Axiomatic complexity in quantum field theory and its applications

# Axiomatic complexity in quantum field theory and its applications

Run-Qiu Yang Quantum Universe Center, Korea Institute for Advanced Study, Seoul 130-722, Korea    Yu-Sen An Institute of Theoretical Physics, Chinese Academy of Science, Beijing 100190, China School of physical Science, University of Chinese Academy of Science, Beijing 100049, China    Chao Niu School of Physics and Chemistry, Gwangju Institute of Science and Technology, Gwangju 61005, Korea    Cheng-Yong Zhang Center for High Energy Physics, Peking University, Beijing 100871, China    Keun-Young Kim School of Physics and Chemistry, Gwangju Institute of Science and Technology, Gwangju 61005, Korea
###### Abstract

Based on three general axioms for complexity, inspired by circuit complexity, and two physical assumptions, we show that the complexity of the SU() group operator is given by the length of a geodesic in bi-invariant Finsler manifolds, of which Finsler metric is uniquely determined. We also provide a new interpretation to Schrödinger’s equation for isolated systems - the quantum state evolves by the process of minimizing “computational cost.” Building on the complexity of the operator, we propose the complexity between states. For pure states, our proposal shows an interesting connection to the complexity of cMERA (continuous multi-scale entanglement renormalization ansatz) tensor networks and implies the results obtained by both the Fubini-Study metric approach and the path-integral approach. Furthermore, our proposal also reproduces the holographic complexity for the thermofield double states by the CV (complexity = volume) and CA (complexity = action) conjectures.

Introduction.– Based on the intuition that the classical spacetime geometry encodes information theoretic properties of the dual quantum field theory (QFT) in the context of gauge/gravity duality, many quantum information concepts have been applied to investigations of gravity theories. A notable example is the holographic entanglement entropy (EE) of a subregion in a QFT Ryu and Takayanagi (2006). Even though EE has played a crucial role in understanding the dual gravity, it turned out that EE is not enough Susskind (2016a), in particular, when it comes to the interior of the black hole. In the eternal AdS black hole, an Einstein-Rosen bridge (ERB) connecting two boundaries continues to grow for longer time scale even after thermalization. Because EE quickly saturates at the equilibrium, it cannot explain the growth of the ERB and another quantum information concept, complexity, was introduced as a dual to the growth of ERB Susskind (2016b); Stanford and Susskind (2014). To ‘geometrize’ the complexity of quantum states in the dual gravity theory, two conjectures were proposed: complexity-volume (CV) conjecture Stanford and Susskind (2014) and complexity-action (CA) conjecture  Brown et al. (2016), which are called holographic complexity. See also Refs Alishahiha (2015); Ben-Ami and Carmi (2016); Couch et al. (2017).

However, note that the complexity in information theory is well-defined in discrete systems like quantum circuits Aaronson (2016). On the contrary, holographic complexity is supposed to be dual to complexity in a QFT, a continuous system. Thus, there is a mismatch in duality. Compared with much progress in holographic complexity, the precise meaning of the complexity in QFT is still not complete. In information theory, one of the useful concepts of the complexity is the circuit complexity, which is the minimal number of simple elementary gates required to approximate a target operator. In order to define complexity in QFT, our strategy is: (i) extract minimal essential axioms for complexity from circuit complexity and (ii) define complexity in continuous QFT systems based on the axioms. Some physical assumptions constrain the structure of complexity and enable us to determine the complexity of the SU() operators uniquely.

Our work is also inspired by a geometric approach by Nielsen et al.  Nielsen et al. (2006); Nielsen (2006); Dowling and Nielsen (2008), where the discrete circuit complexity for a target operator is identified with the minimal geodesic distance connecting the target operator and the identity in a certain Finsler geometry Bao et al. (2000); Shen (2001); Xiaohuan (2006); Asanov (1985), which is just Riemannian geometry without the quadratic restriction.

