Quantum informational representations of entangled fermions and bosons

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1 SPIE Defense nd Security Symposium, Quntum Informtion nd Computtion VII, Vol (009 v1.0 Quntum informtionl representtions of entngled fermions nd bosons Jeffrey Yepez Air Force Reserch Lbortory, Hnscom AFB, Msschusetts 01731, USA ABSTRACT Presented is second quntized technology for representing fermionic nd bosonic entnglement in terms of generlized joint ldder opertors, joint number opertors, interchngers, nd pirwise entnglement opertors. The joint number opertors generte conservtive quntum logic gtes tht re used for pirwise entnglement in quntum dynmicl systems. These re most useful for quntum computtionl physics. The generlized joint opertor pproch provides pthwy to represent the Temperley-Lieb lgebr nd to represent brid group opertors for either fermionic or bosonic mny-body quntum systems. Moreover, the entnglement opertors llow for representtion of quntum mesurement, quntum mps (ssocited with quntum Boltzmnn eqution dynmics, nd for wy to completely nd efficiently extrct ll ccessible bits of joint informtion from entngled quntum systems in terms of quntum propositions. Keywords: joint quntum logic, joint ldder opertors, joint number opertors, entnglement opertors, Temperley-Lieb lgebr, brid group opertors, quntum mesurement, quntum mps, quntum propositions 1. INTRODUCTION A prerequisite to comprehend the richness of quntum dynmics, first for theoreticl nd then for prcticl lgorithmic engineering resons, is cpcity to identify nd quntittively ccount for quntum stte informtion of entngled prticles in mny-body nonliner quntum systems. Furthermore, in ny future prcticl quntum computer, mesurement is lso prerequisite step, which itself my be integrl to nd intertwined with the opertion of the quntum computtionl lgorithm. Tht is, mesurement provides pthwy for exploiting for prcticl purposes the nonlinerities ssocited with wve function collpse. So it is useful to quntify how informtion, prticulrly joint informtion, is generted, trnsfered, nd extrcted during engineered quntum dynmicl evolution nd mesurement processes, respectively. The reltionships between nonlocl quntum entnglement nd nonlocl quntum mesurement re somewht mysterious, 1 so it is useful to hve toolset to explore these reltionships. Presented here is prescription for doing precisely this, with focus on pirwise entnglement. A technology is introduced for quntittive entnglement nlysis bsed on joint number opertors tht in turn re constructed using second quntiztion technology, viz., joint fermionic nd bosonic ldder opertors. Joint ldder opertors crete nd destroy entngled prticle pirs. So joint number opertor counts such entngled pirs. Moreover, these joint number opertors re the genertors of conservtive quntum logic gtes tht re used to represent the most bsic physicl opertions underlying quntum gs dynmics, prticle motion nd prticle-prticle interction. Hence, they serve s building blocks for quntum lttice-gs lgorithms for ccurtely modeling quntum gses. It is possible to hrness quntum entnglement nd mesurment to simulte the locl dynmics of quntum systems tht re otherwise notoriously difficult to efficiently simulte. Remrkbly, this type of quntum logic technology hndles fermions nd bosons eqully well, without ny dditionl complexity or computtionl overhed for nti-commuting lgebrs versus commuting lgebrs. So n importnt future ppliction, nd n outstnding experimentl gol for number of yers now, is to model strongly-coupled fermionic quntum systems on quntum computer using this kind of quntum lgorithmic Further uthor informtion: (Send correspondence to J.Y. J.Y.: E-mil: jeffrey.yepez@gmil.com, Telephone:

