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h(pro) r (ceed) g(with) -150 2893 y(presen) n(ting) k(p) r(ossible) i(new) f (\014nite) h(quan) n(tum) f(algebras) f(and) h(al-) -150 2988 y(gorithms) e(in) i(Sec.) f(I) r(I) r(I.) 43 b(As) 29 b(a) g(result) g(w) n(e) g(obtain) g(a) g(general) f(al-) -150 3084 y(gorithm) 22 b(for) h(obtaining) f(Ko) r(c) n(hen-Sp) r(ec) n(k) n (er) f(v) n(ectors) g(and) i(there-) -150 3179 y(fore) j(an) g (automated) h(pro) r(of) f(of) g(the) h(Ko) r(c) n(hen-Sp) r(ec) n(k) n (er) e(theorem) -150 3275 y(in) j(Sec.) g(IV.) p Fj 534 3599 a(I) r(I.) 89 b(ALGEBRAS) p Fr -67 3821 a(Classical) 21 b(computers) g(standardly) g(manipulate) i(t) n(w) n(o-v) -5 b(alued) -150 3917 y(\(0) 26 b(and) f(1;) h(bits,) p Fi 26 w(bi) p Fr 8 w(nary) g(dig) p Fi(it) p Fr 8 w(s\)) g(elemen) n (ts) f(of) h(information) f(us-) -150 4012 y(ing) 40 b(switc) n(hes) g(\(ph) n(ysical) h(devices\)) f(whic) n(h) g(are) g (called) g(gates.) -150 4108 y(Their) 24 b(design) g(is) h(based) f(on) g(\(a) g(t) n(w) n(o-v) -5 b(alued\)) 24 b(Bo) r(olean) f(algebra,) -150 4203 y(also) h(called) p Fi 25 w(switching) p Fr 35 w(algebra.) 34 b(W) -7 b(e) 26 b(should) f(stress) f(here) g(that) -150 4299 y(a) j(Bo) r(olean) f(algebra) f(based) i(on) p Fh 27 w(n) p Fr(-v) -5 b(alued) 27 b(elemen) n(ts) g(is) g(equiv) -5 b(a-) -150 4394 y(len) n(t) 28 b(to) f(the) h(one) f(based) g(on) h(t) n (w) n(o-v) -5 b(alued) 26 b(elemen) n(ts.) -67 4499 y(A) i(Bo) r(olean) e(algebra) f(is) i(an) g(algebraic) f(structure) g(consisting) -150 4594 y(of) i(a) g(set) g(of) g(elemen) n(ts) g(together) f(with) h(t) n (w) n(o) f(binary) h(op) r(erations) p Fi -150 4690 a(join) p Fr 6 w(,) p Fg 28 w([) p Fr 27 w(and) p Fi 26 w(me) l(et) p Fr 8 w(,) p Fg 27 w(\\) p Fr 27 w(and) e(a) g(unary) g(op) r(eration) p Fi 25 w(ortho) l(c) l(omple-) -150 4785 y(ment) p Fr 8 w(,) p Ff 86 4755 a(?) p Fr 142 4785 a(,) g(suc) n(h) e(that) g(the) h (closure) e(prop) r(ert) n(y) g(holds,) i(the) g(la) n(w) e(of) -150 4881 y(distributivit) n(y) -7 b(,) p Fh 27 w(a) p Fg 15 w(\\) p Fr 15 w(\() p Fh(b) p Fg 15 w([) p Fh 16 w(c) p Fr(\)) 23 b(=) g(\() p Fh(a) p Fg 15 w(\\) p Fh 16 w(b) p Fr(\)) p Fg 15 w([) p Fr 15 w(\() p Fh(a) p Fg 16 w(\\) p Fh 15 w(c) p Fr(\),) k(asso) r(ciativit) n(y) -7 b(,) -150 4976 y(and) 23 b(comm) n(utativit) n(y) g(hold,) h(and) f(the) h(iden) n (tit) n(y) g(and) f(ortho) r(com-) p -150 5146 300 4 v Fp -150 5323 a(\003) p Fk -114 5347 a(Electronic) 53 b(address:) p Fe 90 w(mpavicic@faust.irb.hr;We) q(bpag) q(e:h) q(ttp:) q (//) -150 5425 y(m3k.grad.hr/pavicic) p Fr 2042 1420 a(plemen) n(t) 20 b(exist) g(\(so,) i(it) e(is) h(a) e(distributiv) n (e) h(lattice;) j(a) d(lattice) g(is) g(an) 2042 1516 y(ordered) f(set) h(in) h(whic) n(h) f(all) g(joins) g(and) g(meet) h (exist\).) 34 b(Eac) n(h) 20 b(of) g(the) 2042 1611 y(op) r(erations) e (and) i(their) g(com) n(binations) f(can) h(b) r(e) g(implemen) n(ted) h (in) 2042 1707 y(the) 29 b(form) g(of) h(logic) e(circuits) h(b) n(y) g (means) g(of) g(gates.) 41 b(Hence,) 30 b(one) 2042 1802 y(p) r(erforms) h(a) h(classical) f(task) h(b) n(y) g(\014rst) h (digitizing) f(it,) i(then) f(ma-) 2042 1898 y(nipulating) g(bits,) i (and) e(in) h(the) g(end) f(translating) f(bits) i(bac) n(k) e(to) 2042 1993 y(the) f(original) f(language) g(of) h(the) h(task) e(\(no) i (classical) e(computer) 2042 2088 y(can) 24 b(directly) g(mimic) g(a) g (classical) f(ph) n(ysical) g(pro) r(cess\).) 35 b(In) 25 b(doing) 2042 2184 y(so,) 34 b(one) e(can) h(access) f(the) i(v) -5 b(alues) 32 b(of) h(all) g(bits) h(at) f(an) n(y) f(stage) g(of) 2042 2279 y(their) 27 b(manipulation.) 2125 2390 y(Quan) n(tum) 66 b(computers) g(manipulate) p Fi 67 w(qubits) p Fr 73 w(\() p Fi(qu) p Fr 6 w(an) n(tum) p Fi 2042 2486 a(bits) p Fr 7 w(\)|elemen) n(ts) 35 b(of) g(quan) n(tum) h(information) e (\(whic) n(h) i(are) e(ac-) 2042 2581 y(tually) 26 b(not) g(digits) f (but) i(v) n(ectors) d(\(states\)) i(from) g(Hilb) r(ert) h(space\)) 2042 2676 y(b) n(y) g(means) g(of) p Fi 28 w(quantum) h(gates) p Fr 7 w(.) 2125 2787 y(Closed) 35 b(subspaces) g(of) g(a) h(Hilb) r(ert) g(space) f(form) h(an) f(algebra) 2042 2883 y(called) 22 b(a) g(Hilb) r(ert) h(lattice.) 35 b(A) 23 b(Hilb) r(ert) g(lattice) f (is) g(an) g(orthomo) r(d-) 2042 2978 y(ular) 36 b(lattice) h(whic) n (h,) i(is) d(b) n(y) h(de\014nition) g(a) f(\(relaxed\)) g(Bo) r(olean) 2042 3074 y(algebra) e(in) j(whic) n(h) f(the) g(distributivit) n(y) h (\(see) f(ab) r(o) n(v) n(e\)) f(holds) h(if) p Fh 2042 3169 a(b) p Fg 35 w(\024) p Fh 35 w(a) p Fr 35 w(and) p Fh 35 w(c) p Fg 36 w(?) p Fh 35 w(a) p Fr(.) 