Clifford Algebras, Algebraic Spinors, Quantum Information and Applications

Home , Spacetime algebra

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In relativistic quantum mechanics,1, 2 Clifford algebras naturally appear through Dirac matrices. Covariant bilinears, chirality, CPT symmetries are some of the mathematical objects that play a fundamental role in the theory, built in terms of the spinors and generators of Dirac algebra. The ubiquitous character of Clifford algebras suggests the possibility of using them as a link between quantum computation3, 4 and high energy physics. In fact, recently Martinez et al 5 performed an experimental demonstration of a simulation of a network gauge theory using a low-q-trapped quantum ion computer. A relationship between the particle-antiparticle creation mechanism and the entanglement of the system, measured through logarithmic negativity, was also observed. Moreover there are several works in which Clifford algebra techniques are employed for quantum computing.6–14 Another field in which Clifford algebras and spinors play a key role is supersymmetry.15, 16 An approach through spacetime algebra was obtained in.17 A supersymmetric generalization of qubits has been proposed by18 pari passu to supersymmetric generalizations of entanglement measures through the concept of superdeterminants. Super Bell and super GHZ states have been defined. Local Lie Group Extensions Associated with Classical Commu- nication Local Operations (LOCC), [SU(2)]n, and Stochastic Classical Com- munication Local Operations (SLOCC) were performed with the [Osp(1 2)]n n | and [uOsp(1 2)] supergroups, respectively. In the M theory,| 19 Clifford algebras and octonions have been employed20, 21to derive an eleven-dimensional supersymmetric algebra called M-algebra, a non-trivial extension of the Poincarésuperalgebra. In eleven dimensions, the supercharge Q is a pseudo-Majorana supercharge corresponds to a spinor with 32 components.15 Indeed, possible connections to M-theory are investi- gated by Borsten22 In this work, we use algebraic spinors as main toll for the description of quantum information and their applications. Firstly, in the section 2 we present an algebraic formulation for multipartite q-bit systems in terms of the algebraic spinors in Cl1,3. In section 3 we derive a method for obtaining quantum logic gates in an algebraic setting. In section 4 we presents chirality, charge conjugation, parity and time reversal operators in algebraic formulation. In section 5 we show how the supersymmetry can be explored in the context of spacetime algebra.17 Section 6 contains an approach to obtaining superalgebra by exploring its connection to octonions, qubits and

1 entanglement. Finally, in section 7, we have the conclusions and perspectives.

2 Algebraic spinors

In this section we will review some mathematical facts concerning algebraic spinors23 and we show how to use them to describe qubits. So let us consider A an associative algebra. An left ideal I is a subspace such that AI I L L ⊂ L and analogously, a vector subspace IR is a right ideal if IRA IR. An element ε A is said idempotent if ε2 = ε and ε = 0. An idempotent⊂ is primitive ∈ 6 if there are no idempotents ε1 and ε2 such that ε1ε2 = 0 and ε1 + ε2 = ε. The minimal left ideals in the Clifford algebra Clp,q can be written in the form Clp,qε. Algebraic spinors are defined as elements of minimal left ideals. It is worth mentioning that an alternative way is to use q-bits in terms of operatorial spinors6, 7 . In this paper we will make use of Clifford algebra Cl3,0

σiσj + σjσi =2δij, i, j =1, 2, 3, (1) where δij = 0 if i = j and δij = 1, otherwise. Therefore, σiσj = σjσi, i = j 2 6 − 6 and σi = 1. Any element Γ Cl3,0 can be written as a linear combination of 1 (scalar), σ , σ , σ (vectors),∈ σ σ , σ σ , σ σ , (bivectors) and σ σ σ { } 1 2 3} { 1 2 2 3 1 3} 1 2 3 (trivector). We will also use the Cliﬀord algebra Cl1,3 (or spacetime algebra), deﬁned as

γµγν + γνγµ =2gµν, (2) where g is the Minkowiski metric with signature (+ ). Therefore, µν − −− γ γ = γ γ ,µ = ν and γ2 =1,γ2 = γ2 = γ2 = 1. Any element Γ Cl µ ν − ν µ 6 0 1 2 3 − ∈ 1,3 can be written as a linear combination of 1 (scalar), γ0,γ1,γ2,γ3 (vectors), γ γ ,γ γ ,γ γ ,γ γ ,γ γ ,γ γ , (bivectors),{ } ιγ ,{ ιγ , ιγ , ιγ }(pseu- { 0 1 0 2 0 3 1 2 1 3 2 3} { 0 1 2 3} dovectors) and ι γ0γ1γ2γ3 (pseudoscalar). { ≡ } 24 With the Cl1,3 algebra, the Dirac equation can be written as

ψιγ γ = mψγ (3) ∇ 3 0 0 µ where = γ ∂µ. Due to this fact we will use this algebra as a starting point for future∇ developments in relativistic quantum information theory.

2 Now we will build the isomorphism Cl Cl+ , ξ : Cl Cl+ , given 3,0 ≃ 1,3 3,0 → 1,3 by ζ(σi)= γiγ0. We have the primitive idempotents 1 ε = (1 + σ ) Cl 2 3 ∈ 3,0 1 P = (1 + γ γ ) Cl+ . (4) 2 3 0 ∈ 1,3 Thus we have the following correspondence: 0 σ ε γ γ P, 1 σ ε γ γ P, | i ↔ 3 ↔ 3 0 | i ↔ 1 ↔ 1 0 (5) i 0 σ1σ2ε ιγ3γ0P, i 1 σ2σ3ε ιγ1γ0P. | i ↔ ↔ | i ↔ ↔ Therefore we can write the qubit as an algebraic spinor in Cl3,0:

Ψ = (α1σ3 + α2Iσ3 + α3σ1 + α4Iσ1)ε 1 = (α σ + α σ σ + α σ + α σ σ ) (1 + σ ), (6) 1 3 2 1 2 3 1 4 2 3 2 3 where I = σ σ σ and α R, i =1, 2, 3, 4. The corresponding state in the 1 2 3 i ∈ Hilbert space of (6) is given by Ψ =(α + iα ) 0 +(α + iα ) 1 . (7) | i 1 2 | i 3 4 | i Following this prescription we will consider multipartite q-bits as tensor ⊗n products of algebraic spinors in Cl3,0,i.e., [Cl3,0] : Ψ [Cl ] ε [Cl ] ε [Cl ] ε ∈ 3,0 ⊗ 3,0 ··· 3,0 = [Cl Cl Cl ][ε ε ε] [Cl ]⊗n [ε]⊗n . (8) 3,0 ⊗ 3,0 ⊗···⊗ 3,0 ⊗ ⊗···⊗ ≡ 3,0 For example, the bipartite state Ψ = (α + iα ) 00 +(α + iα ) 01 +(α + iα ) 10 +(α + iα ) 11 | i 1 2 | i 3 4 | i 5 6 | i 7 8 | i (9) is associated to Ψ = [α (σ ) (σ )+ α (σ ) (σ σ ) 1 3 ⊗ 3 2 3 ⊗ 1 2 +α (σ ) (σ )+ α (σ ) (σ σ ) 3 3 ⊗ 1 4 3 ⊗ 2 3 +α (σ ) (σ )+ α (σ ) (σ σ ) 5 1 ⊗ 3 6 1 ⊗ 1 2 +α (σ ) (σ )+ α (σ ) (σ σ )] ε⊗2. (10) 7 1 ⊗ 1 8 1 ⊗ 2 3 with α R, i =1, 2, ..., 8. We deﬁne an entangled bipartite state as a state i ∈ that cannot be written in the form Ψ = Ψ1 Ψ2, otherwise, the state is said to be separable. ⊗

3 3 Algebraic quantum logic gates

We can obtain quantum logic gates based on algebraic elements. For this purpose we will use the algebra Cl3,0. Our construction makes use of the fact that quantum logic gates are elements of the Lie group U(n) and that there is a isomorphism between subalgebras of tensor product of algebras Cl3,0 and the Lie algebra of the group U(2n), denoted by u(2n). The Lie algebra u(n) consists of the space of anti-hermitian complex matrices over R. An arbitrary element of u(n) has the form25

M = αijEij, (11) i,jX=1 where E are matrices of dimension n n with the element i, j equal to 1, and ij × all the other elements are zero. The elements αij are coeﬃcients belonging to the ﬁeld of complex numbers and must satisfy the relation

α = α∗ . (12) ij − ji For the Lie algebra u(n), we have the generators

Mij = i(Eij + Eji), if i > j, M = E E , if i < j, ij ij − ji Mii = iEii. (13)

Note that the elements Mij and Mii are anti-hermitian and linearly independent. For example, in the algebra u(2) we have

i 0 0 1 M = , M = , (14) 11 0 0 12 1 0 −

0 i 0 0 M = , M = . (15) 21 i 0 22 0 i

It is possible to perform a change of basis on the algebra u(2), building a new base in terms of multiples of the Pauli matrices iX, iY, iZ, i1 : { } 0 i 0 1 iX = , iY = , (16) i 0 1 0 − 4 i 0 i 0 iZ = , i1= . (17) 0 i 0 i − Note that if we exclude the element i1, we have the Lie algebra su(2) . In terms of the algebra Cl3,0, whose general element (multivector) is given by

0 1 2 3 12 23 31 123 σ = α 1+α σ1+α σ2+α σ3+α σ1σ2+α σ2σ3+α σ3σ1+α σ1σ2σ3, (18) we can build generators for U(2) through the algebra of bivectors, added to the trivetor I = σ1σ2σ3, i.e.

σ σ , σ σ , σ σ , σ σ σ . (19) { 1 2 2 3 3 1 1 2 3} These elements form an algebra

[σ σ , σ σ ] = 2σ σ , [σ σ , σ σ ] =2 σ σ , [σ σ , σ σ σ ]=0, 1 2 2 3 − 3 1 1 2 3 1 2 3 1 2 1 2 3 [σ σ , σ σ ] = 2σ σ , [σ σ , σ σ σ ]=0, [σ σ , σ σ σ ]=0. 2 3 3 1 − 1 2 2 3 1 2 3 3 1 1 2 3 (20)

We have four generators and this algebra is isomorphic to U(2) Lie algebra. Excluding the trivetor σ1σ2σ3, which is the only constituent of the center of algebra, we have an isomorphism with SU(2) Lie algebra. For u(4), we have 16 generators; these generators can be built as:

σ σ σ , σ σ σ , σ σ σ , σ σ 1, 1 2 ⊗ 1 1 2 ⊗ 2 1 2 ⊗ 3 1 2 ⊗ σ σ σ , σ σ σ , σ σ σ , σ σ 1, 2 3 ⊗ 1 2 3 ⊗ 2 2 3 ⊗ 3 2 3 ⊗ σ σ σ , σ σ σ , σ γ σ , σ σ 1, 3 1 ⊗ 1 3 1 ⊗ 2 3 1 ⊗ 3 3 1 ⊗ σ σ σ σ , σ σ σ σ , σ σ σ σ , σ σ σ 1. (21) 1 2 3 ⊗ 1 1 2 3 ⊗ 2 1 2 3 ⊗ 3 1 2 3 ⊗ We deﬁne the reversal operation of a multivector observing that the elements above are linearly independent and the tensor product between bivectors and vectors for a representation in terms of Pauli’s matrices results in anti-hermitian matrices; a same result occurs for the tensor product between a trivetor (or bivector) and the identity. If we are interested in su(4) algebra, we should exclude the element 1 σ1σ2σ3. For the u(8) algebra, we have 64 generators obtained by taking tensor⊗ products between bivectors (or trivectors) and vectors (identity), with the condition that the bivectors (trivectors) appear an odd number of times. This can be better understood if we refer to the reversal operation in a Cliﬀord algebra, in analogy with the matrix