Recently, inspired by this geometric method, Refs. Brown and Susskind (2017); Jefferson and Myers (2017); Yang (2017); Yang et al. (2018); Khan et al. (2018) explored complexity in QFT. However, in these studies, because the Finsler metric can be chosen arbitrarily, there is a shortcoming that the complexity depends on the choice of the metric. In this letter, we show that, for SU() operators, the Finsler metric and complexity are uniquely determined based on three general axioms (denoted by G1-G3) for complexity and two physical assumptions (denoted by A1-A2). Building on the complexity of the SU() operator, we propose how to compute the complexity between arbitrary two states. Our proposal turns out to be general enough to include other recent developments for the complexity in QFT: cMERA tensor network Evenbly and Vidal (2015a, b); Miyaji et al. (2015a); Molina-Vilaplana and del Campo (2018), Fubini-Study metric Chapman et al. (2017) and path-integral method Caputa et al. (2017a, b). Furthermore, our proposal also correctly reproduces the holographic complexity for thermofield double state (TFD).

Complexity and Finsler manifolds.– We first show that the Finsler metric arises from the minimal and general axioms for the complexity and the smoothness assumption of the complexity. We denote a complexity of an operator in an operators set by . Let us start with three axioms that the complexity should satisfy.

1. , and the equality holds iff is the identity.

2. ,  .

3. If and can be decomposed into the Cartesian product of two sets, i.e., , with the product satisfying for arbitrary , then , where are the identities of and .

G1 and G2 are obvious. G3 can be understood from the schematic explantation in Fig. 1. Intuitively, G3 means that the complexity of two independent tasks is the sum of their complexities.

Next, we restrict ourselves to the SU() group because the quantum circuits are realized by the unitary transformation. For a given operator , as is connected, there is a curve connecting and identity , where the curve may be parameterized by with and . See Fig. 2. The tangent of the curve, , is assumed to be given by a right generator or a left generator :

 ˙c(s)=Hr(s)c(s)   or   ˙c(s)=c(s)Hl(s). (1)

This curve can be approximated by discrete forms:

 ^On=c(sn)=δ^O(r)n^On−1=^On−1δ^O(l)n, (2)

where , , and with or and .

By assuming the complexity of any infinitesimal operator, , is a smooth function of and we have

 A1C(δ^O)=C(^I)+~F(H)δs+O(δs2),

where and by G1. Let us define the cost () of a particular proess , constructed by only or only , as

 Lα[c]:=N∑i=1C(δ^O(α)i)N→∞−−−−→∫10~F(Hα(s))ds. (3)

Geometrically, it is the length of the particular curve and looks like a “Finsler metric”. Indeed, we can prove that satisfies three defining properties of the Finsler metric:

1. (Nonnegativity) and iff

2. (Positive homogeneity) ,

3. (Triangle inequality)

only by using G1, G2 and A1! ( see appendix A of supplemental material for a proof.) Thus, here we introduce a standard notation for the Finsler metric ‘’ and

 Fα(c,˙c):=~F(Hα), with Hr=˙cc−1 & Hl=c−1˙c. (4)

Note that is right-invariant, because is invariant under the right-translation for SU(). Similarly is left-invariant.

Finally, the left or right complexity of an operator () is identified with the minimal length (or minimal cost) of the curves connecting and :

 Cα(^O):=min{Lα[c]| ∀c(s), c(0)=^I, c(1)=^O}. (5)

Path-reversal symmetry and bi-invariant Finsler manifolds.– In general, the left complexity and right complexity are different. Furthermore, because we can construct the operator in some parts by left generators and other parts by right generators we can obtain infinitely many different complexities. We will show that this ambiguity is disappeared if we assume the path-reversal symmetry for the cost of the process:

 A2Lα[c]=Lα[c−1]  or  Fα(c,˙c)=Fα(c−1,dc−1/ds),

where the curve satisfies for . This assumption is inspired by the fact that the complexity in QFT, as a physical observable, should be constrained by the general properties of QFT such as unitarity and CPT (charge, parity, time reversal) symmetry. See appendix B for more details. The condition A2 imposes the following constraints on the Finsler metric:

1. or

2. (bi-invariance)
or

C1 can be understood by A1 and A2. C2 is easily derived from A2, C1, Eq. (4) and . C2 shows that our Finsler manifold is bi-invariant (left-invariant and right-invariant). C2 also implies that the complexity is invariant under the transformation of representation:

1. (representation-independence) for and SU

Reversing the logic, we may first assume C3 for any one of and . Then C2(bi-invariance) is derived without assuming A2. Instead, A2 is derived if we further assume C1. Thus, we see A2C1+C3. From here, we drop the subscripts and for the Finsler metric because of its bi-invariance (C2).