59 b(In) 36 b(an) n(y) e(Hilb) r(ert) i(lattice) f(the) g(op) r(eration) p Fi 2042 3264 a(me) l(et) p Fr 8 w(,) p Fh 43 w(a) p Fg 27 w(\\) p Fh 27 w(b) p Fr(,) 44 b(corresp) r(onds) 39 b(to) h(set) g(in) n(tersection,) p Fg 43 w(H) p Fd 3820 3276 a(a) p Fc 3875 3202 a(T) p Fg 3958 3264 a(H) p Fd 4028 3276 a(b) p Fr 4061 3264 a(,) 2042 3360 y(of) 33 b(subspaces) p Fg 32 w(H) p Fd 2597 3372 a(a) p Fh 2638 3360 a(;) p Fg 14 w(H) p Fd 2745 3372 a(b) p Fr 2811 3360 a(of) h(a) f(Hilb) r(ert) h(space) p Fg 32 w(H) p Fr 1 w(,) h(the) f(ordering) 2042 3455 y(relation) p Fh 34 w(a) p Fg 36 w(\024) p Fh 36 w(b) p Fr 35 w(corresp) r(onds) f (to) p Fg 36 w(H) p Fd 3249 3467 a(a) p Fg 3325 3455 a(\022) j(H) p Fd 3496 3467 a(b) p Fr 3529 3455 a(,) h(the) f(op) r (eration) p Fi 2042 3551 a(join) p Fr 6 w(,) p Fh 34 w(a) p Fg 22 w([) p Fh 22 w(b) p Fr(,) c(corresp) r(onds) f(to) h(the) g (smallest) g(closed) f(subspace) 2042 3646 y(of) p Fg 32 w(H) p Fr 32 w(con) n(taining) p Fg 31 w(H) p Fd 2721 3658 a(a) p Fc 2775 3584 a(S) p Fg 2858 3646 a(H) p Fd 2928 3658 a(b) p Fr 2962 3646 a(,) i(and) e(the) p Fi 33 w(ortho) l(c) l(omplement) p Fh 42 w(a) p Ff 4028 3616 a(?) p Fr 2042 3742 a(corresp) r(onds) g(to) p Fg 34 w(H) p Ff 2682 3712 a(?) p Fd 2681 3762 a(a) p Fr 2738 3742 a(,) k(the) e(set) h(of) f(v) n(ectors) f(orthogonal) f(to) i (all) 2042 3837 y(v) n(ectors) c(in) p Fg 31 w(H) p Fd 2497 3849 a(a) p Fr 2538 3837 a(.) 47 b(One) 31 b(can) f(de\014ne) h (all) g(the) g(lattice) h(op) r(erations) 2042 3933 y(on) e(a) g(Hilb) r (ert) h(space) f(itself) h(follo) n(wing) e(the) i(ab) r(o) n(v) n(e) e (de\014nitions:) p Fg 2042 4028 a(H) p Fd 2112 4040 a(a) p Fg 2177 4028 a(\\) c(H) p Fd 2327 4040 a(b) p Fr 2400 4028 a(=) p Fg 39 w(H) p Fd 2574 4040 a(a) p Fc 2628 3966 a(T) p Fg 2712 4028 a(H) p Fd 2782 4040 a(b) p Fr 2815 4028 a(,) p Fg 40 w(H) p Fd 2948 4040 a(a) p Fg 3013 4028 a([) h(H) p Fd 3164 4040 a(b) p Fr 3237 4028 a(=) 39 b(\() p Fg(H) p Ff 3444 3998 a(?) p Fd 3443 4049 a(a) p Fc 3514 3966 a(T) p Fg 3597 4028 a(H) p Ff 3668 3998 a(?) p Fd 3667 4052 a(b) p Fr 3724 4028 a(\)) p Ff 3756 3998 a(?) p Fr 3812 4028 a(.) 66 b(Also,) 2042 4124 y(the) 40 b(orthogonalit) n(y) e(\(men) n(tioned) j(ab) r(o) n(v) n (e\)) p Fg 39 w(H) p Fd 3521 4136 a(a) p Fg 3605 4124 a(?) j(H) p Fd 3784 4136 a(b) p Fr 3858 4124 a(means) p Fg 2042 4219 a(H) p Fd 2112 4231 a(a) p Fg 2186 4219 a(\024) 34 b(H) p Ff 2356 4189 a(?) p Fd 2355 4243 a(b) p Fr 2412 4219 a(.) g([1,) i(p.) e(175],) h([2,) h(pp.) e(21-29],) g([3,) i(pp.) e(66,67],) g([4,) 2042 4315 y(pp.) 28 b(8-16]) 2125 4425 y(Th) n(us,) d(using) g(the) g(prop) r(erties) f(of) h(Hilb) r (ert) h(space) e(one) g(arriv) n(es) 2042 4521 y(at) 29 b(a) f(de\014nition) h(of) g(the) g(Hilb) r(ert) h(lattice) f(as) f(an) h(orthomo) r(dular) 2042 4616 y(lattice) e(whic) n(h) h(satis\014es:) p Fi 2125 4727 a(Completeness:) p Fr 55 w(The) 34 b(meet) g(and) g(join) g (of) f(an) n(y) g(subset) h(of) g(a) 2042 4822 y(Hilb) r(ert) 28 b(lattice) g(alw) n(a) n(ys) d(exist.) p Fi 2125 4933 a(A) n(tomicity:) p Fr 40 w(Ev) n(ery) c(non-zero) g(elemen) n(t) i(in) g(an) p Fi 23 w(HL) p Fr 22 w(is) g(greater) 2042 5028 y(than) c(or) e(equal) i(to) f(an) g(atom.) 34 b(\(An) 19 b(atom) p Fh 19 w(a) p Fr 18 w(is) g(a) f(non-zero) f(lattice) 2042 5124 y(elemen) n(t) 28 b(with) g(0) p Fh 22 w(<) 23 b(b) p Fg 23 w(\024) p Fh 22 w(a) p Fr 28 w(only) k(if) p Fh 28 w(b) p Fr 23 w(=) p Fh 22 w(a) p Fr(.\)) p Fi 2125 5235 a(Sup) l(erp) l(osition) 36 b(Principle:) p Fr 56 w(\(The) e(atom) p Fh 33 w(c) p Fr 33 w(is) g(a) f(sup) r(erp) r(osi-) 2042 5330 y(tion) 28 b(of) g(the) g(atoms) p Fh 27 w(a) p Fr 28 w(and) p Fh 28 w(b) p Fr 27 w(if) p Fh 29 w(c) p Fg 23 w(6) p Fr(=) p Fh 24 w(a) p Fr(,) p Fh 28 w(c) p Fg 23 w(6) p Fr(=) p Fh 23 w(b) p Fr(,) g(and) p Fh 28 w(c) p Fg 23 w(\024) p Fh 24 w(a) p Fg 18 w([) p Fh 19 w(b) p Fr(.\)) 2042 5425 y(1.) 34 b(Giv) n(en) 22 b(t) n(w) n(o) f (di\013eren) n(t) h(atoms) p Fh 21 w(a) p Fr 22 w(and) p Fh 21 w(b) p Fr(,) h(there) f(is) g(at) f(least) h(one) p 90 rotate dyy eop %%Page: 2 2 2 1 bop Fr 4043 -299 a(2) -150 -83 y(other) 27 b(atom) p Fh 27 w(c) p Fr(,) p Fh 28 w(c) p Fg 23 w(6) p Fr(=) p Fh 23 w(a) p Fr 28 w(and) p Fh 27 w(c) p Fg 23 w(6) p Fr(=) p Fh 23 w(b) p Fr(,) g(that) h(is) g(a) f(sup) r(erp) r(osition) g (of) p Fh -150 12 a(a) p Fr 34 w(and) p Fh 33 w(b) p Fr(;) 36 b(2.) 