Quantum Information Approach to Bosonization of Supersymmetric Yang-Mills Fields

Quan tum Information Approac h to Bosonization of Sup ersymmetric Y ang-Mills Fields Radhakrishnan Balu † and S. James Gates, Jr. Abstract. W e consider b osonization of supersymmetry in the con text of W ess-Zumino quan tum mechanics. Our motiv ation for this inv estigation is the flexibilit y the bosonic fo ck space affords as an y classical probability distri- bution can be realized on it making it a v ersatile framework to w ork with for quantum pro cesses. W e proceed by constructing a minimal bosonization of a system with one bosonic and t wo fermionic degrees of freedom. W e iterate this process to construct a tow er of SUSY systems that is akin to unfolded Adinkras. W e then iden tify an osp (2 | 2) symmetry of the system constructed. T o build an irreducible represen tation of the system we induce represen ta- tions across the sectors, a first to our knowledge, as the previous work hav e focused on induction only within the b osonic sector. First, we start with a fermionic representation using Clifford algebras and then induce a represen- tation to g l (2 | 2) and restrict it to osp (2 | 2). In the second method, we induce a representation from that of the b osonic sector. In b oth c ases, our repre- sentations are in terms of qubit op erators that provide a w a y to solv e SUSY problems using quan tum information based approaches. Depending up on the direction of induction the represen tations are suitable for implemen tation on a hybrid qubit and fermionic or b osonic quantum computers. 1. In tro duction Among the most interesting developmen ts regarding the classes of b osonic ver- sus fermionic field operators are the existences of b osonization or fermionization transformations b etw een the tw o classes. A num b er of historical preceden ts led to this legacy . These include w orks of Jordan-Wigner [1], T omonaga [2], Luther- P eschel [3], and Mattis-Lieb [4]. While these transformations are often referred to as ‘Klein T ransformations,’ it seems this name was app ended fairly late in the history of these developmen ts in quantum field theory [5]. In this lineage of previ- ous works, one of the authors (SJG), considered the Klein T ransformation in the con text of low dimensional sup ersymmetric field theories in the pap ers [6, 7]. Our contribution in this work has three parts. W e constructed an infinite tow er of SUSY systems and the q-deformed version describ es SUSY breaking. W e then 2010 Mathematics Subje ct Classific ation. Primary sup ersymmetry; Secondary Y ang-Mills field theory . Key words and phr ases. sup ersymmetry; Y ang-Mills Fields; deformed Heisenberg algebras; bosonization. † Corresponding Author. 1 2 RADHAKRISHNAN BALU † AND S. JAMES GA TES, JR. built IRRs for the sup er Lie group ops (2 | 2) using the to ols of induced representa- tions. W e carried out the constructions using Mack ey machinery in tw o directions of b osonic-to-fermionic and vice versa as the previous work, in the SUSY context, induced representations only within the b osonic sector [28, 30]. These earlier in- ductions inv olved sup er Lie subgroups to sup er Lie groups with the same sup er Lie algebras and so only within the bosonic sector one could ac hieve generalization of symmetry . W e related the constructions to highest weigh t IRRs that are ubiq- uitous [24] in reduced Lie algebra techniques. Finally , w e cast the c onstructions in the language of quan tum information pro cessing and indicated possible quan tum computing formalisms for realizing them. 2. Preliminaries W e can Z 2 grade a bosonic fock space to describe a supersymmetrical fock space with fermionic and b osonic degrees of freedom. A Klein op erator, K , that satisfies K 2 = 1; { K, a } = 0 = { K , a † } enables the realization of a SUSY system as it splits the fo ck space with the