2 representtion. Yet the primry issue here is to consider, from n nlyticl perspective, how these joint opertors cn themselves be generlized, how they re interrelted to other known lgebrs nd group constructs, nd how we my use joint opertors to identify how entnglement is embedded in mny-body quntum system, nd how we might identify entngled stte invrints modulo topologicl chnges. Entngled pthwys between prticle pirs cused by prticle motion nd -body prticle-prticle interctions in quntum gs cn be represented s the closure of brided strnds, where the qubits tke the plce of the strnds in brid. In this regrd, n overll gend is to ctegorize types of quntum entnglement s geometricl objects with the invrints for knots nd tngles. This rticle is divided into three sections. First, I give derivtion of joint ldder opertors nd generlized joint ldder opertors. These nturlly led to the construction of entnglement opertors, either perpendiculr nd prllel ones with respect to the Bell sttes s described below. Second, I discuss interchngers, respectively for both fermions nd bosons, tht swp the quntum stte between two qubits in the system in tunble fshion. A unitry quntum (bosonic representtion of the brid group hs been used to efficiently compute the Jones Polynomil., 3 So it should not be surprising tht the bosonic joint number opertors cn stisfy the Temperley-Lieb lgebr. Yet both fermionic nd bosonic interchngers cn represent the brid group. An Hilbert spce representtion of these using second quntized mny-body ldder opertors is provided here. Third, I pply the perpendiculr nd prllel entnglement opertors to quntum mesurement nd ttempt to show how these re nturl choices for representing quntum mps nd quntum propositions. Two fetures re highlighted: (1 preservtion of informtion in quntum mesurement intrinsic to quntum mps, nd ( quntum propositionl representtions of ll ccessible informtion in quntum system for efficient retrievl in ccord with Zielinger s principle Joint ldder opertors. GENERALIZED JOINT QUANTUM LOGIC Let us consider entngling two quntum bits, sy q α nd q β, in system comprised of Q qubits nd where the integers α nd β [1, Q] re not equl, α β. Joint pir cretion nd nnihiltion opertors, 5 ct on qubit pir αβ 1 ( α e iξ β, αβ 1 ( α e iξ β, (1 nd re defined in terms of the fermionic ldder opertors α nd α. In turn, these my be represented s the tensor product of Puli mtrices: α 1 number of σ z mtrices, one singleton = 1 ( σx + iσ y or = 1 ( σx iσ y, followed by Q α number of ones: α = ( α 1 ( Q σ z k=1 ( stisfy the nti-commuttion reltions k =α+1 1 The joint number opertor corresponding to (1 is, α = ( α 1 ( Q σ z k=1 k =α+1 1. ( { α, β } = δ αβ, { α, β } = 0, { α, β } = 0. (3 n αβ αβ αβ = 1 ( n α + n β e iξ α β e iξ β α, (4 where the usul qubit number opertor is n α α α. Finlly, I introduce n entnglement number opertor (Hmiltonin with simple idempotent form ɛ δαβ n αβ + (δ 1n α n β, (5 where δ is boolen vrible (0 for the bosonic cse nd 1 for the fermionic cse.... q α... q β..., with qubit of interest locted t α nd nother t β. (5 cts on some stte

3 .. Perpendiculr nd prllel entnglement opertors For convenience, I will use shorthnd for writing stte specifying only two qubit loctions s subscripts qq αβ... q α... q β..., since the opertors ct on qubit pir, regrdless of the respective pir s loction within the system of Q qubits. Then, the entngled singlet sub-stte is 1 ( αβ nd the entngled triplet sub-sttes re 1 ( αβ nd 1 ( 11 ± 00. αβ I will refer to the ket qq αβ s perpendiculr with respect to its constituent qubits q α nd q β when q q nd s prllel when q = q. Exmple or perpendiculr nd prllel -qubit (Fock sttes re depicted in Fig αβ : 1 00 αβ : αth qubit βth qubit Figure 1. Exmple of perpendiculr nd prllel -qubit sub-sttes. The