55 b(If) 34 b(the) g(atom) p Fh 33 w(c) p Fr 33 w(is) g(a) f(sup) r(erp) r(osition) g(of) h(distinct) -150 108 y(atoms) p Fh 29 w(a) p Fr 30 w(and) p Fh 30 w(b) p Fr(,) d(then) f(atom) p Fh 30 w(a) p Fr 30 w(is) g(a) g(sup) r(erp) r (osition) f(of) h(atoms) p Fh -150 203 a(b) p Fr 27 w(and) p Fh 28 w(c) p Fr(.) p Fi -67 303 a(Minimal) h(length:) p Fr 43 w(The) c(lattice) h(con) n(tains) e(at) h(least) g(three) h(el-) -150 398 y(emen) n(ts) p Fh 28 w(a;) 14 b(b;) g(c) p Fr 26 w(satisfying:) 37 b(0) p Fh 22 w(<) 23 b(a) g(<) f(b) h(<) g(c) g (<) p Fr 22 w(1.) -67 497 y(One) i(can) h(also) f(pro) n(v) n(e) f(the) i(other) f(direction) g(and) h(therefore) f(a) -150 593 y(Hilb) r(ert) c(lattice) f(is) g(isomorphic) f(to) h(the) h(set) f(of) g(closed) f(subspaces) -150 688 y(of) 36 b(a) f(Hilb) r(ert) i(space.) e ([5]) h(Here) f(comes) h(a) f(result) h(w) n(e) f(w) n(an) n(t) h(to) -150 784 y(stress:) h(It) 28 b(can) g(b) r(e) g(pro) n(v) n(ed) f(that) h(a) f(Hilb) r(ert) i(lattice) p Fi 28 w(must) p Fr 37 w(con-) -150 879 y(tain) 24 b(in\014nite) g(n) n(um) n(b) r(er) g(of) f (atoms.) g([6]) h(Moreo) n(v) n(er,) e(if) i(w) n(e) f(w) n(an) n(ted) -150 975 y(a) 29 b(Hilb) r(ert) h(lattice) f(to) g(pro) n(vide) f(us) h (with) h(a) f(complex) g(\014eld) g(o) n(v) n(er) -150 1070 y(whic) n(h) 37 b(Hilb) r(ert) h(space) e(can) g(b) r(e) i (de\014ned,) i(w) n(e) c(should) h(assume) -150 1166 y(that) 32 b(the) f(Hilb) r(ert) h(lattice) g(con) n(tains) e(a) h (coun) n(table) g(in\014nite) h(se-) -150 1261 y(quence) 27 b(of) h(orthogonal) d(elemen) n(ts.) -67 1360 y(In\014nite) 31 b(dimensionalit) n(y) f(of) g(a) h(Hilb) r(ert) g(space) e(corresp) r (onds) -150 1456 y(to) f(the) g(space) f(con) n(tin) n(uit) n(y) -7 b(,) 28 b(to) g(the) g(in) n(tegrals) f(instead) h(of) f(sums,) -150 1551 y(to) 39 b(radial) f(functions) h(and) g(spherical) f(harmonics,) j (etc.;) k(in) 40 b(a) -150 1647 y(w) n(ord,) 25 b(to) h(all) g (solutions) f(of) h(the) g(Sc) n(hr\177) -42 b(odinger) 24 b(equation) i(w) n(e) f(are) -150 1742 y(used) k(to.) 43 b(Therefore,) 29 b(the) g(usual) h(space) e(distribution) i(of,) g (e.g.,) -150 1838 y(a) h(w) n(a) n(v) n(e) f(function) j(of) f (electrons) e(within,) k(e.g.,) e(a) g(molecule) f(re-) -150 1933 y(quires) 37 b(an) h(in\014nite) h(dimensional) e(Hilb) r(ert) i (space.) 68 b(Since) 38 b(w) n(e) -150 2028 y(cannot) 32 b(ha) n(v) n(e) f(in\014nite) i(dimensionalit) n(y) f(on) g(a) g(quan) n (tum) g(com-) -150 2124 y(puter) 22 b(w) n(e) g(cannot) f(directly) h (sim) n(ulate) g(quan) n(tum) f(mec) n(hanics) h(on) -150 2219 y(a) 28 b(quan) n(tum) g(computer.) 38 b(But) 28 b(since) g(b) r(oth) g(systems) g(are) f(quan-) -150 2315 y(tum) 35 b(systems,) g(a) f(sim) n(ulation|as) f(opp) r(osed) h (to) g(the) h(classical) -150 2410 y(case|is) 27 b(nev) n(ertheless) f (p) r(ossible.) p Fj 453 2706 a(I) r(I) r(I.) 90 b(ALGORITHMS) p Fr -67 2923 a(In) 30 b(the) g(literature) g(sim) n(ulation) f(of) h (quan) n(tum) g(mec) n(hanics) f(on) -150 3019 y(a) 36 b(quan) n(tum) g(computer) g(has) f(b) r(een) i(approac) n(hed) d(in) j (basically) -150 3114 y(t) n(w) n(o) 22 b(w) n(a) n(ys.) 34 b(The) 23 b(\014rst) g(approac) n(h) e(is) i(to) g(sim) n(ulate) g(one) f(quan) n(tum) -150 3210 y(system) j(b) n(y) g(another) g(whic) n(h) g (resides) g(in) g(a) g(quan) n(tum) h(computer) -150 3305 y(and) 31 b(migh) n(t) g(b) r(e) h(simpler,) f(e.g.,) h(proton) e (spins) h(b) n(y) g(electrons) f(in) -150 3401 y(quan) n(tum) 22 b(dots.) g([7) o(,) i(8) o(]) e(Or) f(ev) n(en) h(\\univ) n(ersal) e (quan) n(tum) i(compu-) -150 3496 y(tation) i(o) n(v) n(er) f(con) n (tin) n(uous) h(v) -5 b(ariables) 23 b(for) h(transformations) f(that) -150 3592 y(are) 18 b(p) r(olynomial) g(in) h(those) f(v) -5 b(ariables.") 18 b([9) o(]) h(This) g(approac) n(h) e(do) r(es) -150 3687 y(not) 27 b(help) h(us,) f(though,) g(since) g(w) n(e) g(ha) n(v) n (e) f(to) h(\014nd) g(the) h(algebra) d(of) -150 3783 y(the) 41 b(quan) n(tum) g(computer) f(system) g(itself.) 76 b(The) 41 b(second) f(ap-) -150 3878 y(proac) n(h) 35 b(is) i(to) g(sim) n(ulate) g(quan) n(tum) g(mec) n(hanics) f(b) n(y) h (means) f(of) -150 3974 y(quan) n(tum) 26 b(gas) e(mo) r(del.) j([10) o (,) f(11) o(]) g(Basically) e(this) i(b) r(oils) g(do) n(wn) g(to) -150 4069 y(application) 20 b(of) h(the) h(corresp) r(onding) d(Sc) n (hr\177) -42 b(odinger) 19 b(equation) i(on) -150 4164 y(p) r(oin) n(ts) 28 b(in) g(a) g(grid.) 38 b(As) 28 b(a) f(result) h(the) h(p) r(oin) n(ts) f(sit) g(in) g(the) h(grid) e (so) -150 4260 y(as) g(to) g(\014t) h(the) f(con) n(tin) n(uous) f(w) n (a) n(v) n(e) g(function.) 37 b(Hence,) 28 b(a) f(discrete) -150 4355 y(set) 38 b(of) f(p) r(oin) n(ts) h(appro) n(ximates) e(a) h(con) n (tin) n(uous) g(w) n(a) n(v) n(e) f(function) -150 4451 y(but) d(w) n(e) g(still) g(do) f(not) h(ha) n(v) n(e) e(a) h(gen) n (uine) h(discrete) f(algebra) f(and) -150 4546 y(discrete) 24 b(Hilb) r(ert) h(space.) 