orthonormal states | n ⟩ = ( n !) − 1 / 2 ( a † ) n | 0 ⟩ , n = 0 , 1 , 2 , . . . ; a | 0 ⟩ = 0 into o dd and even spaces K | n ⟩ = κ ( − 1) n | n ⟩ . Without loss of generality , we c an as- sume κ = +1 and the op erator can b e realized using the Euler’ formula as K = e iπ N = cos ( π N ) + isin ( π N ) = cos ( π N ) = ( − 1) N , N = a † a, N | n ⟩ = n | n ⟩ . In the co ordinate represen tation we can express the creation and annihilation op er- ators as a † = 1 √ 2 ( x − ip ) . (2.1) a = 1 √ 2 ( x + ip ) . (2.2) p = id/dx. (2.3) In this representation K can be though t of as a parity op erator that acts on the w av efunction as K ψ ± = ± ψ ± , ψ ± = 1 2 ( ψ ( x ) ± ψ ( − x )) . (2.4) The abov e bosonic oscillator can b e deformed by a parameter ν that can b e describ ed the Heisenberg algebra satisfying the relation a † = 1 √ 2 ( x − ip ) . (2.5) a = 1 √ 2 ( x + ip ) . (2.6) [ a, a † ] = 1 + ν K . (2.7) p = − i ( d dx − ν 2 x K ) . (2.8) With the v alue of κ as abov e, we get this expression for the num b er op erator op erating in the states | n ⟩ = C n ( a † ) n | 0 ⟩ as N | n ⟩ = [ n ] ν | n ⟩ , [ n ] ν = n + ν 2 (1 + ( − 1) n +1 ) Quantum Information Approach to Bosonization of Supersymmetric Y ang-Mills Fields 3 . It follo ws that when ν > − 1 the deformed algebra has the basis | n ⟩ with the normalization co efficient C n = ([ n ] ν !) − 1 2 , [ n ] ν ! = n Y k =1 [ k ] ν . (2.9) and the num b er op erators are expressed as N = 1 2 { a, a † } − 1 2 ( ν + 1) . (2.10) The deformed annihilation and creation op erators can b e expressed as a ± = 1 √ 2 ( x ∓ ip ) , p = i ( d dx − ν 2 x K ) . (2.11) No w let us consider W ess-Zumino quantum mechanics (WZQM) which has one complex b osonic v ariable ϕ ( t ) = x ( t ) + iy ( t ) and one complex fermionic degrees of freedom ψ ( t ) α , α = 1 , 2. With this set up the Hamiltonian reads as H = 1 2 ( p 2 x + p 2 y ) + | ϕ + g 0 ϕ 2 | 2 + ((1 + 2 h 0 ϕ ) ψ ψ † + h.c. ) . (2.12) The corresp onding sup ercharges are: Q = 1 √ 2 i ( p + ( ϕ + g 0 ϕ 2 )) ψ + ( ϕ + g 0 ϕ 2 )) ψ † . (2.13) Q † = 1 √ 2 i ( p − ( ϕ + g 0 ϕ 2 )) ψ † + ( ϕ + g 0 ϕ 2 )) ψ . (2.14) W ( x ) = ( ϕ + g 0 ϕ 2 ) . (2.15) { ψ , ψ † } = 1 . (2.16) ψ 2 = 0 . (2.17) ( ψ † ) 2 = 0 . (2.18) QQ † = 1 2 ψ ψ † ( p 2 + i [ p, W ′ ( x )] + ( W ′ ( x )) 2 ) . (2.19) Q † Q = 1 2 ψ † ψ ( p 2 − i [ p, W ′ ( x )] + ( W ′ ( x )) 2 ) (2.20) as [p, x] = -i and [p, f(x)] = -idf(x)/dx. (2.21) ψ = a − K. (2.22) ψ † = a + K. (2.23) One w ay to construct a sp ontaneous SUSY breaking system is to compose an o dd op erator and pro jectors Π ± = 1 2 (1 ± K ) [9]. This follo ws from the fact that the op erator 1 + K annihilates odd states, and when comp osed with an o dd op erator, w e get a nilp otent sup erc harge. More formally , ( Q Π ± ) 2 = Q Π ± Q Π ± . = − Q 2 Π ± . = 0 . 4 RADHAKRISHNAN BALU † AND S. JAMES GA TES, JR. As the ab ov e Q op erators are odd, we can set up new sup erc harges as ˆ Q = Q Π ± for an N = 2 SUSY system. The new Hamiltonian will hav e additional terms as; 2 ˆ H = { ˆ Q ∗ , ˆ Q } . 2 ˆ H = Q ( K + 1) Q † (1 − K ) + Q † (1 − K ) Q (1 + K ) , = − QK Q † K + QK Q † − QQ † K + QQ † − Q † K QK − Q † K Q + Q † QK + Q † Q. = QQ † + Q † Q + ( − QQ † + Q † Q ) K + ( − QK Q † − Q † K Q ) K + QK Q † − Q † K Q. = QQ † + Q † Q + ( − QQ † + Q † Q ) K + ( QQ † + Q † Q ) + ( − QQ † + Q † Q ) K. = H + [ Q † , Q ] K. Replacing K with K + 1 and K − 1 in the Hamiltonian (equation (2.12)) we get t wo SUSY systems for WZQM. No w, our fermionic v ariables are a ± ( K ± I ) and it is easy to verify that they satisfy the fermionic relations. Let us lo ok at their sp ectrum and determine whether SUSY is sp ontaneously broken. The original WZQM Hamiltonian has a unique zero-energy ground state, that is, it is an exact SUSY system. So, a − K a + K has to be zero for the n = 0 state only . In the new Hamiltonian, there are three additional terms for each of the