perpendiculr sub-stte 10 (top pir nd the prllel sub-stte 00 (bottom pir re depicted with qubits s unit vectors on the complex circle. Neglecting normliztion fctors, the ction of (5 on the entngled αβ sub-sttes is ɛ δ ( 01 ± 10 αβ = 1 (1 e iξ 01 1 ( 1 ± e iξ 10 ɛ δ ( 11 ± 00 αβ = δ 11. (6 As mtter of convention, the αβ indices re moved from ɛ δ to the stte ket upon which the entnglement number opertor cts nd lso the αβ indices re not repeted on the R.H.S. of n eqution if voiding redundncy does not introduce ny mbiguity. ξ = π nd ξ = 0 re two specil cses of interest. For convenience, let us denote these prticulr ngles with plus nd minus symbols (+ = π nd = 0: n ± αβ 1 ( α ± β ( α ± β, ɛ ± δαβ = n± αβ + (δ 1n αn β. (3 (3 mesures entnglement between qubits tht re perpendiculrly oriented in the 4-dimensionl αβ subspce of the Q -dimensionl Hilbert spce. The sum of these perpendiculr joint number opertors is relted to the qubit number opertors s follows: ɛ + δαβ + ɛ δαβ = n α + n β + (δ 1n α n β. (7 Suppressing indices, ɛ δ hs unity eigenvlue for the singlet stte: ɛ δ ( αβ = ɛ δ ( αβ = 0 ɛ δ ( 11 ± 00 αβ = δ 11, (8 (8b

4 but it hs eigenvlue 0 for the triplet stte, Conversely, ɛ + δ hs zero eigenvlue for the singlet stte: ɛ + δ ( αβ = 0 ɛ + δ ( αβ = ɛ + δ ( 11 ± 00 αβ = δ 11, (9 (9b but it hs eigenvlue 1 for the first triplet stte. To void ny contribution rising from prllel entnglement from the triplet sttes in (8b nd (9b, we must tke δ = 0 in the hermitin opertor (3 to ensure we count only one bit of perpendiculr pirwise entnglement. Hence, denoting the pirwise entngled sttes with qubits of interest t loctions α nd β s ψ ± ± , then these entngled sttes re eigenvectors of the δ = 0 joint number opertors (with unit eigenvlue: ɛ ± 0 ψ± = ψ±. The prllel joint opertors cn be defined in terms of the perpendiculr joint opertors s ε ± 0 1 (1 ɛ +0 ± ɛ 0 ± {σ xσ x, ɛ 1 }, (10 suppressing qubit indices nd σ x σ x σ (α x σ (β x is short-hnd for tensor product. The sum ɛ ɛ 0 + ε+ 0 + ε 0 = 1 (11 is s fundmentl to two-qubit sub-sttes s the singleton number opertor identity n + n = 1 is to one qubit sttes. If the other pirwise entngled sttes with qubits t α nd β re ψ ± ± , then these entngled sttes re eigenvectors of the joint number opertors (with unit eigenvlue: ε ± 0 ψ± summry, the opertors ɛ ± 0 nd ε± 0 re number opertors for the Bell sttes..3. Quntum gtes = ψ±. In A consistent frmework for deling with quntum logic gtes using the quntum circuit model of quntum computtion ws introduced over dozen yers go by DiVincenzo et l. 6 Here we consider n nlyticl pproch to quntum computtion bsed on generlized second quntized opertors which is lso prcticl for numericl implementtions. Let us now consider quntum gtes tht induce entngled sttes from previously independent qubits. The bsic pproch employs conservtive quntum logic gte generted by (5 e iϑ ɛ δ = 1 + (e iϑ 1ɛ δ. (1 Thus, similrity trnsformtion of the number opertors n α nd n β yields generlized joint number opertors ( ( n α(ϑ, ξ e iϑ ɛ δαβ n α e iϑ ɛ δαβ ϑ ϑ = cos n α + sin n β + i sin ϑ ( e iξ α β e iξ β α (13 ( ( n β(ϑ, ξ e iϑ ɛ δαβ n β e iϑ ɛ δαβ ϑ ϑ = sin n α + cos n β i sin ϑ ( e iξ α β e iξ β α. (13b Thus the generlized joint number opertors rotte continuously from n α (0, ξ = n α nd n β (0, ξ = n β s ϑ rnges from 0 to π to the number opertors n α (π, ξ = n β nd n β (π, ξ = n α. Tht informtion is conserved by this similrity trnsformtion is redily expressed by the number conservtion identity obtined by dding (13 nd (13b n α(ϑ, ξ + n β(ϑ, ξ = n α + n β. (14 The L.H.S. counts the informtion in its quntum mechnicl (entngled form wheres the R.H.S. counts informtion in its clssicl form (seprble form. In ny cse, the totl informtion content in the αβ sub-spce is conserved.