35 b(Our) 23 b(aim) h(is) g(to) h(in) n(v) n (estigate) e(whether) -150 4642 y(suc) n(h) k(discretization) g(is) g (p) r(ossible.) -67 4741 y(W) -7 b(e) 51 b(consider) f(\014nite) h (orthomo) r(dular) e(lattices) h(and) g(\014lter) -150 4837 y(them) 22 b(through) f(the) h(conditions) f(stated) h(in) g(Sec.) g(I) r(I) g(and) f(in) n(v) n(esti-) -150 4932 y(gate) 26 b(prop) r(erties) f(whic) n(h) h(hold) g(and) h(whic) n(h) f(fail) g (in) h(the) g(lattices.) -150 5027 y(W) -7 b(e) 28 b(w) n(an) n(t) f (to) h(\014nd) g(classes) e(of) i(suc) n(h) g(lattices) f(whic) n(h) h (w) n(ould) f(ap-) -150 5123 y(pro) n(ximate) i(lattices) h(with) h (in\014nite) g(n) n(um) n(b) r(er) e(of) i(atoms) e(and) h(in) -150 5218 y(the) 24 b(end) f(w) n(e) g(w) n(an) n(t) g(to) g(compare) f (them) i(with) f(lattices) g(w) n(e) g(deriv) n(e) -150 5314 y(from) k(a) g(\014nite) i(dimensional) e(Hilb) r(ert) h(space.) -67 5413 y(The) 23 b(most) f(attractiv) n(e) g(feature) g(of) g(suc) n (h) h(a) f(pro) r(cedure) g(is) g(that) 2042 -83 y(one) i(can) h (de\014ne) g(\014nite) h(lattices) f(b) n(y) g(algorithms) e(whic) n(h) i(are) f(not) 2042 12 y(simply) 29 b(read) g(o\013) g(from) g(the) h (standard) f(Hilb) r(ert) h(space) e(prop) r(er-) 2042 108 y(ties) h(but) i(are) d(deriv) n(ed) h(form) g(highly) g(non) n (trivial) g(theorems) f(de-) 2042 203 y(riv) n(ed) 22 b(in) h(the) h(theory) e(of) h(Hilb) r(ert) h(lattices) f(in) g(the) h (last) f(20) f(y) n(ears.) 2042 299 y(These) 34 b(algorithms) g(also) g (sp) r(eed) h(up) g(calculations) f(for) g(sev) n(eral) 2042 394 y(orders) 29 b(of) i(magnitude.) 46 b(It) 31 b(can) f(b) r(e) h (sho) n(wn) f(that) h(\014nite) g(ortho-) 2042 490 y(mo) r(dular) k (lattices) h(can) g(b) r(e) g(obtained) g(from) f(MMP) h(diagrams) 2042 585 y(whic) n(h) 28 b(are) f(organized) f(as) i(connected) g(blo) r(c) n (ks) f(of) h(m) n(utually) g(or-) 2042 681 y(thogonal) j(atoms.) 52 b(MMP) 32 b(diagrams) f(are) h(diagrams) f(that) i(are) 2042 776 y(de\014ned) 28 b(as) f(follo) n(ws:) 2152 952 y(1.) 32 b(Ev) n(ery) 44 b(v) n(ertex) g(\(i.e.,) 49 b(atom) c(when) g(a) g (diagram) e(corre-) 2249 1047 y(sp) r(onds) 28 b(to) f(a) g(lattice\)) h (b) r(elongs) f(to) h(at) f(least) g(one) h(blo) r(c) n(k;) 2152 1208 y(2.) k(If) 25 b(there) f(are) f(at) h(least) f(t) n(w) n(o) h(v) n (ertices) e(then) j(ev) n(ery) e(blo) r(c) n(k) g(is) 2249 1303 y(at) 28 b(least) f(2-elemen) n(t;) 2152 1464 y(3.) 32 b(Ev) n(ery) 27 b(blo) r(c) n(k) g(whic) n(h) h(in) n(tersects) f(with) i(another) e(blo) r(c) n(k) g(is) 2249 1559 y(at) h(least) f(3-elemen) n (t;) 2042 1734 y(and) i(then) h(generated) f(b) n(y) g(the) h(the) g (isomorph-free) e(generation) 2042 1830 y(pro) r(cedure) e(according) g (to) i(the) g(follo) n(wing) e(algorithm) h([12) o(]:) p Fb 2249 1989 a(pro) s(cedure) p Fr 28 w(scan) g(\() p Fh(D) p Fr 2 w(:) 37 b(diagram;) p Fh 27 w(\014) p Fr 4 w(:) g(in) n(teger\)) p Fb 2432 2150 a(if) p Fh 27 w(D) p Fr 30 w(has) 27 b(exactly) p Fh 27 w(\014) p Fr 32 w(blo) r(c) n(ks) p Fb 27 w(then) 2587 2245 y(output) p Fh 32 w(D) p Fb 2432 2373 a(else) 2587 2500 y(for) p Fr 41 w(eac) n(h) 41 b(equiv) -5 b(alence) 40 b(class) g(of) h (extensions) p Fh 2587 2595 a(D) p Fr 21 w(+) p Fh 18 w(e) p Fb 27 w(do) 2728 2707 y(if) p Fh 28 w(e) p Fg 22 w(2) p Fh 24 w(m) p Fr(\() p Fh(D) p Fr 21 w(+) p Fh 18 w(e) p Fr(\)) p Fb 27 w(then) p Fr 28 w(scan\() p Fh(D) p Fr 20 w(+) p Fh 18 w(e) p Fr(,) p Fh(\014) p Fr 4 w(\)) p Fb 2249 2867 a(end) 32 b(pro) s(cedure) p Fr 2125 3026 a(Without) 22 b(the) g(latter) f(algorithm) g(MMP) g (diagrams) f(w) n(ould) h(b) r(e) 2042 3122 y(nothing) f(but) h(Greec) n (hie) e(diagrams) f([13) o(]) j(with) f(one) g(of) g(the) h(condi-) 2042 3217 y(tions) 31 b(dropp) r(ed.) 48 b(The) 32 b(isomorph-free) d (generation) h(pro) r(cedure) 2042 3313 y(is) 24 b(what) g(mak) n(e) f (them) h(v) n(ery) f(di\013eren) n(t.) 36 b(Greec) n(hie) 23 b(diagrams) f(are) 2042 3408 y(a) f(handy) g(w) n(a) n(y) f(to) h(dra) n (w) f(Hasse) g(diagrams) g(but) h(Hasse) g(diagrams) 2042 3504 y(get) i(more) f(and) h(more) f(in) n(trinsically) g(complicated) h (when) g(w) n(e) f(en-) 2042 3599 y(large) 32 b(the) i(n) n(um) n(b) r (er) f(of) g(atoms.) 54 b(E.g.,) 35 b(a) e(four-atom) f(Greec) n(hie) 2042 3694 y(blo) r(c) n(k) 38 b(has) g(16) f(elemen) n(ts,) k(a) d (\014v) n(e-atom) f(Greec) n(hie) h(blo) r(c) n(k) g(has) 2042 3790 y(32) 33 b(elemen) n(ts,) i(and) e(an) p Fh 34 w(n) p Fr(-atom) g(Greec) n(hie) g(blo) r(c) n(k) g(has) g(2) p Fd 3881 3760 a(n) p Fr 3960 3790 a(ele-) 2042 3885 y(men) n(ts,) 25 b(so) g(they) g(so) r(on) f(b) r(ecome) h(in) n(tractable.) 35 b(MMP) 25 b(diagrams) 2042 3981 y(are) 32 b(ho) n(w) n(ev) n(er) f (just) i(strings.) 