Density Operators in the Multiparticle Spacetime Algebra

Density Operators in the Multiparticle Spacetime Algebra Timothy F. Havel∗ Nuclear Engineering, Massachusetts Institute of Technology, U.S.A.

Chris Doran and Suguru Furuta Cavendish Laboratories, University of Cambridge, U.K. (Dated: January 13, 2003)

Two distinct representations for the density operators of multi-qubit systems are developed and compared within the multiparticle spacetime algebra (MSTA). The two representations arise from two different methods for correlating imaginary structures from different qubit spaces. The first is a direct transliteration of the usual representation by Hermitian matrices, with the complex structure provided by correlated pseudoscalar factors. The second arises from a generalisation of the treatment of spinors as elements of the even subalgebra of the Pauli algebra. In this representation the complex structure for quantum states is provided by correlated bivectors, though the action of this on density operators is more subtle. The advantages of the new representation arise from the ease with which spinor observables are constructed and related to terms in the density operator. These are illustrated by rederiving a well-known expression for the decoherence of a single qubit through its coherent interactions with other qubits in its environment.

∗

Corresponding author; E-mail address: [email protected]

2 I. INTRODUCTION

The density operator determines the expectation values of all possible observables of a quantum system, even when only probabilistic knowledge of its state is available. It plays a central role in the treatment of open quantum systems, where entanglement with an unobserved environment renders the state not merely unknown, but indeterminate [Giulini et al., 1996; Zurek, 1998]. In order to gain deeper insight into the geometric structure underlying the quantum mechanics of open systems, we consider here two schemes for studying density operators within the multiparticle space-time algebra (MSTA). This algebra was introduced by Doran, Lasenby and Gull as a means of representing multi-qubit quantum states within the framework of geometric algebra [Ashdown et al., 1998; Doran et al., 1993, 1996; Somaroo et al., 1999], and constitutes a generalisation of Hestenes’ formulation of quantum mechanics to the multiparticle domain [Hestenes, 1966, 1971, 1979]. Although the MSTA was initially introduced to study pure states [loc. cit.], mixed states are crucial in applications such as NMR and quantum information processing [Cory et al., 2000]. Two generalisations of the MSTA to mixed states of multi-qubit (ergo two-state quantum) systems have been considered. The first is a direct transliteration of the usual Hermitian density matrix into the MSTA, for which detailed presentations and applications are available in papers by Cory, Doran, Havel and Somaroo [Havel et al., 2000a; Havel and Doran, 2002a; Havel et al., 2000b; Somaroo et al., 1998]. The second representation emerges from the way in which observables are constructed from a pure state in the MSTA [Parker and Doran, 2002; Somaroo et al., 1999], and was introduced in a recent study of two-qubit interactions [Havel and Doran, 2002b]. This representation is certainly less familiar, but does bring some new insights into the geometric structure of density operators and the manner in which they evolve. This paper is intended to provide a unified presentation and comparison of both of these representations of density operators, assuming a basic familiarity with geometric algebra (introductions may be found in Doran and Lasenby [2003]; Hestenes [1999]). As an illustration we analyse a simple model [due to Paz and Zurek, 2001] which describes the decoherence of a single qubit through its interactions with an environment consisting of many other qubits.

3 II. SPACETIME ALGEBRA AND QUBIT STATES

Spacetime algebra is the name given to the real Clifford algebra of Minkowski spacetime. This is generated by four vectors γ0 , . . . γ3 which satisfy γµ γν + γν γµ = 2ηµν ,

µ, ν = 0, . . . , 3.

(1)

The metric tensor ηµν is diagonal and our choice of signature is that ηµν = diag(+1, −1, −1, −1).

(2)

˜ and reverses the order of products of vectors The reverse operation is denoted with a tilde, M in any expression. The Pauli algebra sits inside the spacetime algebra by defining σk = γk γ0 ,

k = 1, . . . , 3.

(3)

The {σk } form a set of three spacetime bivectors which satisfy σk σ` = δk` + k`m Iσm ,

(4)

where δ is the Kronecker delta,  the alternating tensor, and I = σ1 σ2 σ3 = γ0 γ1 γ2 γ3 .