Complexity of SU() operators.– Indeed, by using C1, C3, and G3 we can uniquely determine the Finsler metric in the operator space of any SU() groups:

 F(c(s),˙c(s))=~F(H(s))=Tr√H(s)H(s)†, (6)

where or for the curve . For a proof, see appendix C. For example, let us take as a basis of the Lie algebra in the fundamental representation and . At the identity (), and the Finsler metric reads:

 F(^I,H(0))=Tr√12Ha(0)Hb(0)(δabn^I+dabcTc), (7)

where are traceless anti-commutative structure constants of Lie algebra. Because for the Finsler metric is not Riemannian for . Only for SU(2), the Finsler metric is Riemannian. (It is Riemannian at identity and everywhere because of bi-invariance.)

Even though we have the precise Finsler metric, to compute the complexity, we still have to find a geodesic path as shown in (5). This minimization procedure is greatly simplified thanks to bi-invariance, C2. It has been shown that the curve is a geodesic if and only if there is a constant generator such that Latifi and Toomanian (2013); Latifi and Razavi (2011),

 ˙c(s)=¯Hc(s)  or  c(s)=exp(s¯H). (8)

With the condition , we can solve formally . The logarithm of a unitary operator always exists but may not be unique (theorem 1.27 in Ref. Higham (2008)). Finally, the complexity of in Eq. (5) is given by

 C(^O)=min{Tr√¯H¯H† | ∀¯H=ln^O}, (9)

because is constant. The minimization ‘min’ in (5) in the sense of ‘geodesic’ is already taken care of in (8). Here ‘min’ means the minimal value due to multi-valuedness of .

For example, let us consider the SU(2) group in its fundamental representation. For any operator SU(2), there is a unit vector and a real number such that,

 ^O=exp(iθ→n⋅→σ)=^Icosθ+i(→n⋅→σ)sinθ, (10)

where stands for three Pauli matrixes. Because with

 θm=arccos[Tr(^O)/2]+2mπ, (11)

for , the complexity of is given by

 C(^O)=2arccos[Tr(^O)/2]. (12)

where is used.

Complexity principle and Schrödinger’s equation.– In quantum mechanics with a time-independent Hamiltonian , the time evolution operator is governed by the Schrödinger’s equation,

 ˙^U(t)=−iℏ−1H^U(t),   ^U(0)=^I. (13)

Compared with the Eq. (8), can be interpreted as a geodesic generated by .

Let us consider the following question. Assume that after a very short time , the time-evolution operator becomes . Mathematically there will be many different curves connecting the and . How can we find the real curve during ? One answer is that we assume the time evolution operator will obey the Schrödinger’s equation (13). However, we may have an alternative viewpoint, namely ‘complexity principle’: For an isolated system, the time-evolution operator will be along the curve realizing the complexity among all possible curves. In terms of the quantum state acted by a time evolution operator: the quantum state evolves by the process of the minimal ‘computational cost’.

Complexity between states.– Next, we discuss the complexity between states rather than operators. Pervious works Brown and Susskind (2017); Jefferson and Myers (2017); Yang (2017); Yang et al. (2018); Chapman et al. (2017); Caputa et al. (2017a, b); Khan et al. (2018) can deal with only pure states because of their use of state vectors in Hilbert space. We will use density matrix to describe states so that our framework can be applied for pure and mixed states. The complexity between two states described by density matrices and may be defined naturally as

 ¯C(ρ1,ρ2):=min{C(^O)|ρ2=^Oρ1^O†, ∀^O∈SU(n)}. (14)

Let us first consider two pure states (=1,2). One can always find a set of bases vectors such that and (here and ). Thus the complexity is given by an SU(2) operator with and . By Eq. (12), we have

 ¯C(ρ1,ρ2)=C(^O)=2arccosa=2arccos|⟨ψ1ψ2⟩|=2DFS(|ψ1⟩,|ψ2⟩)=2arccosFi(|ψ1⟩,|ψ2⟩), (15)

where is Fubini-Study (FS) distance Chapman et al. (2017); Bengtsson (2006) and Fi is fidelity Braunstein and Caves (1994); Miyaji et al. (2015b). Thus, Eq. (14) includes the complexity defined by Ref. Chapman et al. (2017) as a special case.

Next, for mixed states, we make use of Eq. (15) by constructing a pure state () from a mixed state described by a density matrix i.e. (which is called purification). Here are any orthonormal bases vectors. It is straightforward to show that the complexity is given by the generalized fidelity for mixed states

 ¯C(ρ1,ρ2)=2arccosFi(ρ1,ρ2), (16)

where  Jozsa (1994). (see the supplemental material D for a proof.)