5 Compring the generlized joint number opertor (13 to the joint number opertor (4, we obtin the useful identity in the specil cse of mximl entnglement ( π n α, ξ + π ( π = n β, ξ π = 1 ( n α + n β e iξ α β e iξ β α = n αβ. (15 The tensor product n α n β is invrint under similrity trnsformtion Let us lso construct generlized joint ldder opertors n α n β = e iϑ ɛ δαβ n α n β e iϑ ɛ δαβ. (16 c α = e iϑ ɛ δαβ c α e iϑ ɛ δαβ c α = e iϑ ɛ δαβ c α e iϑ ɛ δαβ. (17 After some lgebric mnipultion these cn be expressed explicitly just in terms of the originl ldder opertors c α = e i ϑ [ cos ( ϑ α + ie iξ sin ( ϑ The product of the generlized joint ldder opertors (.3 ] [ β, c α = e i θ cos yields the generlized joint number opertor (13 s expected..4. Specil cse An importnt specil cse of (1 for hlf ngles ξ = π nd ϑ = π Using (0 s similrity trnsformtion, we find the identities ( ϑ α ie iξ sin ( ϑ ] β. (18 n α = c α c α (19 is n nti-symmetric squre root of swp gte U δ e i π ɛ δ,π/. (0 ɛ + δαβ = U δ n β U δ nd ɛ δαβ = U δ n α U δ. (1 ɛ 1+ αβ = U δ n β U δ nd ɛ 1 αβ = U δ n α U δ. ( ( precisely shows the type of perpendiculr pirwise entnglement induced by (0..5. Generlized interchnger Noting tht the joint number opertor is relted to the generlized joint number opertor ccording to (15 nd lso noting tht n α n β is invrint under the similrity trnsformtion (16, we cn lso write generliztion of (5 s follows ɛ δαβ e iϑ ɛ δαβ [n α + (δ 1n α n β ] e iϑ ɛ δαβ (3 ( ( (13 ϑ ϑ = cos n α + sin n β + i sin ϑ ( e iξ α β e iξ β α + (δ 1n α n β. (3b Now s lst step consider the specil cse when ξ = 0 so tht (3 reduces to genertor of the generlized interchnger n δαβ [ ( ( ɛ ] ϑ ϑ δαβ = cos n ξ=0 α + sin n β + i sin ϑ ( α β β α + (δ 1n α n β. (4 The generlized interchnger is then χ δαβ = ( 1 n δαβ = 1 n δαβ. (5 In the next section the generlized interchnger is relted to quite well known opertors from knot theory. I first found this prticulr interchnger gte, with ϑ = π, in the context of Hubbrd model simultion (δ = 1 fermionic cse with Hugh Pendleton t Brndeis in 1991 nd then multiphse condensed mtter simultion 7 (δ = 0 bosonic cse with Normn Mrgolus t MIT in 1994.

6 3.1. Temperley-Lieb lgebr 3. RELATION TO KNOT THEORY Scling the idempotent genertors, the generlized joint number opertors (4, derived in the lst section by rel number d, we define Q 1 bosonic number opertors with the following form (i.e. for the cse δ = 0 U 1 = d n 01, U = d n 03, (6 over system of Q qubits (ech qubit is like strnd. After some lgebric mnipultions, one finds tht for α = 1,,..., Q 1. The Temperley-Lieb lgebr is defined by U α U α±1 U α = d 4 sin ϑ U α, (7 [ U α, U β ] = 0, for α β > 1 (8 U α U α±1 U α = U α, for 1 α < Q (8b U α = d U α. (8c (8 stting tht U α nd U β re commuting opertors, when the seprtion distnce α β is greter thn one, follows directly from their definition (6 becuse the pir of qubits ffected by U α re n entirely different pir of qubits from the pir ffected by U β in this cse. (8b follows immeditely from (7 provided the scling constnt is chosen to be d = ± csc ϑ, (9 which implies tht d <. Finlly, (8c follows directly from (6 since the n δαβ tht ppers in (4 is idempotent: (n δαβ = n δαβ, for ll vlues of δ, α, nd β. Therefore, we see tht the Temperley-Lieb lgebr cn be strightforwrdly represented by generlized joint number opertors tht generte in tunble fshion, prmeterized by the ngle ϑ, perpendiculr pirwise entnglement within quntum stte. Now if one tkes d = A A. (30 nd does clcultion kin to (7, but strictly in terms of the vrible A, then one finds tht U α U α±1 U α U α = A + A [ ] 6 A 4 A 4 + (A + A cos ϑ U α. (31 8 The R.H.S. must vnish for this to be equivlent to (8b. Thus, one must solve 6 A 4 A 4 + (A + A cos ϑ = 0. (3 This 8th order polynomil eqution hs the following 4 rel nd 4 imginry solutions, respectively [ ( ] ± 1 [ ( ] ϑ A = ± cot 4 ± 1 ϑ, A = ± i cot 4. (33 Inserting these solutions into (30, d = csc ϑ for the rel solutions nd d = csc ϑ for the imginry ones, so this set of solutions is consistent with (9. Thus, in our cse, we see tht Jones t prmeter cn be prmeterized by ϑ s follows ( ϑ t A 4 = tn ±. (34 The U α stisfying the Temperley-Lieb lgebr re used for representing the Kuffmn digrmmtic decompositions of brids.