(5)

The σk therefore clearly generate the Pauli algebra, with the pseudoscalar I generating the complex structure. We use I in place of the more familiar i because the pseudoscalar I is defined as a spacetime entity and does not commute with all elements of the full spacetime algebra. The reverse operation in the Pauli algebra is written with a dagger and is defined by ˜ γ0 . M † = γ0 M

(6)

Within the Pauli algebra a 2-component state vector can be written as Ψ = Ψ 21 (1 + σ3 ),

(7)

where Ψ is a general element of the Pauli algebra. The idempotent factor (or projection) on the right ensures that Ψ has only four real degrees of freedom, as required. (Of course there is nothing sacred about the σ3 in the idempotent; this convention stems from its traditional

4 use as a magnetic field axis, so that spins oriented along σ3 do not evolve under the Zeeman interaction.) The density operator formed from this state is simply ρ = ΨΨ† = Ψ 12 (1 + σ3 )Ψ† .

(8)

In the 1960s Hestenes [1966] pointed out that the same degrees of freedom can be encoded by writing Ψ = ψ 12 (1 + σ3 ),

(9)

where ψ is restricted to the even subalgebra of the Pauli algebra, and so also has four degrees of freedom. The idea is then to remove factors of the idempotent wherever possible, and treat ψ more as an operator. The density operator then becomes ρ =

1 (ψψ † 2

+ ψσ3 ψ † ).

(10)

Working with ψ provides geometric insight into the role of a spinor, as ψ is geometrically just an instruction to rotate.

III. MULTIPARTICLE SPACETIME ALGEBRA

The MSTA is a straightforward generalisation of the spacetime algebra to the algebra generated by the direct sum of N copies of Minkowski space-time. If we let G(p, q) denote the (real) Clifford algebra with signature (p, q), then the MSTA is the algebra G(N, 3N ). The generators of this algebra are the set {γµm } where µ runs from 0 to 3, and m runs from 1 to N . These generators satisfy γµm γνn + γνn γµm = 2 ηµν δmn ,

0 ≤ µ, ν ≤ 3, 1 ≤ m, n ≤ N.

(11)

Each copy of the spacetime algebra contains its own copy of the Pauli algebra generated by the bivectors σkm = γkm γ0m ,

k = 1, . . . , 3, m = 1, . . . , N.

(12)

The set σjm satisfy σjm σkm + σkm σjm = 2δjk and hence generate an algebra isomorphic to the Pauli algebra G(3). Any pair of Pauli generators from different particle spaces commute, σim σjn = σjn σim ,

m 6= n.

(13)

5 It follows that under the geometric product the σkm generate an (8N )-dimensional algebra isomorphic to the tensor product of N copies of the Pauli algebra G(3). We denote this closed subalgebra of the MSTA by P(N ), i.e. P(N ) ≈ G(3) ⊗ G(3) ⊗ · · · ⊗ G(3) . | {z }

(14)

N -fold

Here we see a crucial feature of the MSTA — the tensor product familiar from multiparticle quantum theory is constructed from the geometric product in a relativistic configuration space. This unification of products is a considerable point in favour of the MSTA approach. The tilde symbol extends to define the reverse operation in the full MSTA. Similarly, the dagger symbol extends to denote simultaneous spatial reversion in each particle space particle. Explicitly, this is given by ˜ γN · · · γ1. M † = γ01 · · · γ0N M 0 0

(15)

This operation plays the same role as Hermitian conjugation for tensor products of Pauli matrices. At various points we make use of the spatial inversion operation c = γ1 · · · γN M γN · · · γ1 M 0 0 0 0

(16)

We refer to this simply as the parity operation. The algebra P(N ) is not the same as the algebra generated by the tensor product of the Pauli matrices over the complex field. The difference is that P(N ) contains N distinct pseudo-scalars I n (n = 1, . . . , N ), one from each copy of spacetime, whereas the usual matrix algebra contains a single complex structure. To construct an equivalent algebra we introduce the pseudoscalar correlator, C, defined by C =

N Y

1 2


1 − I 1I n ,

(17)

n=2

where I n is the pseudoscalar from the n-th particle space. Since I m C = I n C for 1 ≤ m, n ≤ N , the corresponding ideal P(N ) mod C contains a single complex structure in accord with conventional quantum mechanics. In fact this ideal, as a subalgebra, is isomorphic to the algebra of 2N × 2N complex matrices, P(N ) mod C ≈ GL(2N , C) .