t wo SUSY systems. Let us first consider the case of K + 1 and the last term inv olving the fermionic v ariables at the Hamiltonian has; a − K a + K + 1 + a − K + a + K . This has a v alue of 1 at n = 0 and again 1 at n = 1 providing degeneracy and th us sp on taneous SUSY breaking. W e can reason similarly for the K − 1 case and find that SUSY is again sp on- taneously broken. Using the deformed algebra, we can c hange the scale of SUSY breaking. The ν -deformed v ersion of the Hamiltonian will hav e the p term suitably mo di- fied b y the ν parameter (equation (2.11). F ermionic op erators are deriv ed from the Jordan-Wigner transformation and deformation applied to the K term as 1 ± K and using (2.10). No w, w e hav e tw o SUSY systems, and the one corresp onding to 1 − K will lead to sp ontaneous symmetry breaking with different degrees. This construction generalizes easily to Hamiltonians with m ultiple b osonic or fermionic v ariables as we can take the corresp onding sup ercharges and create new ones using the Klein pro jectors. In fact, this construction can b e rep eated infinite times and eac h time w e get an additional commutator term in the Hamiltonian. With the aid of a deformed harmonic oscillator w e can rep eat the construction increasing the degree of SUSY breaking with each iteration. 3. O S p (2 | 2) sup ersymmetric YM systems Again, follo wing the constructions of [9] we set up the even and o dd generators of the sup ergroup O S p (2 | 2) as: T 3 = 1 2 ( a + a − + a − a + ) . T ± = ( a ± ) 2 . J = − 1 2 ϵK [ a − , a + ] . Q ± = Q ∓ ϵ . S ± = Q ∓ − ϵ . (3.1) Quantum Information Approach to Bosonization of Supersymmetric Y ang-Mills Fields 5 It is easy to verify that these op erators satisfy the comm utator relationship of the O S p (2 | 2) superalgebra. This description of the sup er YM system co vers b oth exact and brok en SUSY dep eding upon whether the parameter ϵ is ± . Ho wev er, to mak e this useful, that is to build the larger symmetry (exact SUSY) from the smaller one (broken SUSY) we need the represen tation theory of Lie super algebras. W e w ould like to do this b ecause we can construct the sp ectrum of the larger system from that of the smaller one using branching rules or intert winers (generalized Clebsc h-Gordon co efficien ts). In other words, from the sp ectrum of the fermionic (or bosonic) sector of the system w e can compute all the energy levels of the SUSY system. Our goal here is not to describe the formation of sup ersymmetry , rather the computation of SUSY representations from that of the simple symmetric repre- sen tations. 4. Systems of Imprimitivit y Mac key mac hinery to construct systems of imprimitivity (SI) constitute a com- prehensiv e set of to ols to c haracterize the unitary represen tations of Lie groups. SI for a lo cally com pact group (that includes compact groups) suc h as Poincar ´ e is a comp osite ob ject ( G, Ω) of a representation of a group G and its action on a G -space Ω which is usually the configuration space of the system under consid- eration. W e denote the system of imprimitivit y ( G, Ω) lives on the G − orbit Ω. Mac key machinery enables induction of irreducible represen tations of a group G , from that of a subgroup H , that are systems of imprimitivity . The configuration spaces of in terest to us in this work are orbits of Pauli and Clifford groups and the homogeneous space G/H , where H is a closed subgroup of G that consists of left cosets g H , g ∈ G . F rom SI characterizations, we can derive the canonical comm utation relations and infinitesimal forms in terms of equations (Schr ¨ o dinger, Heisen b erg, and Dirac etc) [14] [26] and [16]. In this w ork, we will apply the SI tec hniques to build Heisenberg-W eyl op erators on sup er Hilbert spaces and then construct a represen tation for the sup er Lie group osp (2 | 2). In our earlier w ork, w e hav e constructed cov ariant Quantum Fields via Lorentz Group Representation of W eyl Op erators [27, 25]. Here, we specialize the techniques for compact Lie groups and in particular to osp (2 | 2). 