53 b(A) 33 b(\014v) n(e) g(v) n(ertex) f(blo) r(c) n (k) g(has) h(5) g(ele-) 2042 4076 y(men) n(ts,) 27 b(an) p Fh 28 w(n) p Fr 27 w(v) n(ertex) g(blo) r(c) n(k) g(has) p Fh 27 w(n) p Fr 28 w(elemen) n(ts.) 2125 4172 y(Dep) r(ending) 36 b(on) g(parameters) e(w) n(e) i(use) g(in) g(their) g(generation) 2042 4267 y(\(parameters) f(app) r(ear) g(as) h(options) g(in) g(our) g (programs\)) e(MMP) 2042 4363 y(diagrams) 26 b(can) h(b) r(e) h (represen) n(ted) e(as) h(lattices,) h(but) g(also) e(as) h(par-) 2042 4458 y(tially) 39 b(ordered) f(sets,) 43 b(or) 38 b(as) h(v) n(ectors) f (from) i(a) f(Hilb) r(ert) h(space) 2042 4554 y(whic) n(h) 33 b(do) h(not) f(form) h(a) f(lattice;) k(they) d(can) f(ev) n(en) g(b) r (e) h(used) g(for) 2042 4649 y(represen) n(ting) 29 b(relations) g(b) r (et) n(w) n(een) i(v) n(ectors,) e(planes,) i(and) f(sub-) 2042 4745 y(spaces) 19 b(of) g(an) n(y) p Fh 19 w(n) p Fr(-dim) h(space) f (in) h(classical) e(ph) n(ysics.) 34 b(Whic) n(h) 20 b(dia-) 2042 4840 y(gram) f(will) j(b) r(e) f(appropriate) e(for) h (whic) n(h) h(purp) r(ose) f(is) h(determined) 2042 4936 y(b) n(y) k(a) g(selection) g(pro) r(cedure) g(w) n(e) g(use) g(once) g (they) h(are) e(generated.) 2125 5031 y(So,) 42 b(the) e(3) f(simple) g (aforemen) n(tioned) f(conditions) h(imp) r(osed) 2042 5127 y(on) 27 b(diagrams) g(giv) n(es) f(us) i(all) g(w) n(e) g(need) g (to) f(get) h(all) g(\014nite) g(lattices) 2042 5222 y(of) 36 b(arbitrary) e(complexit) n(y:) 54 b(w) n(e) 35 b(just) i(eliminate) g(diagrams) d(in) 2042 5318 y(whic) n(h) k(Hilb) r (ert) h(lattice) g(prop) r(erties) e(do) h(not) h(hold.) 69 b(W) -7 b(e) 39 b(cur-) 2042 5413 y(ren) n(tly) 23 b(use) h(programs) e (whic) n(h) i(generate) e(and) i(use) g(lattices) g(with) p 90 rotate dyy eop %%Page: 3 3 3 2 bop Fr 4043 -299 a(3) -150 -83 y(up) 31 b(to) g(100) e(atoms) i (but) g(for) f(all) h(results) f(w) n(e) h(ha) n(v) n(e) e(obtained) i (so) -150 12 y(far,) c(15) g(to) g(28) g(atoms) g(su\016ce.) -67 111 y(W) -7 b(e) 29 b(w) n(ere) e(also) g(able) h(to) g(reform) n (ulate) e(Hilb) r(ert) j(lattice) f(prop-) -150 206 y(erties) j(and) h (substitute) g(3) f(\(conjectured) h(all\)) g(classes) e(of) i(p) r (oly-) -150 301 y(nomial) i(equations) g(of) g(the) p Fh 35 w(n) p Fr(-th) h(order) e(for) h(the) h(afore) e(stated) -150 397 y(conditions.) 51 b(One) 33 b(suc) n(h) f(class) f(w) n(as) h(kno) n (wn) g(b) r(efore.) g([14) o(]) h(And) -150 492 y(the) 25 b(other) f(t) n(w) n(o,) h(the) g(orthoarguesian) d(class) i(of) h (n-th) g(order) f(and) p Fi -150 588 a(quantum) c(state) h(e) l (quation) p Fr 28 w(class,) f(w) n(e) e(found) h(only) f(recen) n(tly) -7 b(.) 18 b([15) o({) -150 683 y(17) o(]) 34 b(As) g(the) h(name) f (of) g(the) g(latter) g(class) f(tells) h(us,) i(it) e(is) g(deter-) -150 779 y(mined) 29 b(b) n(y) f(the) h(kind) g(of) f(states) g(imp) r (osed) h(on) f(an) n(y) g(Hilb) r(ert) h(lat-) -150 874 y(tice) 19 b(of) g(Hilb) r(ert) g(space,) h(i.e.,) g(b) n(y) f(p) r (ossible) f(ev) -5 b(aluation) 18 b(of) h(Hilb) r(ert) -150 970 y(lattice) 28 b(elemen) n(ts.) -67 1068 y(Let) f(us) f(see) g(what) h(mak) n(es) e(a) i(di\013erence) f(b) r(et) n(w) n(een) h(a) f (classical) -150 1163 y(and) h(a) h(quan) n(tum) f(state.) -67 1261 y(A) 22 b(state) g(on) g(a) f(lattice) h(L) g(is) g(a) f(function) p Fh 23 w(m) p Fr 23 w(:) i(L) p Fg 23 w(\000) -14 b(!) p Fr 23 w([0) p Fh(;) p Fr 14 w(1]) 21 b(suc) n(h) -150 1357 y(that) p Fh 31 w(m) p Fr(\(1\)) 29 b(=) f(1) i(and) p Fh 31 w(a) p Fg 28 w(?) p Fh 28 w(b) p Fg 59 w(\)) p Fh 60 w(m) p Fr(\() p Fh(a) p Fg 21 w([) p Fh 21 w(b) p Fr(\)) e(=) p Fh 28 w(m) p Fr(\() p Fh(a) p Fr(\)) 21 b(+) p Fh 20 w(m) p Fr(\() p Fh(b) p Fr(\).) -150 1452 y(This) 32 b(yields) p Fh 32 w(m) p Fr(\() p Fh(a) p Fr(\)) 21 b(+) p Fh 21 w(m) p Fr(\() p Fh(a) p Ff 719 1422 a(0) p Fr 743 1452 a(\)) 30 b(=) g(1) i(and) p Fh 31 w(a) p Fg 30 w(\024) p Fh 30 w(b) p Fg 62 w(\)) p Fr 63 w(\(\() p Fh(m) p Fr(\() p Fh(a) p Fr(\)) f(=) -150 1548 y(1) p Fg 50 w(\)) p Fh 51 w(m) p Fr(\() p Fh(b) p Fr(\)) 23 b(=) g(1\).) -67 1646 y(A) 28 b(nonempt) n(y) f(set) p Fh 28 w(S) p Fr 32 w(of) h(states) f(on) g(L) h(is) p Fi 27 w(classic) l(al) p Fr 38 w(if) -150 1744 y(\() p Fg(9) p Fh(m) p Fg 23 w(2) p Fh 24 w(S) p Fr 5 w(\)\() p Fg(8) p Fh(a;) 14 b(b) p Fg 22 w(2) p Fr 23 w(L\)\(\() p Fh(m) p Fr(\() p Fh(a) p Fr(\)) 24 b(=) f(1) p Fg 44 w(\)) p Fh 44 w(m) p Fr(\() p Fh(b) p Fr(\)) g(=) g(1\)) p Fg 44 w(\)) p Fh 44 w(a) p Fg 23 w(\024) p Fh 22 w(b) p Fr(\)) -150 1842 y(and) p Fi 27 w(quantum) p Fr 33 w(if) -150 1940 y(\() p Fg(8) p Fh(a;) 14 b(b) p Fg 22 w(2) p Fr 23 w(L\)\() p Fg(9) p Fh(m) p Fg 24 w(2) p Fh 23 w(S) p Fr 5 w(\)\(\() p Fh(m) p Fr(\() p Fh(a) p Fr(\)) 24 b(=) f(1) p Fg 44 w(\)) p Fh 44 w(m) p Fr(\() p Fh(b) p Fr(\)) g(=) g(1\)) p Fg 44 w(\)) p Fh 44 w(a) p Fg 23 w(\024) p Fh 22 w(b) p Fr(\)) -67 2038 y(No) n(w) k(w) n(e) g(are) g(able) g(to) h(pro) n(v) n (e) e(the) i(follo) n(wing) p Fi -150 2136 a(The) l(or) l(em) p Fr(.) 