5 Hermitian operators. We deﬁne the reversion operation of a multivetor Γ (a p-vetor) as − Γ =( 1)p(p 1)/2 Γ . (22) [p] − [p] For a scalar, a vector, a bivetore and a trivector, we have, respectively

Γ =Γ , Γ =Γ , Γ = Γ , Γ = Γ . (23) [0] [0] [1] [1] [2] − [2] [3] − [3] The analogye can bee seen if we considere a matrixe representation for the multivector Γ, given by (18), in terms of Pauli matrices,

0 3 12 123 1 31 23 2 (α + α )+ i(α + α ) (α + α )+ i(α α ) z1 z2 Γ= 1 31 23 2 0 3 123 − 12 = . (α α )+ i(α + α ) (α α )+ i(α α ) z3 z4 − − − (24)

Performing the reversion operation, we have

z z Γ= 1 3 , z2 z4 e e e that corresponds to conjugate transposee e of Γ. Taking the tensor product of bivectors, we obtain

Γ Γ =( Γ ) ( Γ )=Γ Γ , (25) [2] ⊗ [2] − [2] ⊗ − [2] [2] ⊗ [2] i.e. we have a Hermitiane e element. On the other hand, carrying out the tensor product of a bivector with a vector, we have

Γ Γ =( Γ ) Γ = Γ Γ , (26) [2] ⊗ [1] − [2] ⊗ [1] − [2] ⊗ [1] that result is anti-hermitiane e element; a similar analysis is valid for trivector. In the general case of algebra u(2n), we have

σ σ ... σ , ... σ σ ... 1, 1 2 ⊗ ⊗ 1 1 2 ⊗ ⊗ σ σ ... σ , ... σ σ ... 1, 2 3 ⊗ ⊗ 1 2 3 ⊗ ⊗ σ σ ... σ , ... σ σ ... 1, 3 1 ⊗ ⊗ 1 3 1 ⊗ ⊗ σ σ σ ... σ , ... σ σ σ ... 1, (27) 1 2 3 ⊗ ⊗ 1 1 2 3 ⊗ ⊗ with bivectors and trivectors appearing an odd number of times, in order to ensure that the tensor product results in an anti-hermitian element. For

6 example, in the case of algebra u(23) = u(8), the terms in which the ﬁrst factor of the tensor product is σ1σ2, and the last factor is σ1, following an analogous procedure to that of u(4), we get

σ σ σ σ , (28) 1 2 ⊗ 1 ⊗ 1 σ σ σ σ , (29) 1 2 ⊗ 2 ⊗ 1 σ σ σ σ , (30) 1 2 ⊗ 3 ⊗ 1 σ σ 1 σ . (31) 1 2 ⊗ ⊗ 1 The exponential of a multivector Γ is deﬁned as

∞ Γn exp Γ = . (32) n! Xn=0 Then, for a multivector Γ

Γ = α...1σ σ ... σ + ... + α...0σ σ ... 1 (33) 12 1 2 ⊗ ⊗ 1 12 1 2 ⊗ ⊗ +α...1σ σ ... σ + ... + α...0σ σ ... 1 23 2 3 ⊗ ⊗ 1 23 2 3 ⊗ ⊗ +α...1σ σ ... σ + ... + α...0σ σ ... 1 31 3 1 ⊗ ⊗ 1 31 3 1 ⊗ ⊗ +α...311 ... σ σ + ... + α...1231 ... σ σ σ , 0 ⊗ ⊗ 3 1 0 ⊗ ⊗ 1 2 3 a general element of U(2n) is given by

U = exp α...1σ σ ... σ + ... + α...0σ σ ... 1 12 1 2 ⊗ ⊗ 1 12 1 2 ⊗ ⊗ +α...1σ σ ... σ + ... + α...0σ σ ... 1 23 2 3 ⊗ ⊗ 1 23 2 3 ⊗ ⊗ +α...1σ σ ... σ + ... + α...0σ σ ... 1 31 3 1 ⊗ ⊗ 1 31 3 1 ⊗ ⊗ +α...311 ... σ σ + ... + α...1231 ... σ σ σ . (34) 0 ⊗ ⊗ 3 1 0 ⊗ ⊗ 1 2 3 where the lower indices refer to the ﬁrst factor of the tensor product, assigning the index 0 for the unit, and the upper indices indicate the sequence in which j the multivectors appear at from the second factor. The αi coeﬃcients are real. With this mathematical development we show how it is possible to build logic gates within the algebraic formulation through a U(2n) Lie algebra structure contained within of the algebra product Cl3,0 Cl3,0 ... Cl3,0. These logic gates can be expressed through the algebra⊗ generato⊗rs.⊗ In fact, when these logic gates operationalize over the states, the resultant state can

7 be obtained easily since we have relations between the generators and the states are also expressed in terms of these generators. Consider initially, the quantum logic gate NOT denoted by the Pauli matrix X, whose eﬀect on a qubit is given by

α β X = . β α

Since Cl3,0 is deﬁned on ﬁeld of reals, we should write the state of qubit as

ψ =(α1 + α2I)σ3ε1 +(β1 + β2I)σ1ε1. (35)

Then the element associated to gate X is given by σ1, since

σ1 [(α1 + α2I)σ3ε1 +(β1 + β2I)σ13ε1] = (β1 + β2I)σ3ε1 +(α1 + α2I)σ1ε1. (36)

The element σ1 can be obtained through of product

Ux = exp(ασ1σ2σ3) exp(θσ2σ3)

= exp(ασ1σ2σ3 + θσ2σ3), (37) where α = π/2, θ = π/2 and the second equality occurs because the trivetor − σ1σ2σ3 commutes with the bivector σ2σ3. To better understand this result, we will analyze each one of the above factors. As Ux1 = exp(ασ1σ2σ3), we have