7 3.. Brid group A representtion of the brid group B Q cn be formed from the generlized joint number opertors s we did for the Temperley-Lieb lgebr representtion in the previous section b α = A 1 Q + A 1 d n δα,α+1, (35 for α = 1,,..., Q 1 where the rel number A is relted to the scling prmeter d by (30. Notice tht in (35 it is not necessry to restrict the vlue of the vrible δ nd, therefore, for the specil cse of A = 1, (35 is conservtive quntum gte, exctly the generlized interchnger (5. Furthermore, one finds tht b α b α+1 b α b α+1 b α b α+1 = (1 δδ cos ϑ csc 6 ( ϑ tn 3 ( ϑ. (36 Since δ is boolen, the R.H.S. vnishes for both fermionic nd bosonic brid opertors. following defining lgebr for the brid group Thus, we hve the [ b α, b β ] = 0, for α β > 1 (37 b α b α+1 b α = b α+1 b α b α+1, for 1 α < Q. (37b In generl the brid opertors (35 re not unitry; they re unitry only for specil vlues of A. However, scled by the vrible A they cn be formlly relted to conservtive quntum logic gtes generted by n δα,α+1 Υ δα,α+1 (z = e z n δα,α+1 = 1Q + (e z 1n δα,α+1. (38 From (35 we see tht b α A (30 = 1 Q + ( A 4 1 n δα,α+1. (38b Thus, equting these expressions, we find provided e z = A 4, or b α = A Υ δα,α+1 (z (39 z (34 = iπ + ln t. (40 Inserting this vlue of z bck into (39, we see tht in generl the brid opertor cn be expressed s b α (38 = A e (iπ+ln t n δα,α+1 = A ( t n δα,α+1. (41 Compring this expression with the form of the generlized interchnger (5, one sees how the brid opertor is similr to n interchnger gte Reprmeteriztion Strting with the form of the generlized joint interchnger with the form (4, mpping between the Temperley- Lieb lgebr nd the brid group requires one to restrict the possible vlues of d ccording to (9, nd in turn the possible vlues of A ccording to (33. In this section, we will consider n lternte prmeteriztion of the generlized joint interchnger by mpping the ngle ϑ s follows ϑ rccos(sec ϑ. (4 With this prmeteriztion n δα,α+1 mps over to new interchnger tht I will denote here by E δα. Tht is, (4 tkes the following form E δα = 1 [ ] 1 + i tn(ϑ n α + 1 [ ] 1 i tn(ϑ n α+1 + i sec(ϑ ( α α+1 α+1 α + (δ 1n α n α+1. (43

8 The new interchnger retins the property of idempotency nd, for δ = 0, interlevency s well In the new prmeteriztion, the scling prmeter (9 is now One my consider reprmeterized version of (35 too, Eδα = E δα (44 E 0α E 0α±1 E 0α = 1 4 cos (ϑ E 0α. (44b d = cos(ϑ. (45 b α = A 1 Q + A 1 d E δα, (46 s the strting opertionl definition of the brid opertors. There re two possible constrints tht we cn impose on the inverse brid opertor (35, nd they re not necessrily complementry. In the first cse, if we require b α to be unitry, then we hve the constrint tht the inverse is the djoint (trnspose conjugte b 1 α = b α. (47 In the second cse, s is typiclly done in the literture on the brid group, one imposes the following constrint One the one hnd, considering the ltter constrint (48 first, we hve provided b 1 α = A 1 1 Q + A d E δα. (48 b α b α 1 (46 = 1 Q + (d E δα + ( A + A d E δα (44 = 1 Q, (49 d = A A. (50 This implies the vlue of A. To be consistent with (45, one chooses A to be complex, A = ±1 e iϑ. Inserting this vlue into (41, we see tht the brid opertor tkes the form b α = (±1 1 e iϑ e i(π 4ϑE δα. (51 The reson for choosing the reprmetriztion (4 is to cst the brid opertor in wht ppers to be strictly unitry form. Yet, on the other hnd, considering the former cse (47, the unitrity of b α depends on the hermiticity of E δα. Note tht (43 is hermitin with respect to conjugtion i i, α α, nd α α, nd trnsposition of indices α α + 1. However, the trnspose of the brid group opertor, b T α, does not follow from the trnsposition of its α indices. If we choose complete set of bsis sttes, nd