5. Stabilizer subgroups (Little groups) In this section w e describ e some examples of systems of imprimitivity that live on the orbits of the stabilizer subgroups of Poincar ´ e. It is goo d to k eep in mind the picture that SI is an irreducible unitary representation of Poincar` e group P + induced from the represen tation of a subgroup such as S O 3 . This is a subgroup of homogeneous Lorentz, as ( U m ( g ) ψ )( k ) = e i { k,g } ψ ( R − 1 m k ) where g b elongs to the R 4 p ortion of the Poincar` e group, m is a member of the rotation group. The duality b etw een the configuration space R 4 and the momentum space P 4 is 6 RADHAKRISHNAN BALU † AND S. JAMES GA TES, JR. expressed using the character the irreducible representation of the group R 4 as: { k , g } = k 0 g 0 − k 1 g 1 − k 2 g 2 − k 3 g 3 , p ∈ P 4 . (5.1) ˆ p : x → e i { k,g } . (5.2) { Lx, Lp } = { x, p } . (5.3) ˆ p ( L − 1 x ) = ˆ Lp ( x ) . (5.4) In the ab ov e, L is a matrix represen tation of homogeneous Loren tz group acting on R 4 as w ell as P 4 and it is easy to see that p → Lp is the adjoint of L action on P 4 . The R 4 space is called the configuration space and the dual P 4 is the momentum space of a relativistic quan tum particle. The stabilizer subgroup of the P oincar ` e group P + (Space-lik e particle) can b e describ ed as follo ws: [19]: The Loren tz frame in whic h the particle is at rest has momen tum prop ortional to (0,1,0,0) and the little group is again SO(3, 1) and this time the rotations will c hange the helicity . In a similar fashion, in this w ork w e are interested in subgroups of Pauli group that ha ve stability p oints (states). In the Lie algebraic settings they play the role of Cartan subalgebras or maximal tori. Definition 5.1. A super Hilb ert space is a Z 2 -graded sup er v ector space H = H 1 ⊕ H 2 o ver C with a scalar product ( ., . ), where the H i ( i = 0 , 1), referred as ev en and o dd, are closed m utually orthogonal subspaces. W e set up the parity op erator as p ( x ) = ( 0 , if x ∈ H 0 , 1 , if x ∈ H 1 . W e define an even sup er Hilb ert form ⟨ x, y ⟩ =      0 , if x and y are of opp osite parity ( x, y ) , if x and y are even i ( x, y ) , if x and y are o dd . W e hav e ⟨ y , x ⟩ = ( − 1) p ( x ) p ( y ) ⟨ x, y ⟩ . If T ( H → H ) is a b ounded linear op erator, we denote by T ∗ its Hilb ert space adjoin t and by T † its sup er adjoint giv en b y ⟨ T x, y ⟩ = ( − 1) p ( T ) p ( x ) ⟨ x, T † y ⟩ . Here, p ( T † = p ( T ) and the parity of the op erator as T is even or o dd. Definition 5.2. A super Lie group is ( G 0 , g ) is a sup er Harish-Chandra pair if G 0 is a classical Lie group and g is a sup er Lie algebra with an action of G 0 on it such that (i) Lie( G 0 )) = g 0 = the even part of g . (ii) The action of G 0 on g is the adjoint action of G 0 ; more precisely , the adjoin t action of G 0 on g is the differen tial of the action of G 0 on g . A represen tation of a sup er Lie group is a triplet ( π , γ , H ), where π is an even representation of G 0 in a sup er Hilb ert space H and γ is a sup er represen tation of g in H . Quantum Information Approach to Bosonization of Supersymmetric Y ang-Mills Fields 7 Definition 5.3. A sup er Lie algebra is a sup er vector space g with a bilinear brac ket [ , ] such that g 0 is an ordinary Lie algebra with [ ., . ] and g 1 is a g 0 -mo dule for the action a → ad ( a ) : b → [ a, b ] , ( b ∈ g 1 ). F urther, a ⊗ b → [ a, b ] is a symmetric g 0 -mo dule map from g 1 ⊗ g 1 to g 0 . It also satisfies the nonlinear condition [ a, [ a, a ]] = 0 , ∀ g ∈ g 1 . One w ay , to ensure this last condition is met, is to require that the range of the o dd brack et g 2 is a subset of g 0 whic h acts on g 1 trivially