When the degrees of freedom are very large in chaotic systems, the fidelity will rapidly decrease to near zero (e.g., see Refs. Jacquod et al. (2001); Wisniacki (2012)), which means that will approach to rapidly as shown in Eq. (16). In such a case, it is more convenient to introduce a slightly different definition of the complexity as

 C(ρ1,ρ2):=−ln{cos2[12¯C(ρ1,ρ2)]}=−2lnFi(ρ1,ρ2), (17)

which is reduced to

 C(|ψ1⟩,|ψ2⟩)=−ln(|⟨ψ1|ψ2⟩|2), (18)

for pure states. The ‘’ here also can be understand as follows. Let us consider the complexity between two product states and . As two states belong to two different sub-Hilbert spaces, it is natural to expect that . The ‘’ is a simple way to realize this expectation.

Relations to other field theoretic proposals.– For example, let us consider the complexity between the field operator eigenstate and ground state for a given Hamiltonian. denotes the field eigenvalue of the field operator acting on . By using a Euclidean path integral representation of the inner product the complexity reads

 e−C=|⟨φ0|Ω⟩|2=∫φ(x,0)=φ0(x)D[φ(x)]exp{−1ℏSE[φ]}, (19)

where is Euclidean action for a given Hamiltonian. In the classical limit , the leading term of is approximated by,

 C≈1ℏmin{SE[φ]−SE[0] | ∀φ,s.t.,φ(x,0)=φ0(x)}, (20)

where with the ground energy and the Euclidian time . This may serve as a proof of the conjecture that complexity can be given by a classical on-shell action Caputa et al. (2017a, b). A Euclidean path integral description of a vacuum wave functional can be well approximated by a cMERA tensor network Evenbly and Vidal (2015a, b); Miyaji et al. (2015a). As inner product is just the vacuum wave functional, the Eqs. (21) and (22) may build a bridge to unify our framework and previous considerations based on tenser networks.

Relation to path-integral complexity.– For example, let us consider the complexity between the field operator eigenstate and ground state for a given Hamiltonian. denotes the field eigenvalue of the field operator acting on . By using a Euclidean path integral representation of the inner product the complexity reads

 e−C=|⟨φ0|Ω⟩|2=∫φ(x,0)=φ0(x)D[φ(x)]exp{−1ℏSE[φ]}, (21)

where is Euclidean action for a given Hamiltonian. In the classical limit , the leading term of is approximated by,

 C≈1ℏmin{SE[φ]−SE[0] | ∀φ,s.t.,φ(x,0)=φ0(x)}, (22)

where with the ground energy and the Euclidian time . This may serve as a proof of the conjecture that complexity can be given by a classical on-shell action Caputa et al. (2017a, b). A Euclidean path integral description of a vacuum wave functional can be well approximated by a cMERA tensor network Evenbly and Vidal (2015a, b); Miyaji et al. (2015a). As inner product is just the vacuum wave functional, the Eqs. (21) and (22) may build a bridge to unify our framework and previous considerations based on tenser networks.

Comparison with holographic conjectures.– Our complexity between states (18) not only reproduces other field theory approaches but also holographic results. The complexity of the thermofield double (TFD) states attracts particular interests in holographic setups. The TFD state of the free scalar field can be written as, with  Yang (2017),

 |TFD⟩=∏→k1Z(ω→k,T)∞∑n=0e−nω→k2T|n,→k⟩L|n,→k⟩R, (23)

where is temperature, is momentum, is the particle number, is the energy of a particle with momentum (here we consider a conformal theory so .) and is the normalization factor. According to Eq. (18), the complexity between the ground state (the TFD state at zero temperature limit) and reads,

 C=Sd−2Γ(d−1)ζ(d)(2π)d−1VTd−1=ζ(d)(2π)d−1dET, (24)

where is the average energy of , is the spatial volume and is the spacetime dimension. Eq. (24) agrees with the holographic complexities based on CV and CA conjectures Yang et al. (2018) up to unimportant overall factors.