Ux1 = exp(ασ1σ2σ3) α2 α3 = 1+ ασ σ σ + (σ σ σ )2 + (σ σ σ )3 + ... 1 2 3 2! 1 2 3 3! 1 2 3 α2 α3 α4 α5 = 1+ ασ σ σ 1 σ σ σ + 1+ σ σ σ 1 2 3 − 2! − 3! 1 2 3 4! 5! 1 2 3 α2 α4 α3 α5 = (1 + ...)+(α + ...)σ σ σ − 2! 4! − − 3! 5! − 1 2 3 = cos(α)1 + sin(α)σ1σ2σ3. (38)

For Ux2 = exp(θσ2σ3), we obtain

Ux2 = exp(θσ2σ3)=

8 θ2 θ3 = 1+ θσ σ + (σ σ )2 + (σ σ )3 + ... 2 3 2! 2 3 3! 2 3 θ2 θ3 θ4 θ5 = 1+ θσ σ 1 σ σ + 1+ σ σ 2 3 − 2! − 3! 2 3 4! 5! 2 3 θ2 θ4 θ3 θ5 = (1 + ...)+(θ + ...)σ σ − 2! 4! − − 3! 5! − 2 3 = cos(θ)1 + sin(θ)σ2σ3. (39) Then, with α = π/2 and θ = π/2 we get −

U = Ux1 Ux2 = sin( π/2) σ σ σ sin(π/2)σ σ − 1 2 3 2 3 = σ1. (40) Another logical gate of interest is the gate CNOT , which performs the trans- formation 00 00 , | i → | i 01 01 , | i → | i 10 11 , | i → | i 11 10 . (41) | i → | i To determine the algebraic element in this case, consider U = exp[α(σ σ σ 1)] 1 2 3 ⊗ θ exp[ (σ σ σ 1+ σ σ 1+ σ σ σ σ σ σ σ )].(42) × 2 1 2 3 ⊗ 1 2 ⊗ 1 2 3 ⊗ 1 − 1 2 ⊗ 1 Since (σ σ σ 1)2 = 1 1 (43) 1 2 3 ⊗ − ⊗ and 1 2 (σ σ σ 1+ σ σ 1+ σ σ σ σ σ σ σ ) = 1 1, (44) 2 1 2 3 ⊗ 1 2 ⊗ 1 2 3 ⊗ 1 − 1 2 ⊗ 1 − ⊗ we can rewrite U by performing a procedure analogous to that of the logic gate X, i.e. U = [cos α(1 1) + sin α[(σ σ σ 1)] [cos θ(1 1) ⊗ 1 2 3 ⊗ ⊗ 1 + sin θ(σ σ σ 1+ σ σ 1+ σ σ σ σ σ σ σ )] . (45) 2 1 2 3 ⊗ 1 2 ⊗ 1 2 3 ⊗ 1 − 1 2 ⊗ 1 9 And, if we set α = π/2 and θ = π/2, we have − 1 U = (σ σ σ 1) (σ σ σ 1+ σ σ 1+ σ σ σ σ σ σ σ ) − 1 2 3 ⊗ 2 1 2 3 ⊗ 1 2 ⊗ 1 2 3 ⊗ 1 − 1 2 ⊗ 1 1 = (1 1+ σ 1+1 σ σ σ ) 2 ⊗ 3 ⊗ ⊗ 1 − 3 ⊗ 1 1 = [(1 + σ ) 1+(1 σ ) σ ] 2 3 ⊗ − 3 ⊗ 1 1 1 = (1 + σ ) 1+ (1 σ ) σ . (46) 2 3 ⊗ 2 − 3 ⊗ 1 This U is the algebraic element correspondent of the gate CNOT . In fact,

U(σ σ )(ε ε ) = (σ σ )(ε ε ), 3 ⊗ 3 1 ⊗ 1 3 ⊗ 3 1 ⊗ 1 U(σ σ )(ε ε ) = (σ σ )(ε ε ), 3 ⊗ 1 1 ⊗ 1 3 ⊗ 1 1 ⊗ 1 U(σ σ )(ε ε ) = (σ σ )(ε ε ), 1 ⊗ 3 1 ⊗ 1 1 ⊗ 1 1 ⊗ 1 U(σ σ )(ε ε ) = (σ σ )(ε ε ). (47) 1 ⊗ 1 1 ⊗ 1 1 ⊗ 3 1 ⊗ 1 An important aspect to note in our algebraic formulation is that we obtained two logic gates as examples - the logic gate (X) and the logic gate CNOT . The others logic gates of a q-bit, (Y ) and (Z), can be generated with an identical procedure to that used for gate (X). In consequence, our results are that its possible to build the logic gates of a q-bit and the logical gate CNOT algebraically. Hence, as any other quantum logic gate can be built from these gates,3 it follows that our proposal for quantum algebraic computing can be interest to this research area.

4 Chirality, charge conjugation, parity and time reversal operators

In this section we explore some concepts of the relativistic quantum me- + chanics by using Cl1,3 Cl3,0 and algebraic spinors. Similar results were obtained with Dirac algebra.≃ 27 Here however we explore the structure of Cliﬀord algebra and algebraic spinors. For example, the chirality operator γ5 = iγ0γ1γ2γ3, that in terms of Weyl representation is given by

12×2 0 γ5 = , (48) 0 12×2 − 10 in our formulation γ5 can be written as

Cl+ ⊗Cl+ Cl+ ⊗Cl+ Γ 1,3 1,3 = γ γ 1= Γ 1,3 1,3 . (49) 5 3 0 ⊗ 5 The spinors that are eigenvectors of Γ withe eigenvalues 1 and 1 are called 5 − right-handed spinors ΨR, and left-handed spinors ΨL, respectively: + ⊗ + + + Cl1,3 Cl1,3 Cl1,3 Cl1,3 Γ5 ΨR = ΨR , Cl+ ⊗Cl+ Cl+ Cl+ Γ 1,3 1,3 Ψ 1,3 = Ψ 1,3 . (50) 5 L − L + We can write a general state of a qubit as an algebraic spinor in Cl1,3 + ⊗ Cl1,3 using the prescription of (5),