as g 2 ⊂ g 0 ⇒ [ g 1 , g 0 ] = 0 . A super algebra A is an algebra of endomorphisms of linear maps on a super v ector space V . The maps that preserv e he grading of V are designated as even and those rev erse it are called o dd. T o get a sup er Lie algebra from A we ca use the brack et [ a, b ] = ab − ( − 1) p ( a ) p ( b ) ba. Let us recollec t the notions of systems of imprimitivit y (SI) and an imp ortant result b y Mac key that characterizes suc h systems in terms of induced represen- tations, k ey notions in Clifford algebras, spinor fields, and Sch wartz spaces [10], b efore discussing our main result in the sup er context. W e provide the SUSY gen- eralizations along with their classical coun terparts using the notations and notions from the works of V aradara jan [10, 11, 30]. Definition 5.4. [10] A G-space of a Borel group G is a Borel space X with a Borel automorphism ∀ g ∈ G, t g : x → g · x, x ∈ X suc h that t e is an identit y (5.5) t g 1 ,g 2 = t g 1 t g 2 . (5.6) The group G acts on X transitively if ∀ x, y ∈ X , ∃ g ∈ G such that x = g · y . Definition 5.5. [10] A system of imprimitivity , for a group G acting on a Hilb ert space H , is a pair (U, P), where P : E → P E is a pro jection v alued measure defined on the Borel space X with pro jections defined on the Hilb ert space and U is a representation of G satisfying U g P E U − 1 g = P g · E (5.7) Systems may b e decomp osed in to SI ( G 0 , Ω = G 0 /H 0 ), where H 0 is a closed subgroup (for example a stabilizer subgroup) of G 0 , and a stabilizer at ω 0 ∈ Ω on orbits b y the transitive actions of the group and there exists a functor b et w een the category of unitary representations of H 0 and the category of SI ( G 0 , Ω). In the case of Poincar ´ e group, with stabilizer subgroups as the three little groups with constan t momentum in a reference frame, a transitive SI is of in terest to us and so w e use the sp ecialized version of the Mac key machinery . If σ be a representation of H 0 on a Hilb ert space K σ , then there is a canonical SI ( π σ , P σ ) for G 0 based on Ω with the represen tation induced by that of H 0 and the natural pro jection v alued measure on K σ . The Hilb ert space is the set of equiv alence classes of measurable 8 RADHAKRISHNAN BALU † AND S. JAMES GA TES, JR. functions f : G 0 → K σ satisfying: f ( xη ) = σ ( η ) − 1 f ( x ) , for almost all η ∈ H 0 . (5.8) Z | f ( x ) | 2 K σ dx < ∞ . (5.9) The representation π σ acts by left translations and the SI relation σ → ( π σ , K σ ) states that there is a functor that exists b et ween the category of unitary repre- sen tations of H 0 and the category of SI based on Ω. One can develop an in tuition [30] for this construction as the Hilbert space K σ is attached to the fixed point ω 0 . F or all the non-fixed p oints ω = g [ ω 0 ] a Hilb ert space K σ ω is asso ciated via an unitary isomorphism. This results in a fib er bundle V σ = K σ × G 0 / ∼ , where the equiv alence relation is defined b y ( g , ψ ) ∼ ( g η , σ ( η ) − 1 ψ ). The group G 0 has a a natural right action on the bundle. // F or constructing the irreducible representations of osp (2 | 2) we pro ceed by in- v oking the standard version of systems of imprimitivit y techniques that use rep- resen tations of closed subgroups to induce representations of a larger group. This means, we induce a represen tation of [28] I nd osp (1 | 2) osp (0 | 2) : osp (0 | 2) → osp (1 | 2) . W e can repeat the process to get the representation for osp (2 | 2), though w e will not detail the steps here. More precisely , we induce a representation of I nd g l (1 | 2) osp (0 | 2) : osp (0 | 2) → g l (1 | 2) and then use the fact that the group osp ( n | m ) embeds in the sup er Lie group g l ( n | m ). When an IRR is restricted to a subgroup the resulting represen tation ma y not b e irreducible. Here, the representations (fundamental) are on the same base