The product state with no spatial correlations is frequently used as a reference state for the circuit in cMERA Haegeman et al. (2013); Nozaki et al. (2012). It is also treated as a candidate for the reference state in both the CV and CA conjectures. For free scalar field, this state can be approximately given by Chapman et al. (2017); Yang (2017),

 |R(Λ)⟩=⎛⎜⎝∏|→k|≥Λ|0,→k⟩L|0,→k⟩R⎞⎟⎠×∏|→k|<Λ1~Z(→k)∞∑n=0(tanhξ→k)n|n,→k⟩L|n,→k⟩R, (25)

where is a UV cut-off for momentum, and . According to Eq. (18) the complexity between the ground state and reads

 C∝VΛd−1. (26)

This agrees to the holographic results based on the CV and CA conjectures Carmi et al. (2017); Yang et al. (2017) up to a overall factor. It also agrees to the results from cMERA Molina-Vilaplana and del Campo (2018) and the Fubini-Study metric method Chapman et al. (2017). However, in the former, norm is assumed and, in the latter, the generators set is assumed to be SU(1,1) and norm is assumed. In our framework, Eq. (26) naturally appears without such additional assumptions.

Outlook.– In this letter we have obtained the complexity of the SU() operator without ambiguity: Eq.(9). Building on it, we proposed the complexity between arbitrary states: Eqs. (16) and (17). We have shown that Eqs. (16) and (17) reproduce other field theoretic approaches and agree with the holographic results for TFD states. However, we need to investigate in more detail to understand concrete relations between several conjectures and their physical implications. The bi-invariant complexity for operators here is different from the only right-invariant complexity for qubit systems Nielsen et al. (2006); Nielsen (2006); Dowling and Nielsen (2008); Brown and Susskind (2017) due to the locality and isotropy. One can easy to check, if we consider only the isotropic local interactions, the right-invariant metric in Refs. Dowling and Nielsen (2008); Brown and Susskind (2017) becomes bi-invariant, too.

We also want to emphasize that our complexity (16) and (17) can be used for mixed states contrary to previous proposals. It is also important to check if our complexity shows a linear growth for chaotic systems at the late time limit 111This point has been confirmed at the classical limit in our forthcoming work. Ref. Czech (2018) showed that the classical Einstein equation can be obtained from a classical on-shell Liouville action identified with complexity Caputa et al. (2017a, b). Because, we showed that such a complexity indeed equals to the full partition function rather than a classical on-shell action, it will be interesting to ask if we can see any relation between quantum gravity and complexity in our setup.

###### Acknowledgements.
The work of K.-Y. Kim and C. Niu was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT Future Planning(NRF- 2017R1A2B4004810) and GIST Research Institute(GRI) grant funded by the GIST in 2018. C.Y. Zhang is supported by National Postdoctoral Program for Innovative Talents BX201600005.

## I Appendix

### i.1 A. Proof for the Finsler metrics of Fα(c,˙c)

In this appendix, we first show three properties F1-F3 of implied by the general axioms G1, G2 and the assumption A1. F1-F3 readily show that the Finsler geometry emerges.

F1: , and .
Proof:
1⃝ If , so for by G1. Thus, by A1.
2⃝ If , such that so by G1

 0

where . By G2 we have the following inequality

 C[(exp(Hλ/N))N]≤NC[exp(Hλ/N)].

Thus,

 0

With , Eq. (27) means

 0

where is defined in A1. Thus, .

F2: , .
Proof: By A1, for an arbitrary generator and infinitesimal parameter

 C(exp(Hε))=~F(H)ε+O(ε2). (29)

For an arbitrary ,

 C[exp(H⋅λε)]=C[exp(λH⋅ε)], (30)

which implies by Eq. (29).

F3: and ,
Proof:

By G2, for arbitrary generators and

 C(exp(H1ε))+C(exp(H2ε))≥C(exp(H1ε)exp(H2ε)).

It yields, up to order ,

 C(exp(H1ε))+C(exp(H2ε))≥C[exp((H1+H2)ε)], (31)

which implies by Eq. (29).

By the relation between and in Eq. (4), we can also prove that satisfies the following properties for SU() and two arbitrary tangent vectors at :
F1’: and  .
F2’: , we have  .
F3’: .
These imply that is Finsler metric.

### i.2 B. Arguments for assumption A2

As we commented in the main text, one can show that A2C1+C3. Thus, the assumption A2 can be replaced by assuming C1 and C3. However, interestingly enough, if we consider the complexity of quantum field theory (QFT), C1 and C3 need not be assumed because they can be derived from general properties of QFT. It turns out that (1) the CPT(Charge, Parity, and Time reversal) symmetry of QFT implies C1 and (2) the unitarity of QFT implies C3. Therefore, in this sense, A2 is not an assumption but a consequence of the QFT. Here, we will explain (1) and (2) one by one.