36 b(An) n(y) 22 b(orthomo) r(dular) e(lattice) h(that) h(admits) g(classical) -150 2232 y(states) 27 b(is) h(a) f(Bo) r(olean) f (algebra.) p Fi -150 2330 a(The) l(or) l(em) p Fr(.) 68 b(An) n(y) 37 b(Bo) r(olean) g(algebra) e(admits) j(classical) e (states) -150 2425 y(and) 27 b(an) n(y) g(Hilb) r(ert) h(lattice) g (admits) g(quan) n(tum) f(states.) p Fi -150 2523 a(The) l(or) l(em) p Fr(.) 37 b(An) 24 b(orthomo) r(dular) e(lattice) h(that) h(admits) f (quan) n(tum) -150 2619 y(states) k(is) h(still) g(not) f(necessarily) f (a) h(Hilb) r(ert) h(algebra) e(\(lattice\).) -67 2717 y(The) 39 b(pro) r(of) g(of) g(the) h(latter) f(theorem) g(is) g (simple:) 61 b(man) n(y) 38 b(of) -150 2812 y(the) 23 b(orthoarguesian) d(equations) i(\(c) n(haracteristic) f(of) i(an) n(y) f(3) h(and) -150 2908 y(more) k(dimensional) g(Hilb) r(ert) h(space\)) f (fail) g(in) h(man) n(y) f(MMP) g(dia-) -150 3003 y(grams) h(with) i (lo) r(ops) f(of) g(at) h(least) f(5) g(blo) r(c) n(ks) g(and) g(in) n (terpreted) g(as) -150 3099 y(Hasse) e(diagrams) f(whic) n(h) h(allo) n (w) g(quan) n(tum) g(states.) -67 3197 y(T) -7 b(ak) n(en) 29 b(together,) g(w) n(e) g(conjecture) h(that) g(an) f(in\014nite) h (dimen-) -150 3292 y(sional) d(Hilb) r(ert) i(space) f(can) g(b) r(e) h (represen) n(ted) e(b) n(y) h(a) g(p) r(olynomial) -150 3388 y(quan) n(tum) f(algebra) f(\(Hilb) r(ert) i(lattice) f(reform) n (ulated) f(b) n(y) h(means) -150 3483 y(of) h(orthoarguesian) d(and) j (state) g(equations\)) g(of) g(the) p Fh 29 w(n) p Fr(-th) g(order) -150 3579 y(with) p Fh 40 w(n) p Fg 42 w(!) 42 b(1) p Fr 40 w(and) d(for) f(\014nite) p Fh 40 w(n) p Fr 39 w(suc) n(h) h(an) g (algebra) e(can) i(b) r(e) -150 3674 y(implemen) n(ted) 30 b(on) f(a) g(w) n(ould-b) r(e) g(quan) n(tum) g(computer.) 42 b(Qubits) -150 3770 y(as) 25 b(the) h(elemen) n(ts) f(of) g(the) h (algebra) e(ob) r(ey) h(sup) r(erp) r(osition) g(princi-) -150 3865 y(ple) i(but) g(do) g(not) f(allo) n(w) g(a) g(\014xed) h(ev) -5 b(aluation) 26 b(\(see) h(ab) r(o) n(v) n(e:) 35 b(there) -150 3961 y(is) k(no) h(state) f(for) p Fi 39 w(every) p Fr 48 w(elemen) n(t) g(of) h(the) g(algebra\).) 71 b(This) 40 b(is) -150 4056 y(due) 28 b(to) g(particular) f(w) n(a) n(y) g(in) i (whic) n(h) f(the) g(orthogonalit) n(y) e(can) i(b) r(e) -150 4151 y(de\014ned) 38 b(in) g(MMP) f(diagrams,) h(i.e.,) i(in) e(Hilb) r (ert) g(lattices) g(and) -150 4247 y(Hilb) r(ert) 22 b(space,) f(and) g(this) g(orthogonalit) n(y) e(turns) h(out) h(to) g (b) r(e) h(v) n(ery) -150 4342 y(promising) j(in) i(solving) e (problems) h(b) r(ecause) f(it) i(can) f(b) r(e) h(reduced) -150 4438 y(to) g(linear) f(equations) g(as) g(opp) r(osed) g(to) h(the) g (standard) f(approac) n(h) -150 4533 y(to) k(the) g(orthogonalit) n(y) e (whic) n(h) h(is) h(nonlinear.) 43 b(The) 30 b(details) f(are) -150 4629 y(presen) n(ted) e(in) h(the) g(next) g(section.) p Fj -64 4919 a(IV.) 88 b(OR) -7 b(THOGONALITY,) 30 b(ENT) -7 b(ANGLEMENT,) 85 5006 y(AND) 29 b(K) n(OCHEN-SPECKER) i(THEOREM) p Fr -67 5222 a(In) 41 b(1993) e(\\[w]e) h(prop) r(os[ed]) g(a) h(new) g (exp) r(erimen) n(t) f(emplo) n(y-) -150 5318 y(ing) e(t) n(w) n(o) g (indep) r(enden) n(t) i(sources) d(of) i(spin) g(correlated) e(photon) -150 5413 y(pairs.) 63 b(Tw) n(o) 36 b(photons) g(from) g(di\013eren) n (t) h(unp) r(olarized) f(sources) 2042 -83 y(eac) n(h) f(pass) g (through) g(a) g(p) r(olarizer) g(to) h(a) f(detector.) 61 b(Although) 2042 12 y(their) 38 b(tra) 5 b(jectories) 37 b(nev) n(er) h(mix) h(or) f(cross) f(they) i(exhibit) g(4th{) 2042 108 y(order{in) n(terference{lik) n(e) 32 b(correlations) i(when) j (the) f(other) g(t) n(w) n(o) 2042 203 y(photons) 30 b(in) n(terfere) h(on) g(a) g(b) r(eam) g(splitter) g(ev) n(en) g(when) g(the) h(lat-) 2042 299 y(ter) e(t) n(w) n(o) f(do) g(not) h(pass) f (an) n(y) h(p) r(olarizers) e(at) i(all") f([18) o(,) h(19],) g(inde-) 2042 394 y(p) r(enden) n(tly) 24 b(of) g([20) o(,) h(21) o(]) f(and) g (sim) n(ultaneously) f(with) h([21) o(].) 36 b(Later) 2042 490 y(the) 19 b(obtained) h(results) e(ha) n(v) n(e) g(b) r(een) i(v) n (eri\014ed) f(\\exp) r(erimen) n(tally) f(...) 