Hilb ert space (carrier space of the representation) of C 2 ⊗ s C 2 ⊕ C 2 ⊗ a C 2 and so an IRR of g l (2 | 2) when restricted to osp (2 | 2) will still be an IRR. W e accomplish this generalization of symmetry by reversing the pro cess describ ed ab ov e. That is, we start with a representation of the smaller algebra osp (0 | 2) and increase the dimension by 1 using the matrix unit e i i and construct the larger algebras as previously . Let us now state and discuss the main result for the case of osp (2 | 2) with Clifford group by constructing a strict co cycle from the represen tation of a subgroup follow- ing the prescription (lemma 5.24) in V aradara jan’s text. The SI is a consequence of strict co cycle prop ert y and the construction is not canonical. This construc- tion enables the application of quan tum information pro cessing techniques to the sup ersymmetric con text. That is, starting from qubit system w e can generate a represen tation of the group osp (2 | 2). Alternately , w e can start from the b osonic qubits with P auli group as the symmetry and construct a representation for the orthosymplectic group. Although, we c hose to use tw o-qubit representations to make the actions on the b osonic and fermionic sectors explicit. This means, instead of defining the action for g ∈ G we define the action ( g , g ) ∈ G × G as the constructions easily generalizes to n-qubit representations on the b´ eb ´ e (finite dimensional) fo ck space. Quantum Information Approach to Bosonization of Supersymmetric Y ang-Mills Fields 9 Definition 5.6. The Pauli group P 1 = g l (2 | 0) has the generators I 2 , ± i, ± 1 and X =  0 1 1 0  , Z =  1 0 0 − 1  , Y =  0 − i i 0  . The generalized v ersion is P n = ⟨ i I , X, Y , Z ⟩ ⊗ n . The Clifford group corresp onding to the namesake algebra C 1 = g l (0 | 2) with the generators H = 1 √ 2  1 1 1 − 1  , S =  1 0 0 i  . The generalized version is C n = { U | U g U † ∈ P n , ∀ g ∈ P n } . Matrix units: M ij = | e i ⟩ ⟨ e j | ; Matrix with a 1 at the ij p osition . A n = I n − 1] ⊗ | e 0 ⟩ ⟨ e 1 | ⊗ I [ n +1 . A † n = I n − 1] ⊗ | e 1 ⟩ ⟨ e 0 | ⊗ I [ n +1 . A n A † n = I n − 1] ⊗ | e 0 ⟩ ⟨ e 0 | ⊗ I [ n +1 . Λ n = A † n A n = I n − 1] ⊗ | e 1 ⟩ ⟨ e 1 | ⊗ I [ n +1 Num b er op erator . The matrix units form the canonical basis the Pauli operators can b e defined in terms of them and the stability states remain the same. Eac h of the four Bell states are stabilizer states of the Pauli subgroup { Z 1 Z 2 , X 1 X 2 } that satisfies the relation [ Z 1 Z 2 , X 1 X 2 ] = 0 making it a comm utative one. This is a p ossible example of a subgroup to induce represen tations as detailed b elow. Theorem 5.7. R epr esentation of the sup er gr oup g l (2 | 2) on the Hilb ert sp ac e sp ac e C 2 ⊗ s C 2 ⊕ C 2 ⊗ a C 2 is a tr ansitive system of imprimitivity living on the orbit Ω 2 ⊂ C 2 ⊗ C 2 ⊕ C 2 ⊗ a C 2 (antisymmetric tensor) induc e d fr om a r epr esentation of the Cliffor d sub gr oup C 2 . Pr o of. W e start with this equiv alence gl (0 | 2) ≃ sl (2) ⊕ k I , k ∈ C and consider the corresp onding Clifford algebra and its spinor representation S P . In this represen- tation the P auli op erators act on the complex an ti-symmetric space. As a compact Lie group is completely characterized by its Lie algebra we use these tw o notions in terchangeably . Instead of using the matrix units, that are canonical basis for the g l group, for the b osonic op erators w e use the Pauli op erators. The Pauli subgroup P A of P 2 with the fermionic (to distinguish from the b osonic ones) stabilizer states as the stability points is a closed subgroup of gl (1 | 2) with the representation S P . The usual SI techniques, as opp osed to the sup er v ersion of SI, allo w us to set up the fib er bundle across the fermionic-b osonic 10 RADHAKRISHNAN BALU † AND S. JAMES GA TES, JR. sectors as: F B 1 = { ( v , p ) ∈ C 2 ⊕ C 2 ⊗ a C 2 , v ∈ C 2 , p ∈ Ω 1 ⊗ C 2 ⊂ C 2 ⊗ a C 2 , (5.10) H ( v , p ) = 0 , H is a SUSY Hamiltonian. (5.11) Ω 1 = S × S − orbits of fermionic stabilizer states . (5.12) π :( v , p ) → p. (5.13) and we can define the fib ers as C 2 at a stability point x and isomorphic fib ers at the x-orbit of P A 2 . The Z 2 − graded fiber bundle has the p olarity and inner pro duct defined according to Definition 5.1. W e take the s ections ϕ : p → ϕ ( p ) of the fib er as the Hilb ert space for the representation. The inner pro ducts of the sup er Hilbert space C 2 ⊕ C 2 ⊗ a C 2 are defined on the bosonic and fermionic sectors in the usual wa y . F or example, the norm is defined as: ∥ ϕ ( p ) ∥ 2 = Z Ω p − 1 0 ⟨ ϕ ( p ) , ϕ ( p ) ⟩ dα ( p ) , where α is the Haar me asure of the compact Lie group . The FB is cov arian t with resp ect to the op erators gl (1 | 2) by transforming ( v , p ) ( g | h ) as ( P 2 ( g ) − 1 v , S P 2 ⊗ S P 2 ( h ) p ). Then, the unitaries representing gl (1 | 2) are U g ,h ( p ) = ϕ ( S P − 1 2 ( h ) p ) ( g | h ) and the irreducible representation is an SI living on Ω 1 = Orbits of the stabilizer states. W e can rep eat the step from going to one level higher by using again C 2 as the fiber on the new bundle and taking the symmetric tensor with the fib er of the previous bundle. W e no w hav e several c hoices in selecting the stabilizer p oin ts of the b osonic P auli op erators to induce the representations from the subgroups generated by P 1 . Let us detail the representation induced from the subgroup generated by X as: F B 2 = { ( w , ( v , p )) ∈ C 2 ⊗ s C 2 ⊕ C 2 ⊗ a C 2 , w ∈ C 2 , v ∈ Ω 2 ⊂ C 2 , (5.14) H ( w , ( v , p )) = 0 , H is a SUSY Hamiltonian. (5.15) Ω 2 = X × X − orbits of the boso nic stabilizer states ⊗ Ω 1 (5.16) π :( v , p ) → p. (5.17) Again, the FB is co v ariant with resp ect to the op erators gl (2 | 2) b y transforming ( w , ( v , p )) ( f ,g | h ) as ( P 1 ( f ) − 1 w , ( P 1 ( g ) − 1 v , S P 2 ⊗ S P 2 ( h ) p )). Then, the unitaries represen ting gl (2 | 2) are U f ,g ,h ( p ) = ϕ ( S P − 1 2 ( h ) p ) ( f ,g | h ) is an SI living on Ω 1 ⊕ Ω 2 . The creation and annihilation op erators of the b osonic sector can b e constructed using the Pauli X and Y op erators, that is by reversing co ordinate representation discussed ab ov e. □ No w, we pro ceed in the other direction by starting from a represen tation of the Heisenberg-W eyl group and induce a representation. This time we use a b´ eb ´ e Quantum Information Approach to Bosonization of Supersymmetric Y ang-Mills Fields 11 fo c k space and the matrix units basis to construct the annihilation, creation, and n umber operators so that we can get b osonized represen tations of the system. Theorem 5.8. Bosonize d r epr esentation of the gr oup g l (2 | 2) on the b´ eb ´ e fo cke sp ac e L n i =1 H 2 ⊗ i s , the Hilb ert sp ac e H wil l b e define d as p art of the pr o of, Z 2 gr ade d by the Klein op er ator K i = cos 2 iπ Λ i , is a tr ansitive system of imprimitivity on the orbit of a sele cte d stability state and induc e d fr om a r epr esentation of the Pauli sub gr oup P n on the b osonic se ctor. Pr o of. As b efore, let us consider the subgroup group of Pauli group P n with stabilit y points (states) that is a closed subgroup of g l (2 | 2). This time, w e will induce the representation in one step as g l (2 | 0) → g l (2 | 2). The SI mac hinery allo ws us to set up the base of the fib er bundle as: F B = { ( v , p ) ∈ K n ( C 2 ⊗ n s ) , v ∈ fermionic sector b ecause of Klein op erator } , (5.18) p ∈ Ω , in the orbits of the selected stability p oint (5.19) H ( v , p ) = 0 , H is a SUSY Hamiltonian. (5.20) π :( v , p ) → p. (5.21) and w e can define the fib ers as C 2 ⊗ j s and compose the Klein op erator with the Hamiltonian to get the fermionic and b osonic sectors. The Z 2 − graded fib er bun- dle F B i has the polarity and inner pro duct defined according to