(1) Let us consider an arbitrary CPT symmetric Lorentz invariant local quantum field in Minkowski spacetime. In Heisenberg picture, the dynamic of the quantum field is given by a time evolution operator :

 Φ(→x,t)=c(t)†Φ(→x,0)c(t), (32)

where is assumed to be an arbitrary constant generator for convenience and is the Hamiltonian of the quantum field (in this case, left and right generators are the same so we do not distinguish them). Denoting the CPT partner of by we have

 Φm(→x,t)=C∘P∘T[c(t)†Φ(→x,0)c(t)]=c(−t)†Φm(→x,0)c(−t)=cm(t)†Φm(→x,0)cm(t), (33)

where and the evolution of the CPT parter is given by . Here we used the fact that is invariant under P and C. Because and are not distinguishable in the sense that any physical observable will be invariant under the transformation , it is natural to assume that the costs of and are the same, i.e.,

 Lα[c]=Lα[cm]. (34)

The generators of are given by . Thus, we see that implies that

 ∫10~F(−H)dt=∫10~F(H)dt (35)

holds for an arbitrary constant generator , which implies that (C1).

(2) Let us consider an arbitrary quantum field with Hilbert space and vacuum , which are denoted by . Its unitary partner is , and , which are denoted by . Their time evolutions, Eq. (32), are related as

 ~Φ(→x,t)=^U†c(t)†^U^U†Φ(→x,0)^U^U†c(t)^U=~c(t)†~Φ(→x,0)~c(t). (36)

where so the evolution for the unitary partner is given by . On the other hand, we cannot distinguish and its unitary partner in the sense that any physical observable will be invariant under the transformation . Then it is natural to assume that we also cannot use complexity to distinguish them. Thus, we have to assume that the cost satisfies

 Lα[c]=Lα[^U†c(t)^U],     ∀^U, (37)

which is C3.

As an interesting comparison with other previous works, let us make a short discussion on the case where the complexity does not only depend on the operator but also depends on some quantity . In this case, the cost function becomes

 Lα[c;σ]=∫10~F(Hα,σ)ds. (38)

Let us assume under the unitary transformation , we have . Then the Eq. (37) becomes,

 Lα[c;σ]=Lα[^U†c(t)^U;σ′],     ∀^U, (39)

which implies

 ~F(^U†Hα^U,σ′)=~F(Hα,σ). (40)

As in general we have , we will find that and the complexity will not be invariant under the transformation . In this case, with the unitary equivalence we still can find some complexity geometries which are only left or only right invariant, but not bi-invariant.

### i.3 C. General Finsler metric for SU(n): proof of Eq. (6)

Proof: We prove (6) by three steps.
(1) Let us first show that is only the function of eigenvalues of and independent of the permutations of the eigenvalues. Notice that always can be diagonalized under the transformation of representation by an SU() operator :

 ^UH^U−1=diag(γ1,γ2,⋯,γn), (41)

where are the eigenvalues of . Thus

 ~F(H)=~F(^UH^U−1)=~F(diag(γ1,γ2,⋯,γn)). (42)

where the first equality comes from the self-adjoint invariance C3. To see the independence of the permutations of the eigenvalues, we consider a SU() operator yielding

 ^U′diag(γ1,γ2,⋯,γn)^U′−1=diag(γ2,γ1,γ3,⋯,γn). (43)

where and for . Again by C3 is invariant under the permutation . Similarity, is invariant under any kind of permutation on .

(2) Next, let us prove that if then , where , , Dim and stands for the -dimensional zero matrix.

Assume (=1,2) to be the representations of and one matrix representation of can be given by . Then the axiom G3 implies that the complexity should satisfy

 C(^O1⊕^O2)=C1(^O1)+C2(^O2). (44)

Here and . To be specific, let us consider two 1-parameter subgroups and with . Then is the -dimensional subgroup of . As , . From Eq. (44), we have

 C[exp(ϵH)]=C1[exp(ϵH1)]+C2[exp(ϵH2)], (45)

which, in the limit , reduces to

 ~F(H)=~F(H1⊕H2)=~F1(H1)+~F2(H2). (46)

In general, so Eq. (46) can be generalized as .

(3) Finally, we combine the results in previous two steps. Noting the fact that