+ Cl , Ψ 1 3 = (α1γ3γ0 + α2ιγ3γ0 + α3γ1γ0 + α4ιγ1γ0)P 1 = (α γ γ + α γ γ γ γ + α γ γ + α γ γ γ γ ) (1 + γ γ ), (51) 1 3 0 2 1 0 2 0 3 1 0 4 2 0 3 0 2 3 0 with αi R, i = 1, 2, 3, 4. Eq. (51) is an element of the minimal left ideal + ∈ Cl1,3P and corresponds to Hilbert state

Cl+ Ψ 1,3 =(α + iα ) 0 +(α + iα ) 1 . (52) | i 1 2 | i 3 4 | i + ⊗n In the multipartite systems we need to build the product algebra Cl1,3 . We can write multipartite q-bits as tensor products of algebraic spin ors in + + ⊗n Cl1,3, i.e. as elements of minimal left ideals in Cl1,3 : ⊗n Cl+ Ψ[ 1,3] Cl+ P Cl+ P Cl+ P ∈ 1,3 ⊗ 1,3 ··· 1,3 + + + = Cl1,3 Cl1,3 Cl1,3 [P P P ] ⊗⊗ ⊗···⊗ ⊗ ⊗···⊗ Cl+ n [P ]⊗n . (53) ≡ 1,3 so that a bipartite state Ψ = (α + iα ) 00 +(α + iα ) 01 +(α + iα ) 10 +(α + iα ) 11 | i 1 2 | i 3 4 | i 5 6 | i 7 8 | i (54) corresponds to

Cl+ ⊗Cl+ Ψ 1,3 1,3 = [α (γ γ γ γ )+ α (γ γ γ γ γ γ ) 1 3 0 ⊗ 3 0 2 3 0 ⊗ 1 0 2 0 +α (γ γ γ γ )+ α (γ γ γ γ γ γ ) 3 3 0 ⊗ 1 0 4 3 0 ⊗ 2 0 3 0 +α (γ γ γ γ )+ α (γ γ γ γ γ γ ) 5 1 0 ⊗ 3 0 6 1 0 ⊗ 1 0 2 0 +α (γ γ γ γ )+ α (γ γ γ γ γ γ )] P ⊗2. (55) 7 1 0 ⊗ 1 0 8 1 0 ⊗ 2 0 3 0 11 where αi R, i =1, 2, ..., 8. The chiral∈ projection operators are given by

1 Cl+ ⊗Cl+ 1 P R = 1+Γ 1,3 1,3 = (1 1+ γ γ 1) 2 5 2 ⊗ 3 0 ⊗ 1 Cl+ ⊗Cl+ 1 P L = 1 Γ 1,3 1,3 = (1 1 γ γ 1) , (56) 2 − 5 2 ⊗ − 3 0 ⊗ with P R +P L =1 1. So the Dirac spinor is a superposition of these spinors ⊗ + + + + + + ⊗ Cl , ⊗Cl , Cl , ⊗Cl , Cl1,3 Cl1,3 1 3 1 3 1 3 1 3 Ψ = ΨR +ΨL , (57) wherein

+ + + ⊗ + Cl , ⊗Cl , R Cl1,3 Cl1,3 1 3 1 3 P Ψ = ΨR = [α (γ γ γ γ )+ α (γ γ γ γ γ γ ) 1 3 0 ⊗ 3 0 2 3 0 ⊗ 1 0 2 0 +α (γ γ γ γ )+ α (γ γ γ γ γ γ )]P ⊗2 (58) 3 3 0 ⊗ 1 0 4 3 0 ⊗ 2 0 3 0 and

+ + + + ⊗ Cl , ⊗Cl , L Cl1,3 Cl1,3 1 3 1 3 P Ψ = ΨL = [α (γ γ γ γ )+ α (γ γ γ γ γ γ ) 5 1 0 ⊗ 3 0 6 1 0 ⊗ 1 0 2 0 +α (γ γ γ γ )+ α (γ γ γ γ γ γ )]P ⊗2, (59) 7 1 0 ⊗ 1 0 8 1 0 ⊗ 2 0 3 0 + + Cl1,3 Cl1,3 using that γ3γ0P = P . It is easy to see that ΨR and ΨL are not entangled states, and if state (10) is entangled the chiral projectors P R and P L always project such a state into separable states. We will now analyze in the present context the charge conjugation, parity and time reversal operations. In our formulation, they are unitary operators and can represent quantum logic gates. The parity operation, usually de- iφ scribed in Dirac algebra by ΦP = e γ0, in our formulation is given by

Cl+ ⊗Cl+ Φ 1,3 1,3 = [1 (cos(φ)+ ι sin(φ))] [γ γ 1] , (60) P ⊗ 3 0 ⊗ + + Cl , ⊗Cl , + ⊗ + 1 3 1 3 Cl1,3 Cl1,3 so the action of ΦP on Ψ is given by:

12 + ⊗ + + + Cl1,3 Cl1,3 Cl ⊗Cl Φ Ψ 1,3 1,3 = [1 (cos(φ)+ ι sin(φ))] P ⊗ [α (γ γ γ γ )+ α (γ γ γ γ γ γ ) × 1 3 0 ⊗ 3 0 2 3 0 ⊗ 1 0 2 0 +α (γ γ γ γ )+ α (γ γ γ γ γ γ ) 3 3 0 ⊗ 1 0 4 3 0 ⊗ 2 0 3 0 α (γ γ γ γ ) α (γ γ γ γ γ γ ) − 5 1 0 ⊗ 3 0 − 6 1 0 ⊗ 1 0 2 0 α (γ γ γ γ ) α (γ γ γ γ γ γ )] P ⊗2, − 7 1 0 ⊗ 1 0 − 8 1 0 ⊗ 2 0 3 0 (61) and we have that

Cl+ ⊗Cl+ Φ 1,3 1,3 = [1 (cos(φ) ι sin(φ))](γ γ 1). (62) P ⊗ − 3 0 ⊗ f The charge conjugation C = iγ2γ0 can be written as:

Cl+ ⊗Cl+ C 1,3 1,3 = ( ι 1)(γ γ γ γ ) − ⊗ 1 0 ⊗ 2 0 = γ γ γ γ γ γ , (63) 3 0 2 0 ⊗ 2 0 Cl+ ⊗Cl+ Cl+ ⊗Cl+ with C 1,3 1,3 = C 1,3 1,3 . The action of the charge conjugation oper- − ator on the base states is given by: e Cl+ ⊗Cl+ ⊗2 ⊗2 C 1,3 1,3 (γ γ γ γ )P = (γ γ γ γ γ γ )P 3 0 ⊗ 3 0 − 2 0 ⊗ 2 0 3 0 = (γ γ γ γ )P ⊗2, (64) 1 0 ⊗ 1 0

Cl+ ⊗Cl+ ⊗2 ⊗2 C 1,3 1,3 (γ γ γ γ )P = (γ γ γ γ γ γ γ γ γ γ )P . 1 0 ⊗ 1 0 1 0 2 0 3 0 ⊗ 1 0 2 0 = (γ γ γ γ )P ⊗2. (65) − 3 0 ⊗ 3 0

Cl+ ⊗Cl+ ⊗2 ⊗2 C 1,3 1,3 (γ γ γ γ )P = (γ γ γ γ γ γ )P 3 0 ⊗ 1 0 2 0 ⊗ 1 0 2 0 = (γ γ γ γ )P ⊗2 (66) − 1 0 ⊗ 3 0

Cl+ ⊗Cl+ ⊗2 ⊗2 C 1,3 1,3 (γ γ γ γ )P = (γ γ γ γ γ γ γ γ γ γ )P 1 0 ⊗ 3 0 − 1 0 2 0 3 0 ⊗ 2 0 3 0 = (γ γ γ γ )P ⊗2 (67) 3 0 ⊗ 1 0

13 using equivalent representations for the algebraic state. We can verify that they conduct entangled Bell states in themselves:

+ + 1 1 Cl , ⊗Cl , ⊗2 ⊗2 C 1 3 1 3 [γ3γ0 γ3γ0 + γ1γ0 γ1γ0]P = [γ1γ0 γ1γ0 γ3γ0 γ3γ0]P √2 ⊗ ⊗ √2 ⊗ − ⊗ + + 1 1 Cl , ⊗Cl , ⊗2 ⊗2 C 1 3 1 3 [γ3γ0 γ1γ0 + γ1γ0 γ3γ0]P = [γ3γ0 γ1γ0 γ1γ0 γ3γ0]P . √2 ⊗ ⊗ √2 ⊗ − ⊗ (68)

The time reversal T = iγ γ is given by − 1 3 Cl+ ⊗Cl+ Cl+ ⊗Cl+ T 1,3 1,3 =1 γ γ = T 1,3 1,3 . (69) ⊗ 2 0 The eﬀect of time reversal on base states ise given by:

Cl+ ⊗Cl+ ⊗2 ⊗2 T 1,3 1,3 (γ γ γ γ )P = (1 ι)(γ γ γ γ )P 3 0 ⊗ 3 0 ⊗ 3 0 ⊗ 1 0 Cl+ ⊗Cl+ ⊗2 ⊗2 T 1,3 1,3 (γ γ γ γ )P = (1 ι)(γ γ γ γ )P 1 0 ⊗ 1 0 − ⊗ 1 0 ⊗ 3 0 Cl+ ⊗Cl+ ⊗2 ⊗2 T 1,3 1,3 (γ γ γ γ )P = (1 ι)(γ γ γ γ )P 3 0 ⊗ 1 0 − ⊗ 3 0 ⊗ 3 0 Cl+ ⊗Cl+ ⊗2 ⊗2 T 1,3 1,3 (γ γ γ γ )P = (1 ι)(γ γ γ γ )P . (70) 1 0 ⊗ 3 0 ⊗ 1 0 ⊗ 1 0 Entangled states are also preserved by the action of time reversal

+ + 1 (1 ι) Cl , ⊗Cl , ⊗2 T 1 3 1 3 [γ3γ0 γ3γ0 + γ1γ0 γ1γ0]P = ⊗ [γ3γ0 γ1γ0 √2 ⊗ ⊗ √2 ⊗ γ γ γ γ ] P ⊗2 − 1 0 ⊗ 3 0 + + 1 1 Cl , ⊗Cl , ⊗2 T 1 3 1 3 [γ3γ0 γ1γ0 + γ1γ0 γ3γ0]P = (1 ι)[γ1γ0 γ1γ0 √2 ⊗ ⊗ √2 ⊗ ⊗ γ γ γ γ ] P ⊗2. − 3 0 ⊗ 3 0 (71)

We explore here the fact that Cliﬀord algebras appear naturally in relativistic quantum mechanics through Dirac matrices γ. We believe that these results in terms of Cliﬀord algebras and algebraic spinors without the use of a particular representation can allow a better analysis of problems related to quantum information.

14 5 Supersymmetry

Clifford algebras are essential tools in the construction of supersymmetric theories. Supersymmetry connects bosons and fermions and exhibits charac- teristics of major importance for theoretical physics, such as, for example, the solution to the hierarchy problem. Despite their theoretical relevance, supersymmetric partners were not observed. However, supersymmetry techniques have been used successfully in condensed matter physics, atomic physics and nuclear physics, for example. Clifford’s algebras allow the construction of supersymmetric extensions of Poincaré’s algebra, that are Lie superalgebras. In this section, motivated by the development carried out in the refer- 17 ence, in which an association between Cl1,3 algebra and supersymmetry was presented, we propose a new construction showing its relationship with quantum computing. Thus, consider the Qα supersymmetry generators in Lie-Poincarésuperalgebra given by:17