Definition 5.1. The P i unitaries act on the fib ers of the bundle. W e tak e the sections of the fib er bundle as the Hilb ert space H i for the representation with the action of en tire g l (2 | 2) defined on the b ´ eb´ e fock space ⊕ n i =1 H 2 ⊗ i s . The fiber bundles F B i are cov ariant with resp ect to the op erators gl (2 | 2) by transforming ( v , p ) ( g | h ) as ( P i ( g ) − 1 v , P i ( h )). Then, the unitaries represen ting gl (2 | 2) are U g ,h ( p ) = ϕ ( P − 1 i ( h ) p ) ( g | h ) is an SI living on Ω = Orbits of the stabilizer states. □ Sev eral remarks are in order based on the ab o ve results: (1) In the ab o ve using the subspace S 2 ⊂ C 2 will describ e the qubit represen- tations. The qubit normalization constrain ts can b e incorp orated while defining the bundle instead of the plain cross pro duct. (2) The fiber at the stability p oin t, and thus at all the p oin ts in the orbit, is an one dimensional space b ecause of the c hoice of our p oin t. These details are imp ortant while designing quantum circuits for implemen tation of the bundle. (3) F or a sp ecific SUSY with a Hamiltonian (2.12) this relationship can be enco ded in defining the bundle. (4) W e used the antisymmetric and symmetric tensor pro ducts to describ e the b osonic and fermionic qubits respectively hiding the details such as the Jordan-Wigner transformations if a fermionic quantum computer is not used. 12 RADHAKRISHNAN BALU † AND S. JAMES GA TES, JR. (5) W e can generate an IRR by disjoint union of of multi-particle sector as M n ≥ 0 P n ⊕ C n . (6) With the aid of the constructions discussed earlier and using the Klein op erator K we hav e em b edded the fermionic fo ck space in to the bosonic one. (7) As the Clifford group is a normalizer of Pauli group (that is it embeds in it), the ab ov e theorems can b e combined in to single one using b osoniza- tion. As the op erators of b oth sectors are the usual qubit operators and with the b osoniztion, the system can b e studied using circuit quantum electro dynamics (C-QED) based quantum computation [31]. (8) W e induced represen tations that are stability subgroups generated b y a single op erator. So, they are the center (Casimir op erators) of the algebra or the maximal torus of the algebra. The stability p oin ts are eigen vectors with the eigen v alue 1. In essence, we hav e constructed IRRs with highest w eight ( λ = 1) with resp ect the Cartan subalgebras thar are generated by a single op erator. This connection is significant as sev eral Lie algebraic represen tation approac hes are cast in terms of highest weigh t v ectors while reducing the symmetries. This connection enables to go other w ay around b y considering the corresp onding groups and induce representations of larger symmetries. (9) The q-deformed generalization of the ab o v e construction is rather straight forw ard and so we omitted the details. 6. Summary and conclusions W e approached bosonization of SUSY using the to ols of group representation theory . W e illustrated the tec hniques of Mack ey machinery for SUSY WZQM sys- tem by inducing representations from the fermionic to b osonic and the vice verza. T o illustrate the p o w er of b osonization, we constructed a tow er of SUSY systems. W e connected our induced representation to highest w eight represen tation with λ = 1 and the Pauli subgroup with stability states serving as the maximal torus or the Cartan subalgebra. This connection op ens up the wa y to lift the widely a v ailable represen tation results on reductive Lie algebraic metho ds on compact Lie groups to go in the opp osite direction, that is constructing a larger symmetry from smaller ones. In the next step, w e will fermionize an Adinkra and develop circuits for its implementation suitable for a fermionic quantum computer. 7. App endix Ac kno wledgmen t. 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