2 = 2/7 and ι''2 = 4/7.

--- Page 4 ---

The configuration can be produced from a basic vector (1,0,0,0) by application different transformations from A˜7. Each vector is included together with an opposite one and it corresponds to configuration ΛA7 with 120 different quantum states represented on Table 1 with notation for coordinates from Eq. (3) and indexing explained below. Complex Witting polytope also has 240 vertices, but only 40 different quantum states can be obtained from such vectors due to six common phase multipliers ±ωk, k = 0,..., 5.

--- Page 5 ---

Despite of such difference both configurations considered as 240 8D real vectors up to some orthogonal transformation are equivalent with E8 root system or minimal vectors of E8 lattice denoted further simply as E8. For Witting polytope relation with the 8D real uniform polytope 421 (equivalent with E8) was mentioned by Coxeter [5, 8]. The construction of a map between realifications of both complex configurations is provided below. For certainty let us fix 40 vectors producing whole Witting configuration by multiplication on six ‘phases’ ±ωk Eq. (5), k = 0,..., 5. Let us note, that four vectors Eq. (6b) with µ = ν = 0 have pure imaginary coordinates i√3(0, 1, -1, 1), i√3(1, 0, -1, -1), i√3(1, -1, 0, 1), and i√3(1, 1, 1, 0) (Page 5).

--- Page 6 ---

All states are normalized on unit and two copies of Table 2 with opposite signs produces complete set Λ˜A7 with 240 complex vectors. Index t in last column of Table 1 denotes type of four states in a row similar with used for description of lattice for group A˜7 denoted as 2A7 in Ref. [9, p.

1. This work enables the construction of complex configurations derived from complexification of 240 minimal vectors of 8D real lattice corresponding to root system of Lie algebra E8, which can serve as a new mathematical framework for modeling quantum systems beyond standard Hilbert space representations.

2. The improved AI system can perform nonlocality and contextuality analysis by providing an analysis of properties of suggested configuration of quantum states using many analogies with properties of Witting configuration, allowing it to test hypotheses about quantum mechanics using the structure derived from the E8 root system.

3. The AI can model entanglement in two configurations by utilizing specific unitary transformations, such as those satisfying (A ⊗ A)omegaJ i = ±omegaJ i =⇒ AJ¯ = ±JA for generators of subgroups like A˜5 and S˜5, enabling the prediction of pair of such ‘J-opposite states for entangled systems.

4. The system can analyze measurement bases by applying transformations from group A˜7 to relate measurement bases between two systems, as described by (A ⊗ B)omegaJ i = omegaJ i =⇒ BJ = JA, B ¯ = JAJ¯ −1, leading to a more efficient scheme where different transformations are used for preparation of measurement bases.

5. The AI can utilize the derived 210 different measurement bases and the properties of N-cliques to identify potential noncontextual classical models, as the paper notes that if a noncontextual model were to reproduce quantum mechanics, it would require finding maximal 4-cliques in these configurations.