(18)

6 It follows that we have a natural representation for the operators on N -particle Hilbert space. As in the single-particle case, the multiparticle spinors can be identified with the subspace of the MSTA defined by Ψ = Ψ C 12 (1 + σ31 ) · · · 12 (1 + σ3N ) ≡ Ψ C P31 · · · P3N ≡ Ψ C P3

(19)

for all Ψ ∈ P(N ). The idempotents P3n on the right-hand side ensure that the spinor subspace spans the same number of degrees of freedom as the complex column spinors usually used to represent the states of multi-qubit systems: 2N +1 . The Hermitian density operator is then formed from Ψ in much the usual way, ρ ≡ ΨΨ† = Ψ P3 Ψ† C .

(20)

The factor of C does nothing save to allow us to identify the imaginary units from different particle spaces, and hence it is safe to omit explicit reference to it from most expressions. The imaginary unit is now the correlated pseudoscalar ι ≡ I 1 C = · · · = I N C (where ι is Greek iota). Since this specific square-root of −1 commutes with everything in P(N ), it can be identified with the abstract imaginary unit i of traditional quantum mechanics. The end result is that we have a direct representation of a density operator within the C-correlated algebra P(N ) mod C. The density operator of a general mixed state is simply a weighted sum of terms of the same form as ΨΨ† , which corresponds to an arbitrary positive-semidefinite Hermitian matrix in the Pauli matrix representation. The representation naturally places the density operator in a more geometric setting while staying close to the matrix formulation. But a question remains — how do we generalise Hestenes’ operator view of a pure state to the multiparticle regime? It would be extremely disappointing if the insights this idea brings to single-particle quantum mechanics did not carry over to the multiparticle theory. Fortunately there is a simple answer to this question. In the single-particle theory the degrees of freedom in a 2-state spinor are carried by an element of the even subalgebra of the Pauli algebra. We denote this space by G + (3). The complex structure for this state is denoted by right multiplication by the bivector Iσ3 . The natural generalisation of G + (3) in the MSTA is the 4N -dimensional product space Q(N ), Q(N ) = G + (3) ⊗ G + (3) · · · ⊗ G + (3) . | {z } N -fold

(21)

7 This space currently has too many degrees of freedom to represent a quantum state, so again we have to consider that quantum mechanics involves a single complex structure. We thus introduce the quantum correlator : E =

1 2

1 − Iσ31 Iσ32


1 2


1 − Iσ31 Iσ33 · · ·

1 2


1 − Iσ31 Iσ3N .

(22)

Here we have adopted a useful convention that the superscript particle indices label the entire object to their immediate left, so Iσ3n = I n σ3n = γ2n γ1n .

(23)

The E correlator defines a left ideal in which the complex structure is given by rightmultiplication by any one of the bivectors Iσ3n . We give the resulting factor the symbol J, so J = JE = EJ = E Iσ3n ,

∀ n, 1 ≤ n ≤ N .

(24)

The elements of this left ideal Q(N ) mod E span a real subspace of dimension 2N +1 , and this space maps naturally onto the usual N -qubit complex Hilbert space. Given a state ψ = ψE, ψ ∈ Q(N ), the next question is how to generalise the terms in the single particle density operator of equation (10). The answer is surprisingly simple. First, the scalar term (ψψ † ) is replaced by ψEψ † , which contains terms of grade 0, 4, 8, . . .. This leaves the vector term in ψσ3 ψ † in equation (10). To generalise this we first replace the vector by its dual bivector ψIσ3 ψ † . This generalises immediately to the multiparticle object ψJψ † , which contains terms of grade 2, 6, 12, . . .. That is, we can form the object % =

1 2

ψEψ † + ψJψ †




= ψ 12 (E + J) ψ † ,

(25)

and % contains precisely the same information as the Hermitian density matrix. Moreover, the density operator for a mixed state is just a weighted sum of terms of the same form. This procedure constructs an arbitrary element of Q(N ), which in turn has precisely the correct number of dimensions to represent a general density operator. The MSTA therefore provides two natural representations of a density operator. The first of these, the ρ defined by equation (20), is a fairly literal translation of the matrix approach. We therefore refer to this as the Hermitian density operator. The second “even” object, % ∈ Q(N ), is rather more unusual. The map between the two representations is easy to spot

8 with the aid of the relation P3 C = P31 · · · P3N −1 21 (1 + σ31 σ3N )C = P31 · · · P3N −1 12 (1 − Iσ31 Iσ3N )C = P31 · · · P3N −2 12 (1 − Iσ31 Iσ3N −1 ) 12 (1 − Iσ31 Iσ3N )C = · · · = P31 E C =

1 (1 2

− I 1 Iσ31 )E C =

1 (E 2

(26)

− ιJ)C ,

where ι = I 1 C denotes the unit imaginary in the C-correlated algebra P(N ) mod C. It follows that ρ ≡ ψ P3 C ψ † =

1 2


ψ E ψ † − ι ψJψ † C .