represents b α in this bsis, then the resulting mtrix trnspose (exchnging rows nd columns is n opertion tht is different thn the trnsposition of its α indices. Thus, b α is not mnifestly unitry s one would hope. Unfortuntely, the chnge of vribles (4 cuses E δα not to be hermitin for rbitrry ϑ even while the generlized interchnger (4 from which it derives is strictly hermitin. It is only in the specil cse of ϑ = π, nd multiples thereof, tht E δα both stisfies the Temperley-Lieb lgebr (for d = nd is hermitin. And in this specil cse E δα reduces to the originl interchnger. We rrive bck to where we strted. As quick lgebric check, expnding (51 using the exponentil series, mking use of (44, nd collecting terms gives b α = i (±1 1 he iϑ 1 Q e iϑ cos(ϑ E δα (5 (45 = i (±1 1 he iϑ 1 Q ± e iϑ d E δα, (5b which is just our strting point expression (46, with A 1 = ±(±1 1 e iϑ s expected. Modulo the phse fctor (±1 1, (5 hs form similr to the elementry unitry gtes generted by the degree representtion of the Temperley-Lieb lgebr given by Kuffmn et l. in the 3-strnded quntum lgorithm for the Jones Polynomil. 3

9 Remrkbly, however, hermitin joint number opertors stisfying TL Q over system of Q qubits, for set of ngles ϑ (0, π, does exist for mp such s: ϑ rccos ( tn ϑ. A complementry pir of interchnger genertors which re hermitin for the rnge of ngles 0 < ϑ < π re the following E δα = tn ϑ n α + cos ϑ sec ϑ ( n β + i d 1 α β β α + (δ 1n α n β (53 E δβ = cos ϑ sec ϑ n α + tn ϑ ( n β i d 1 α β β α + (δ 1n α n β, (53b where A = i ( 1 sec ϑ 1 4 nd d = A A sec ϑ+1 = sec ϑ 1, for d <. The resulting Temperley-Lieb lgebr, tht is lso hermitin, is the following E δα = E δα, α = 1,,..., Q 1 (54 E 0α E 0α±1 E 0α = d E 0α (54b E 0α E 0β = E 0β E 0α, α β (54c E δα = E δα, 0 < ϑ < π. (54d Other prmeteriztions re possible. Informtion conservtion in this cse is E δα +E δβ = n α +n β +(δ 1n α n β Occuption probbilities 4. QUANTUM MEASUREMENT Suppose q α nd q β in mny-body quntum system re initilized such tht with the seprble stte ψ q α q β. Following unitry evolution ψ n α ψ nd b ψ n β ψ, (55 ψ = U δ ψ, (56 let us denote the mesurement outputs s ψ n α ψ nd b ψ n β ψ. Since the ψ is entngled, mesurement tht determines the vlue of (yielding one clssicl bit likewise determines b. Also, the conservtion of the probbilities nd b is ensured becuse (0 is conservtive quntum logic gte. Let us now consider process of extrcting single bit upon mesurement, such tht + b = + b. (57 This is sttement bout quntum mesurement tht is the mesoscopic representtion of ( Quntum mps A projective mp, ssocited with quntum mesurement of qubits α nd β, going from Hilbert spce to kinetic spce is ( ψ nβ ψ P : ψ. (58 ψ n α ψ Oppositely, tensor product opertion is n injective mp from kinetic spce to Hilbert spce, ssocited with the initil preprtion of independent qubits: ( ( ( 1 1 I : b. (59 b b The quntum evolution tht entils stte preprtion, n ( entngling opertion, ( nd quntum mesurement cn be seen s kinetic spce trnsformtion of probbilities 8 b = P U δ I, which cn be written s the mp b C: ( ( ( ( qα q β ɛ 1+ b b αβ = q αq β q α q β ɛ 1 αβ q. (60 αq β

10 Entnglement drives the mesoscopic quntum dynmics, leding to governing quntum Boltzmnn eqution. Cn we invert this mp to retrieve the incoming probbilities (, b only from the outgoing ones (0, b0? Inverting (60 is not possible, becuse the mp (i.e. unitry gte (0 plus one mesurement induces n irreversible trnsition between kinetic spce points. Yet, it is informtive to see exctly where the inversion breks down. The first step towrds this end is to write (60 explicitly in terms of the kinetic spce vribles.9 The mp C is: 0 ρ p + p( (b b, (61 0 = ρ b b ( (b b where ρ + b is the number density. We will define the number velocity s