′ ∂ µ α Q = i iσ ′ θ ∂ , (72) α − ∂θα − αα µ α where µ is a spatial index, θα and θ are Grassmann variables. It can be shown that by deﬁning Γ, Γ′ 1 (ΓΓ′ + ΓΓ′) with { } ≡ 2 θ1 = 1+eγ3γ0e θ = 1 γ γ 1 − 3 0 θ2 = γ1γ0 + ιγ2γ0 θ = γ γ + ιγ γ , (73) 2 − 1 0 2 0 we have θ , θ = θ , θ ; θ , θ =2δ , with α, β =1, 2. (74) { α β} { α β} { α β} α,β It is interesting to verify the action of these objects on the base elements γ3γ0P,γ1γ0P : 1 θ γ γ P = γ γ P, 2 1 3 0 3 0 1 θ γ γ P = 0 γ γ P, 2 1 3 0 · 3 0 1 θ γ γ P = 0 γ γ P, 2 2 3 0 · 1 0 1 θ γ γ P = γ γ P. (75) 2 2 3 0 − 1 0 15 and 1 θ γ γ P = 0 γ γ P, 2 1 1 0 · 1 0 1 θ γ γ P = γ γ P, 2 1 1 0 1 0 1 θ γ γ P = γ γ P, 2 2 1 0 3 0 1 θ γ γ P = 0 γ γ P. (76) 2 2 1 0 · 1 0 A matrix representation for these operators using the algebraic base γ3γ0P and γ3γ0P is given by 2 0 0 0 θ = , θ = (77) 1 0 0 1 0 2 and 0 0 0 2 θ2 = , θ2 = − . (78) 2 0 0 0 Still in a supersymmetric context, using an octonionic formulation of the M-theory,20 the charge conjugation matrix C is given by: b 0 0 10  0 0 01  C = . (79) 1 0 00  −  b  0 1 0 0   −  In our formalism we have + + Cl , ⊗Cl , Cl+ ⊗Cl+ 1 3 1 3 C 1,3 1,3 = (1 ι)(γ γ 1) = C . (80) − ⊗ 2 0 ⊗ − Its action onb Bell states is given by eb + + 1 1 Cl , ⊗Cl , ⊗2 ⊗2 C 1 3 1 3 [γ3γ0 γ3γ0 + γ1γ0 γ1γ0]P = [γ1γ0 γ3γ0 γ3γ0 γ1γ0]P √2 ⊗ ⊗ √2 ⊗ − ⊗ b + + 1 1 Cl , ⊗Cl , ⊗2 ⊗2 C 1 3 1 3 [γ3γ0 γ1γ0 + γ1γ0 γ3γ0]P = [γ1γ0 γ1γ0 γ3γ0 γ3γ0]P . √2 ⊗ ⊗ √2 ⊗ − ⊗ b (81) Therefore these operators preserve entanglement. This prescription is valid for a larger number of dimensions, extending, as mentioned above, the number of factors in the tensor product of algebras.

16 6 M-superalgebra

In M-theory, the most general supersymmetry algebra in D = 11 Minkowskian spacetime is given by20, 21, 28

Q , Q =(CΓ ) P µ +(CΓ ) Zµν +(CΓ ) Zµ1...µ5 , (82) { r s} µ r,s [µν] r,s [µ1...µ5] r,s where Q is a pseudo-Majoranab spinorb that has 32b components. Here we must have in the left hand side a symmetric matrix with 528 components; so, in the right-hand side, elements of rank 1, 2 and 5 of Cliﬀord algebra must constitute a basis for symmetric matrices.20 In fact, we have21 11 11 11 + + = 528. 1 2 5

C in equation (82) is charge conjugation matrix, P µ is Poincar´egenerator of translation, Zµν is tensorial central charge of rank 2, Zµ1...µ5 is the tensorial b central charge of rank 5 and 1 Γ = ǫ(σ)Γ ...Γ , (83) [µ1...µl] l! µσ(1) µσ(l) σX∈Gl where ǫ(σ) is the signature of permutation σ, Gp is the symmetric group and, in the present section, Γ is used to denote Dirac matrices (see below). The µσ1 charge conjugation must satisfy the relations15 CΓµCT = ξ(Γµ)T CT = αC, (84) where ξ = ( 1)p−1, CbT isb the transposedb matrixb C and α = ( 1)p(p−1)/2, − − with (p, q) is the signature of the algebra. It was obtained in the reference21 b b representations of Clp,q through

0 γi 0 1d 1d 0 Γi = , τ2 = , τ3 = , (85) γi 0 1d 0 0 1d − − if (p, q) (p +1, q + 1), deﬁning d as the dimension of γ, or through 7→

0 γi 0 1d 1d 0 Γi = , τ2 = , τ3 = . (86) γi 0 1d 0 0 1d − − if (p, q) (q +2,p), i.e. these Γ , τ and τ form a 2d dimensional represen- 7→ i 2 3 tation of Clp,q. Indeed our construction can be applied to algebra Cl0,7 and

17 to obtain the algebras Cl9,0 and Cl10,1. We can build generators for Cl10,1 + through of generators of Cl1,3 as

γ ιγ γj 1,0 ⊗ 2,0 ⊗ 0,7  γ γ 1 1 1 1,0 ⊗ 1,0 ⊗ ⊗ ⊗ Γµ =  γ γ 1 1 1 (87)  1,0 ⊗ 3,0 ⊗ ⊗ ⊗  ιγ 1 1 1 1 3,0 ⊗ ⊗ ⊗ ⊗  γ3,0 1 1 1 1  ⊗ ⊗ ⊗ ⊗ where  1 γ0,7 = ιγ2,0 γ1,0 1 2 ⊗ ⊗  γ0,7 = ιγ2,0 γ3,0 1  3 ⊗ ⊗  γ0,7 =1 ιγ2,0 γ1,0  4 ⊗ ⊗  γ0,7 =1 ιγ2,0 γ3,0 (88)  5 ⊗ ⊗ γ0,7 = γ1,0 1 ιγ2,0 6 ⊗ ⊗  γ0,7 = γ3,0 1 ιγ2,0  7 ⊗ ⊗  γ0,7 = ιγ2,0 ιγ2,0 ιγ2,0  ⊗ ⊗ are generators of Cliﬀord algebra Cl , where γ γ γ . The charge conju- 0,7 i,0 ≡ i 0 gation in our formulation is given by