2042 585 y([b) n(y]) 23 b(t) n(w) n(o) f(pairs) g(of) h(p) r(olarization) e(en) n (tangled) h(photons) h(and) f(sub-) 2042 681 y(ject[ing]) 30 b(one) f(photon) h(from) f(eac) n(h) g(pair) h(to) f(a) h(Bell-state) f (mea-) 2042 776 y(suremen) n(t.) 34 b(This) 22 b(results) g(in) g(pro) 5 b(jecting) 21 b(the) i(other) e(t) n(w) n(o) h(outgo-) 2042 872 y(ing) i(photons) g(in) n(to) g(an) h(en) n(tangled) e(state.") h ([22) o(]) h(The) f(v) n(ery) g(same) 2042 967 y(sc) n(heme) j(w) n(as) f(also) h(used) h(for) f(telep) r(ortation.) g([18) o({21) n(,) h(23) o (,) g(24) o(]) 2125 1073 y(That) f(\\en) n(tangle[men) n(t]) g(and) h (correlat[ion]) e(in) i(p) r(olarization) 2042 1169 y([of) 38 b(the) g(other) g(t) n(w) n(o) f(photons]) h(ev) n(en) f(when) h(w) n (e) g(do) g(not) g(mea-) 2042 1264 y(sure) f(p) r(olarization) g(on) h (the) g(\014rst) g(t) n(w) n(o) g(at) g(all") f([19) o(]) h(and) g (also) 2042 1360 y(our) 24 b(disco) n(v) n(ery) f(of) i(a) f(100\045) g (p) r(olarization) f(correlation) g(b) r(et) n(w) n(een) 2042 1455 y(unp) r(olarized) 33 b(photons) g([25) o(]) g(w) n(e) g(arriv) n (ed) f(at) i(b) n(y) f(in) n(v) n(estigating) 2042 1551 y(creation) h(and) g(annihilation) h(op) r(erators) e(when) i(acting) f (on) g(or-) 2042 1646 y(thogonal) 17 b(states) i(in) g(the) g(second) f (quan) n(tization) g(formalism.) 33 b(W) -7 b(e) 2042 1742 y(realized) 22 b(that) i(this) f(orthogonalit) n(y) e(is) i (crucially) f(di\013eren) n(t) i(from) 2042 1837 y(the) 33 b(classical) e(orthogonalit) n(y) -7 b(.) 50 b(In) 33 b(en) n(tanglemen) n(t) f(w) n(e) h(mak) n(e) f(a) 2042 1933 y(tensor) d(pro) r(duct) g(state) g(and) h(then) g(extract) f (just) h(a) f(part) g(of) h(the) 2042 2028 y(state.) 76 b(Since) 41 b(in) g(the) g(obtained) f(Hilb) r(ert) h(subspace) f(the) h (or-) 2042 2124 y(thogonalit) n(y) 26 b(of) i(the) g(one) g (dimensional) f(subspaces) g(con) n(taining) 2042 2219 y(relev) -5 b(an) n(t) 21 b(v) n(ectors) f(means) i(that) g(they) g (are) f(included) h(in) g(the) h(span) 2042 2315 y(of) 30 b(the) g(other) g(one) f(dimensional) h(subspaces) f(of) h(the) g (subspace,) 2042 2410 y(i.e.,) i(that) f(the) g(v) n(ectors) f(are) g (orthogonal) e(to) j(eac) n(h) f(other,) i(span) 2042 2506 y(the) c(considered) e(subspace,) h(and) h(mak) n(e) e(it) i(a) g (Hilb) r(ert) g(space.) 2125 2612 y(Exactly) 18 b(this) i(prop) r(ert) n (y) e(of) h(the) g(quan) n(tum) h(orthogonalit) n(y) c(en-) 2042 2707 y(ables) 26 b(us) g(to) h(use) f(linear) g(instead) h(of) f (classical) f(nonlinear) h(equa-) 2042 2803 y(tions.) 45 b(T) -7 b(o) 30 b(see) g(the) h(meaning) f(of) h(the) g(di\013erence) f (w) n(e) g(will) h(con-) 2042 2898 y(sider) 38 b(the) h(old) g(problem) f(of) h(\014nding) g(\014nite) g(Ko) r(c) n(hen-Sp) r(ec) n(k) n(er) 2042 2994 y(v) n(ectors) 26 b(whic) n(h) h(pro) n(v) n(e) f(the) i(Ko) r (c) n(hen-Sp) r(ec) n(k) n(er) e(theorem.) 2125 3100 y(Recen) n(tly) 63 b(prop) r(osed) f(exp) r(erimen) n(tal) h(tests) g (of) g(Ko) r(c) n(hen-) 2042 3196 y(Sp) r(ec) n(k) n(er) 36 b(theorem) h([26) o(,) j(27) o(]) d(and) h(disputes) f(on) g (feasibilit) n(y) g(of) 2042 3291 y(suc) n(h) 23 b(exp) r(erimen) n(ts) h([28) o({32) n(]) g(prompted) g(a) f(renew) n(ed) h(in) n(terest) f (in) 2042 3386 y(the) 32 b(theorem) g(and) g(this) g(an) g(additional) g (reason) e(for) i(reconsid-) 2042 3482 y(ering) 27 b(the) h(theorem.) 2125 3588 y(The) g(original) e(Ko) r(c) n(hen-Sp) r(ec) n(k) n(er) f (theorem) i([33) o(]) h(pro) r(duced) g(a) 2042 3684 y(set) h(of) g(117) f(3-dimensional) g(Hilb) r(ert) i(space) e(v) n (ectors) g(for) g(whic) n(h) 2042 3779 y(there) i(is) g(no) h(w) n(a) n (y) e(to) h(assign) f(1's) h(and) h(0's) f(to) g(their) g(states) g (and) 2042 3875 y(therefore) f(no) g(w) n(a) n(y) g(to) h(pro) n(vide) e (quan) n(tum) i(space) f(with) i(a) e(clas-) 2042 3970 y(sical) i(Bo) r(olean) f(mo) r(del.) 50 b(The) 32 b(pro) r(of) f(w) n (as) g(tedious) g(and) h(subse-) 2042 4066 y(quen) n(t) 24 b(attempts) h(to) g(reduce) f(the) g(n) n(um) n(b) r(er) h(of) f(v) n (ectors) f(ga) n(v) n(e) g(the) 2042 4161 y(follo) n(wing) 29 b(minimal) i(results:) 43 b(33) 30 b([34) o(]) h(and) f(31) g([35) o(,) i(p.) f(114]) e(3-) 2042 4257 y(dim) 21 b(v) n(ectors,) g(18) f([36) o (]) i(and) e(14) h([37) o(]) g(4-dim) f(v) n(ectors,) h(29,) h(31,) f (and) 2042 4352 y(34) 26 b(5-dim,) h(6-dim,) g(and) g(7-dim) g(v) n (ectors,) f(resp) r(ectiv) n(ely) g([38],) h(36) 2042 4447 y(8-dim) i(v) n(ectors) f([39) o(],) i(etc.) 43 b(Reducing) 29 b(the) h(n) n(um) n(b) r(er) f(of) h(v) n(ectors) 2042 4543 y(turn) 23 b(out) h(to) f(b) r(e) h(imp) r(ortan) n(t) f(b) r (ecause) h(a) f(direct) g(connection) g(b) r(e-) 2042 4638 y(t) n(w) n(een) i(suc) n(h) g(v) n(ectors) f(and) h(an) g(exp) r (erimen) n(tal) g(setup) h(can) f(b) r(e) h(es-) 2042 4734 y(tablished.) 21 b([38) o(]) g(Ho) n(w) n(ev) n(er,) f(no) g (general) g(metho) r(d) h(for) f(construct-) 2042 4829 y(ing) 26 b(sets) g(of) g(Ko) r(c) n(hen-Sp) r(ec) n(k) n(er) e(v) n (ectors) g(has) i(b) r(een) h(prop) r(osed) e(so) 2042 4925 y(far) i(and) g(here) g(w) n(e) h(giv) n(e) e(one.) 2125 5031 y(The) e(main) h(idea) f(of) g(our) g(approac) n(h) e(is) i(to) g (\014rst) h(sho) n(w) e(that) i(for) 2042 5127 y(particular) 20 b(set) i(of) f(orthogonal) f(Hilb) r(ert) i(space) f(v) n(ectors) f (one) i(can) 2042 5222 y(imp) r(ose) i(no) g(0) 14 b(-1) 24 b(state) g(on) g(the) h(v) n(ectors.) 