follows: p v 0 b0 = ( (b b. (6 The number density nd number velocity re joint conjugte vribles to the output vribles. Squring (6 gives v4 = ( (b b = b(1 ρ+(b. The quntity b stisfies the qudrtic eq. (b (ρ 1b v4 = 0, with the single physicl solution p ρ 1 + (ρ 1 + v b =. (63 We hd to tke the positive root becuse b 0. Finlly, writing the number density s ρ = + (b, we cn solve for the input vlue in terms of the known output quntities ρ nd (b. We hve nother qudrtic eqution ρ + (b = 0, which hs solution pirs p p ρ ρ 4(b ρ ± ρ 4(b or b, =. (64, b = To dismbigute the possible orderings, we need one dditionl clssicl bit. Remrkbly, we hve found the bit tht ws lost upon mesurement. It is ssocited with the ordering of (, b. Thus, the mp (61 is irreversible, s we hd nticipted. C 1 [S ] C[S ] Figure. Informtion preserving mp C nd C 1, both cting on Riemnn sphere S of configurtions. (Left The incoming preimge C 1 [{0, b0 }] = {, b} nd c = c0, with {0, b0, c0 } S, is topologiclly torus with four cusps. (Right The outgoing imge {0, b0 } = C[{, b}] nd c0 = c, with {, b, c} S, is topologiclly doubly pinched sphere. It is possible to generlize (61 so this bit of ordering is not lost. Just encode the ordering of the input (, b in the output (0, b0. This is ccomplished generlizing (61 with the following nonliner reversible mp C: 0 ρ p p( (b b Θ(b 0 = σx, (65 ρ b b ( (b b + where the unit step function is Θ(x = 1, for x 0, nd Θ(x = 0, for x < 0. Here is the inverse mp C 1 : p 0 Θ(b0 0 σx ρ + pρ 4(b =, (66 b0 b ρ ρ 4(b

11 which hs the property C 1 C = 1. The quntity (b is computed from (, b ccording to (63, since ρ = + b nd v = b. The mps C nd C 1 re topologiclly expressed in Fig. s one-to-one mpping between distinct Riemnn surfces torus with cusps C 1 sphere C doubly pinched sphere. The unit step Θ(b = 0, 1 encodes single bit. (65 is type II quntum mp 8 tht conserves nd loclizes informtion, but otherwise indistinguishble from coherent evolution followed by stte demolition. The term loclize denotes n intrinsic informtion-conserving clss of wve function collpse without uncertinty in the ordering of the kinetic vribles. The distinction between ordinry projective mesurement nd extrordinry reversible locliztion is quntified by the trnsference of one bit rther peculir nonliner quntum opertion tht induces stte demolition while conserving ll kinetic-spce informtion in the quntum stte Quntum propositions It is possible to identify entngled sttes using either clssicl propositions or quntum propositions. 4 Here we consider the ltter. The eigenvlues of the sum of two joint opertors re equivlent to the true (1 or flse (0 vlue of proposition. Consider Q = 3 n s exmple ɛ ε 1 (q 1q q 1 q or (q q 3 q q 3, ε ε 1 (q 1 = q, ε ε 3 (q = q 3, (67 worked out in Tble 1 for mximlly entngled sttes. Csting propositions such s those in (67 with joint opertors determines (clssiclly multivlued properties in one mesurement. In system of size Q, there re only Q clssicl number opertors while there re Q(Q 1 joint opertors. In generl, using entnglement number opertors in lieu of usul number opertors offer more wys to express prticulr propositionl vlue. This is the bsis of the remrkble efficiency of quntum versus clssicl computtion. 10 Pthwys fster thn the clssicl pthwys re the interesting ones of course. Entngled stte Propositions Vlues (ɛ ε 1 (ε+ 1 + ε 1 (ε+ 3 + ε 3 (q 1q q 1q (q 1 = q (q = q = = = = = = = = 5 Tble 1. Eight mximlly entngled sttes in Q = 3 system. Fock stte ordering is q 1q q 3, with the 1st qubit is on the left nd the lst on the right. The propositions re cst in terms of joint opertors to identify ech entngled stte. 5. CONCLUSION In the lte 1940 s second quntized ldder opertors (quntum prticle cretion nd opertors were originlly developed to quntize field theories in momentum spce. Their uses quickly expnded within the physics community, with primry ppliction to condensed mtter systems, where there re plethor of pplictions. Quntum informtion theory is reltively new entrnt in theoreticl physics with n entirely new lexicon for understnding how informtion is creted, trnsferred, nd retrieved from quntum system. The ppliction of second quntized ldder opertors in quntum informtion theory is reltively recent development the new