Cl+ C 10,1 = γ 1 1 1 1. (89) 3,0 ⊗ ⊗ ⊗ ⊗ If we think in termsb of an algebraic spinor we can associate these spinors +⊗5 ⊗5 Cl10,1 with a state of 5 qubits. Note that Cl10,1 Cl1,3 Cl3,0. If S is the ⊂ ≃ 15, 16, 26 spinor representation for Cl10,1, by using the results of references, we have

End(SCl10,1 ) SCl10,1 SCl10,1 ≃ ⊗ (1) (2) (5) (R10,1) (R10,1) (R10,1) (90) ≃ ^ ⊕ ^ ⊕ ^ where End designates endomorphism. So we have, 1 Γ = ǫ(σ)Γ Γ (91) [µ1µ2] 2! µσ(1) µσ(2) σX∈Gp and 1 Γ = ǫ(σ)Γ Γ Γ Γ Γ . (92) [µ1µ2µ3µ4µ5] 5! µσ(1) µσ(2) µσ(3) µσ(4) µσ(5) σX∈Gp

18 We can also build

(0) (2) (SCl3,0 SCl3,0 )⊗5 ( (R3,0) (R3,0))⊗5 ⊗ ≃ ^ ⊕ ^ We can show that SCl10,1 SCl10,1 (SCl3,0 SCl3,0 )⊗5 for some spinor ⊗5 ⊗ ⊂ ⊗ Cl , Cl10,1 space S 3 0 . For this we note that we have S I10,1, a minimal left ideal ⊗5 ≃ ⊗ Cl3,0 5 of Cl10,1 and similarly S ICl⊗5 , a minimal left ideal of Cl3,0. Every ideal ≃ 3,0 of an algebra A may be written as AE, where E is a primitive idempotent. ⊗5 Therefore, we have I10,1 = Cl10,1E10,1 and I ⊗5 = Cl E ⊗5 Since both al- Cl3,0 3,0 Cl3,0 gebras have the same dimension related to irreducible representation, each ⊗5 primitive idempotent of Cl10,1 corresponds to a primitive idempotent of Cl3,0. Remembering that primitive idempotents always admit a diagonal represen- +⊗5 tation and Cl10,1 Cl1,3 the proof is ﬁnished. The M-superalgebra also ⊂ 20, 21, 29 admits an octonionic representation. Consider octonions oi satisfying

o o = δ + C , i, j, k =1, ..., 7 (93) i ◦ j − i,j i,j,k 21 where Ci,j,k are totally antisymmetric octonionic structure constants. We set o = γi Cl . A general octonion is given by i 0,7 ∈ 0,7

o = λ0o0 + λ1o1 + λ2o2 + λ3o3 + λ4o4 + λ5o5 + λ6o6 + λ7o7, (94) with λ R; k =0, ..., 7. The conjugate o of octonion is given by k ∈ ∗ o = λ o λ o λ o λ o λ o λ o λ o λ o . (95) 0 0 − 1 1 − 2 2 − 3 3 − 4 4 − 5 5 − 6 6 − 7 7 The norm is deﬁned as

2 2 2 2 2 2 2 2 o = √oo = λ0 + λ1 + λ2 + λ3 + λ4 + λ5 + λ6 + λ7 k k ∗ q

In our formulation an octonionic representation of Cl10,1 can be build as

0 0 0 oi

oi  0 0 oi 0  Γ = for i =1, ..7, (96) 10,1 0 o −0 0  i   o 0 0 0   − i 

19 0001 0 0 1 0 o,8  0010  o,9  0 0 0 1  Γ = , Γ = , (97) 10,1 0100 10,1 1 0 0− 0      1000   0 10 0     − 

10 0 0 0 0 10 o,10  01 0 0  o,0  0 0 01  Γ = , Γ = , (98) 10,1 0 0 1 0 10,1 1 0 00  −   −   00 0 1   0 1 0 0   −   −  and we can write the Majorana spinor in D = 11 as an 4 4 octonionic column vector ×

ζ1 o,0  ζ2  ζ = , (99) 10,1 ζ  3   ζ   4  with ζ O, k =1, .., 4. We deﬁne the octonionic qubit as k ∈

ϑ1 ϑ o = , (100) | i ϑ2

2 2 where ϑk O, k =1, 2 and ϑ1 + ϑ2 = 1. Consequently it is possible to identify∈ the spinor (99) ask a statek ofk twok octonionic qubits. Obtaining a separability criterion for octonionic qubits is not so simple because octonions are noncommutative and nonassociative. However they constitute an algebra without zero divisor so that we can verify, for example, that the state

ζa o,0  0  ζ = , (101) 10,1 0    ζ   b  is entangled.

20 7 Conclusions

In this paper, we have derived a formulation based on tensor product of Clif- ford algebras and minimal left ideals. In this context, we have show how to build quantum logic gates and general states of a qubits. Furthermore, we have shown as concepts from particle physics such as chirality, charge conjugation, parity and time reversal are easily introduced with our formulation and can be expressed in connection to quantum information theory. Such a relationship is possible by noting that Dirac algebra is a Cliﬀord algebra. Subsequently, we have shown how to obtain Grassmann variable in a supersymmetry context as well as the charge conjugation matrix in an algebraic formulation of M-theory. We note that the derived operators are Hermitian or unitary operators corresponding to quantum observables and logic gates, respectively. All states and operators are described in terms of the algebra + generators. The M-algebra were derived from Cliﬀord algebra Cl1,3 Cl3,0 and its relationship between qubits and octonions was analyzed. It≃ is our view that these results may be useful in future developments of quantum computing related to simulations of relativistic systems, high energy physics, and M-theory.

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