(27)

This map holds for a general mixed-state density operator as well, so we can write ρ = (%+ − ι%− ) C,

(28)

where %+ and %− are given by %+ = h%i0 + h%i4 + · · · =

1 (% 2

%− = h%i2 + h%i6 + · · · =

1 (% 2

+ %† ) , †

(29)

− % ).

(The angle brackets denote the projection onto terms of a given grade in the MSTA.) This defines a one-to-one map between % ∈ Q(N ) and the traditional Hermitian density operator ρ.

IV. EVOLUTION EQUATIONS

Evolution of the Hermitian operator ρ under a the action of a unitary operator U is carried out in precisely the standard way, so ρ 7→ U ρU † . The result of this can easily be mapped to % if required. But it is interesting to see what happens if we construct evolution equations directly for %. The action of the Pauli operators and the abstract imaginary i on multiparticle spinors ψ = ψE, ψ ∈ Q(N ) is expressed by [Doran et al., 1993] σkn |ψi ↔ −Iσkn ψ J ,

i |ψi ↔ ψ J .

(30)

This representation has some unexpected consequences for the action of propagators on density operators %. The problem is that there is no obvious representation of the unit imaginary when acting on %. This is first seen in two-particle propagators such as exp(−iσ31 σ32 t) [Havel and Doran, 2002a], the argument of which acts on states in the left-ideal Q(N ) mod E via −iσ31 σ32 |ψi ↔ Iσ31 Iσ32 ψJ ,

(31)

9 keeping us entirely within Q(N ) mod E. If we now consider the action of the propagator on a general density operator we will again need an appropriate representation of i. This may be discerned simply by assuming that it is associative and acts on spinors in the usual way:
(32) i ψ(E + J)ψ † = (iψ)(E + J)ψ † = ψJ(E + J)ψ † = ψ(E − J)ψ † . (Here we have used the generic symbol i to denote the complex structure in our chosen representation.) So in terms of %+ and %− we can write the action of i on % as i: i:

%+ → %−

(33)

%− → − %+ .

Applied twice, i results in the map % 7→ −%, as expected. This somewhat unexpected role of the imaginary in 2-particle propagators is hidden in more traditional approaches. Similar considerations apply to the von Neumann equation. The von Neumann evolution equation for the Hermitian density operator ρ is unchanged from its matrix equivalent, i.e. ι ρ˙ = [ H, ρ ] ,

(34)

where H is a Hermitian operator in the C-correlated algebra P(N ) mod C. For a timeindependent Hamiltonian H this equation can be integrated by computing the exponential exp(−ιHt). For low-dimensional and/or highly symmetric problems this may be done analytically within the MSTA by using Cayley–Hamilton theorem to collapse its Taylor series; with larger or less structured problems an explicit matrix representation must be numerically diagonalized. The evolution equation for % is more complicated, though also more revealing. We must first express the degrees of freedom in iH in terms of suitable objects in Q(N ). To achieve this we first decompose H into its parity-even and parity-odd parts, i.e. H = H+ + H− =

1 (H 2

b + 1 (H − H) b . + H) 2

(35)

(This decomposition is unaffected by the presence of C.) The even part contains terms like σi1 σj2 C = −Iσi1 Iσj2 C

(36)

(recall that Iσi1 = I 1 σi1 ). The factor of −Iσi1 Iσj2 is contained in Q(N ), so we can remove the factor of C and work entirely with an object in Q(N ). Working through all of the terms in H+ defines the multivector G+ . Similarly, iH− contains terms like iσj1 C = Iσj1 C .

(37)

10 Again the correlator C can be removed to leave an object in Q(N ). This procedure defines the multivector G− . The various terms are related by G + C = H+ C = H+ ,

G− C = iH− C = iH− ,

(38)

and hence H = H+ + H− = G+ C − iG− C .