12 informtionl prdigm provides impetus to retool second quntized ldder technology. So, quntum informtion not only provides new ppliction for this tried nd true toolset but provides motivtion to upgrde it s well. Therefore, I hve introduced vrious generlized second quntized opertors, including generlized joint ldder opertors, generlized joint number opertor, nd entnglement opertors. Some pplictions include, inter li, representing the Temperley-Lieb lgebr nd relted brid group in terms of these generlized second quntized opertors. The ppliction to knot theory is mnifest. And the ppliction to quntum computtionl physics is lso mnifest. Some spects ssocited with quntum mesurement were briefly treted. Now finl remrk on quntum dynmics: decoherence is sufficient to explin mcroscopic dissiption. Yet, in its purest form, quntum dynmics conserve informtion: decoherence cn be modeled s succession of projections of the stte of pirs of entngled prticles whereby the lost degrees of freedom in the Hilbert spce mplitudes re precisely gined in the orderings of the degrees of freedom in the ffected vlues of the kinetic vribles (probbilities following the projection. Joint informtion is trnsfered, not bsolutely lost. A simple mesurement rchetype ws offered: one joint bit extrcted from the destruction of pirwise entnglement is inserted in the ordering of two ffected kinetic vribles. This hs direct ppliction to reversible simultions of quntum processes driven by the quntum Boltzmnn eqution, even when the collision hierchy is cutoff to only hndle locl entnglement. The relevnt ppliction is to the quntum computtion of nonliner physicl dynmics, viz., mesoscopic quntum simultion tht respects detiled-blnce nd Onsger reciprocity reltions. ACKNOWLEDGMENTS I would like to thnk Hns von Beyer for helpful discussions. REFERENCES 1. S. Bndyopdhyy, S. Kimmel, nd W. K. Wootters, The quntum cost of nonlocl mesurement, in Interntionl Conference on Quntum Informtion, Interntionl Conference on Quntum Informtion, p. QTuA1, Opticl Society of Americ, D. Ahronov, V. Jones, nd Z. Lndu, A polynomil quntum lgorithm for pproximting the Jones polynomil, in STOC 06: Proceedings of the thirty-eighth nnul ACM symposium on Theory of computing, pp , ACM, (New York, NY, USA, L. H. Kuffmn nd J. Smuel J. Lomonco, A 3-strnded quntum lgorithm for the Jones polynomil, in Quntum informtion theory, E. J. Donkor, A. R. Pirich, nd H. E. Brndt, eds., Quntum Informtion nd Computtion V 6573, p T, SPIE, Brukner nd A. Zeilinger, Opertionlly invrint informtion in quntum mesurements, Phys. Rev. Lett. 83, pp , Oct J. Yepez, Quntum lttice-gs model for computtionl fluid dynmics, Physicl Review E 63, p , Mr A. Brenco, C. H. Bennett, R. Cleve, D. P. DeVincenzo, N. H. Mrgolus, P. W. Shor, T. Sletor, J. Smolin, nd H. Weinfurter, Elementry gtes for quntum computtion, Physicl Review A 5(5, pp , J. Yepez, A lttice-gs with long-rnge interctions coupled to het bth, Americn Mthemticl Society 6, pp , J. Yepez, Type-II quntum computers, Inter. J. Mod. Phys. C 1(9, pp , J. Yepez, Open quntum system model of the one-dimensionl Burgers eqution with tunble sher viscosity, Physicl Review A 74, p. 043, Oct D. Deutsch nd R. Jozs, Rpid solutions of problems by quntum computtion, Proc. R. Soc., London A439, pp , 199.