(39)

On substituting this expression for H and ρ = %+ − i%− into equation (34) one can derive an equation of motion for %. This in turn may be simplified somewhat by separating it into its reverse-even and odd parts, to get:   %+ , G − +   = %− , G − −

%˙ + = %˙ −

  %− , G + ;   %+ , G + .

(40)

(Observe that all the factors of C have been dropped so that everything lies entirely within Q(N )). In addition, the usual von Neumann equation has been decomposed into a pair of coupled differential equations containing no explicit imaginary unit. The advantage of this decomposition is that it highlights the significance of terms involving even numbers of particles (contained in G+ ), which are usually interactions between pairs of particles. Such terms interchange degrees of freedom between ρ+ and ρ− . When forming reduced density matrices these interactions can cause decoherence, as described in the following section. The evolution equation (40) is mainly useful when there is a simple decomposition of the Hamiltonian into parity-even and parity-odd terms. In particular if one of H+ or H− vanishes, or the two terms commute, then integrating equation (40) is often straightforward. For more complicated problems, however, it appears that a direct integration of equation (34) is the preferable route. The unitary operator thus obtained can be converted into a pair of operators with an equivalent action in Q(N ), if desired.

V. ZUREK’S SOLVABLE MODEL OF DECOHERENCE

In this section we shall illustrate the use of the “even” density operator % by analyzing a well-known model of a “system” qubit which decoheres through its interactions with an environment also consisting of (a large number N of) other qubits. This model admits an analytic solution due to the fact that the internal Hamiltonian of the system is assumed to

11 commute with the Hamiltonian for its interaction with the environment. The model was first considered in Zurek [1982], although we shall be following the more recent and compact derivation in Paz and Zurek [2001]. The system qubit’s Hamiltonian is assumed to have the form Hsys = ω 0 σ30 rad/sec, as for example if the qubit is a spin in Zeeman interaction with a magnetic field along the σ3 axis. We use a superscript zero to label the system of interest, and let 1, . . . , N label the environment qubits. The initial state of the system qubit is an arbitrary superposition of its two energy eigenstates. We write this as the spinor φsys = φsys E in the 0-th (leftmost) factor space of Q(N + 1) [Doran et al., 1993, 1996], φsys = a − Iσ20 b ,

(41)

where a = a0 + a1 Iσ30 and b = b0 + b1 Iσ30 , with a0 , a1 , b0 , b1 ∈ R. So a and b are the usual complex coefficients, written in terms of the G + (3) representation. It will be assumed that the environment qubits are initially in an arbitrary product state, which is similarly written as ϕenv =

N Y


αk − Iσ2k β k E ,

(42)

k=1

where E is the quantum correlator of the system-plus-environment, the αk & β k are “complex” magnitudes like a & b above, and the product on the right is the geometric product in the MSTA. The interaction Hamiltonian is simply Hsys,env = −

1 2

Iσ30

N X

Iσ3k Ωk ,

(43)

k=1

where the Ωk ∈ R are the strengths of the couplings between the system and environment qubits, again in units of rad/sec. By passing into a suitable rotating or interaction frame of reference, the system’s Hamiltonian can be transformed to zero without any change in the system’s evolution due to its interaction with the environment [Havel et al., 2000b]. Although the environment’s internal Hamiltonian is not expected to commute with the interaction Hamiltonian, it is assumed to be far larger so that its effect on the system qubit via the coupling is averaged out, allowing it to be regarded as “non-secular” (i.e. ignored). Let us first see how the state of the system and environment evolve under their interaction. Since it is a sum of mutually commuting terms each of which squares to a scalar, the

12 propagator of Hsys,env is readily seen to be U (t) =

N Y

N Y

exp i Iσ30 Iσ3k Ωk t/2 = cos(Ωk t/2) + i sin(Ωk t/2) Iσ30 Iσ3k .

k=1

(44)

k=1

When working in Q(N ) the complex structure for pure states is generated by rightmultiplication by J. As the i symbol is currently unemployed, it is convenient to keep the i symbol as a shorthand for right-multiplication by J. The initial state of the system plus environment is ψ(0) = φsys ϕenv = a ϕenv − Iσ20 b ϕenv .

(45)

The propagator commutes with a and b, and the interaction Hamiltonian anticommutes with Iσ20 , so the time evolution of ψ is determined by ψ(t) = a U (t) ϕenv − Iσ20 b U (−t) ϕenv

(46)

= a ϕenv (t) − Iσ20 b ϕenv (−t) .

(47)

Applying the propagator to the environment’s state gives N  Y

ϕenv (t) =

αk e − i Ω

k

t/2

− Iσ2k β k e i Ω

k

t/2



E,

(48)

k=1

where we have used the the fact that Iσ30 Iσ3k E = −E to “soak up” these factors in the odd terms of the Taylor expansions of the corresponding exponentials. Putting this back into equation (46) now yields ψ(t) = a

N  Y

αk e −i Ω

k

t/2

− Iσ2k β k ei Ω

k

t/2



E

k=1

−

Iσ20

b

N  Y

αk ei Ω

k

t/2

− Iσ2k β k e −i Ω

k

t/2



E.

(49)

k=1

The final step is to take a “trace” over the environment (or equivalently, the projection onto the 0-th qubit’s subspace), and so learn what the system looks like “on average” to an observer with no access to information on the state of the environment [Havel and Doran, 2002a]. To do this it is necessary to form the density operator corresponding to the pure state ψ(t), namely:
%(t) = ψ(t) 12 E + J ψ † (t).

(50)

13 We are only interested in the terms in %(t) in the space of particle 0, and the scalar term which must be 1/2 (after normalisation). We are therefore interested only in the grade-2 terms in %(t), which we express by writing

0

0 %(t) 2 = ψ(t) J ψ † (t) 2 ,

(51)

where the superscript 0 on the angle brackets denotes the projection onto particle space 0, and the subscript 2 denotes the projected grade. This expression now yields

0 0

0

ψ(t) J ψ † (t) 2 = |a|2 ϕenv ( t ) Jϕ†env ( t ) 2 − |b|2 Iσ20 ϕenv ( −t ) Jϕ†env ( −t ) 2 Iσ20

0 + 2 ϕenv ( t ) Jϕ†env ( −t ) ab† Iσ20 2 . (52)

We have already significantly reduced the number of terms we need to compute. A considerable further simplification may be obtained by noting that factors depending only on the qubits over which we are tracing may be rotated within the partial trace, just like any of the factors in a full trace, hence ϕenv ( t ) J ϕ†env ( t )

0

ϕenv ( t ) J ϕ†env ( −t ) ab† Iσ20

0



2 2

=



=



ϕ†env ( t ) ϕenv ( t ) J

0 2

,

ϕ†env ( −t ) ϕenv ( t ) J ab† Iσ20

(53) 0 2

.

(54)

2 2 We may assume that the environment states are normalised such that αk + β k = 1. Since hEi0 = 2−N , it follows that

ϕenv ( t ) Jϕ†env ( t )

0 2

=



ϕenv (−t) Jϕ†env (−t)

0 2

= 2−N Iσ30 .

(55)

The decoherence effects are contained in the cross terms, for which we find that

ϕ†env (−t) ϕenv (t) J ab† Iσ20

0 2

= 2−N ab†

N  Y k 2 −2Iσ0 Ωk t k 2 2Iσ0 Ωk t  0 α e 3 + β e 3 Iσ1 .

(56)

k=1

The final result for the normalised reduced density operator of the system qubit is therefore

0 %(t) =

where r(t) =

1 2

N  Y

1 + 2a b† r(t) Iσ10 +



|a|2 − |b|2 Iσ30 ,

2 2
 cos(Ωk t) − sin(Ωk t) αk − β k Iσ30 .

(57)

(58)

k=1

The decoherence effects are contained entirely in r(t), which describes a rotation / contraction about the z-axis due to the interaction of the system with its environment [Paz and Zurek, 2001].

14 VI. CONCLUSIONS

We have shown how the multiparticle spacetime algebra provides for novel representations of the degrees of freedom held in a density operator. Precisely which representation is appropriate will depend on the application. If one is interested in the eigenstructure of the density operator, or its entropy, it will usually be easiest to use the “literal” Hermitian representation and, if necessary, to convert it to an explicit matrix representation. For applications involving pure state evolution and entanglement, where we are ultimately interested in reduced density matrices, the “even” representation developed here has a lot to recommend it. Many calculations can be streamlined by employing the advantages of the embedding within a geometric algebra, which can help both to simplify and understand many computations. In future work we aim to utilise this representation to construct more detailed models of decoherence formed from more realistic interactions between the system and environment.

Acknowledgments

This work was supported in part by the Cambridge–MIT Institute, Ltd.

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