34 b(Ho) n(w) n(ev) n(er,) 24 b(w) n(e) g(do) g(that) 2042 5318 y(using) e(the) h(Hilb) r(ert) g (space) e(orthogonalit) n(y:) p Fh 32 w(a) p Fg 23 w(\024) p Fh 23 w(b) p Fg 8 w([) p Fh 8 w(c) p Fg([) p Fr 23 w(.) 14 b(.) g(.) g(,) 23 b(not) 2042 5413 y(the) 30 b(standard) f(one:) 40 b(\() p Fh(a;) 14 b(b) p Fr(\)) 27 b(=) f(0,) k(\() p Fh(a;) 14 b(c) p Fr(\)) 27 b(=) f(0,) k(.) 14 b(.) f(.) h(,) 31 b(whic) n(h) e(b) r(oils) p 90 rotate dyy eop %%Page: 4 4 4 3 bop Fr 4043 -299 a(4) -150 -83 y(do) n(wn) 22 b(to) g(a) g (non-linear) f(system:) p Fh 34 w(a) p Fa 960 -71 a(1) p Fh 997 -83 a(b) p Fa 1033 -71 a(1) p Fr 1078 -83 a(+) p Fh 8 w(a) p Fa 1195 -71 a(2) p Fh 1232 -83 a(b) p Fa 1268 -71 a(2) p Fr 1313 -83 a(+) p Fh 8 w(a) p Fa 1430 -71 a(3) p Fh 1467 -83 a(b) p Fa 1503 -71 a(3) p Fr 1548 -83 a(+) p Fg 8 w(\001) 14 b(\001) g(\001) p Fr 22 w(=) 23 b(0,) p Fh -150 12 a(a) p Fa -106 24 a(1) p Fh -69 12 a(c) p Fa -33 24 a(1) p Fr 28 12 a(+) p Fh 23 w(a) p Fa 160 24 a(2) p Fh 197 12 a(c) p Fa 233 24 a(2) p Fr 294 12 a(+) p Fh 23 w(a) p Fa 426 24 a(3) p Fh 463 12 a(c) p Fa 499 24 a(3) p Fr 560 12 a(+) p Fg 23 w(\001) 14 b(\001) g(\001) p Fr 36 w(=) 36 b(0,) h(.) 14 b(.) g(.) g(But) 35 b(ev) n(en) g(this) h(Hilb) r(ert) -150 108 y(orthogonalit) n(y) 31 b(w) n(e) i(do) g(not) h(\\calculate"|it) e(is) h(\\built) h(in") f(in) -150 203 y(the) 40 b(MMP) g(diagrams) e(b) n(y) i(its) g(generations) e (algorithm.) 73 b(W) -7 b(e) -150 299 y(only) 36 b(c) n(hec) n(k) f (whether) h(one) g(can) g(or) f(cannot) h(imp) r(ose) g(classical) -150 394 y(0) 14 b(-1) 34 b(state) h(on) h(the) f(diagrams.) 59 b(W) -7 b(e) 36 b(then) g(only) f(ha) n(v) n(e) g(to) g(\014nd) -150 490 y(the) k(one) f(whic) n(h) h(do) r(es) f(not) h(allo) n(w) e(suc) n (h) i(a) f(state) g(and) h(this) g(is) -150 585 y(done) 25 b(b) n(y) g(a) g(simple) g(program) f(whic) n(h) h(follo) n(ws) f(the) i (de\014nition) f(of) -150 681 y(the) j(classical) e(state.) -67 776 y(In) 38 b(order) e(to) h(con) n(vince) g(the) h(reader) e(w) n(e) h (will) h(\014nd) g(a) f(mini-) -150 872 y(mal) h(set) g(of) g(v) n (ectors) e(from) i(a) f(5) h(dim) g(space) g(as) f(an) h(example.) -150 967 y(The) 29 b(smallest) f(MMP) g(diagram) f(w) n(e) i(\014nd) g(not) f (to) h(allo) n(w) e(a) h(clas-) -150 1063 y(sical) h(state) h(0) 14 b(-1) 28 b(is) h(the) h(follo) n(wing) f(one:) p Fh 41 w(abc;) 14 b(cde;) g(e) -14 b(f) -5 b(a;) 14 b(eg) s(b;) g(dg) -11 b(f) p Fr -150 1158 a(\(where,) 23 b(e.g,) p Fh 22 w(abc) p Fr 21 w(means) e(an) g(orthogonal) f(triple:) p Fh 34 w(a) p Fg 23 w(?) p Fh 22 w(b) p Fr(,) p Fh 23 w(a) p Fg 23 w(?) p Fh 22 w(c) p Fr(,) -150 1254 y(and) p Fh 20 w(b) p Fg 23 w(?) p Fh 23 w(c) p Fr(\).) 35 b(Then) 20 b(w) n(e) h(form) f(equations) g(corresp) r(onding) f(to) h(the) -150 1349 y(inner) 33 b(pro) r(ducts) g(of) g(5) f(dim) i(orthogonal) c(v) n (ectors) i(b) r(eing) h(equal) -150 1445 y(to) 28 b(0) h(and) f(solv) n (e) g(the) h(system.) 39 b(W) -7 b(e) 29 b(deal) f(with) i(triples) e (and) g(not) -150 1540 y(with) 36 b(quin) n(tuples) f(since) g(w) n(e) g (only) g(ha) n(v) n(e) f(to) h(\014nd) g(a) g(set) g(whic) n(h) -150 1636 y(do) r(es) 40 b(not) h(allo) n(w) f(0) 14 b(-1) 39 b(v) -5 b(aluation.) 76 b(I.e.,) 45 b(w) n(e) 40 b(follo) n(w) g(the) h (t) n(w) n(o) -150 1731 y(Ko) r(c) n(hen-Sp) r(ec) n(k) n(er) 25 b(\(actually) -7 b(,) 28 b(Gleason's) e([40) o(]\)) i(conditions:) -40 1893 y(1.) 33 b(No) 22 b(t) n(w) n(o) g(orthogonal) f(ra) n(ys) g(are) h (b) r(oth) i(assigned) d(the) j(v) -5 b(alue) 58 1988 y(1;) -40 2150 y(2.) 33 b(In) 24 b(an) n(y) f(group) f(of) p Fh 24 w(n) p Fr 24 w(m) n(utually) h(orthogonal) f(ra) n(ys,) h(not) h (all) 58 2246 y(of) j(the) h(ra) n(ys) e(are) h(assigned) f(the) i(v) -5 b(alue) 28 b(0.) -67 2408 y(Suc) n(h) 34 b(triples) g(are) f(in) i (principle) f(just) h(a) e(part) h(of) g(a) g(p) r(ossible) -150 2503 y(exp) r(erimen) n(t.) 78 b(What) 42 b(is) f(imp) r(ortan) n(t) g (is) g(that) h(for) f(particular) -150 2599 y(orthogonalities) 19 b(b) r(et) n(w) n(een) j(c) n(hosen) f(v) n(ectors) f(w) n(e) h(cannot) g(ascrib) r(e) -150 2694 y(0) 14 b(-1) 26 b(v) -5 b(alues) 27 b(to) h(them.) -67 2790 y(Since) 36 b(only) g(the) g(directions) f(of) h (the) g(v) n(ectors) e(\(\\ra) n(ys") g([34) o(]\)) -150 2885 y(are) 27 b(relev) -5 b(an) n(t) 28 b(they) h(m) n(ust) g(b) r(e) f (real.) 39 b(Since) 29 b(w) n(e) f(did) h(not) f(care) g(to) -150 2980 y(\014nd) 37 b(\\nice) f(lo) r(oking") f(v) n(ectors) g(some) h(v) n(ectors) f(are) g(\\big") g(due) -150 3076 y(to) 26 b(a) g(recursion) e(pro) r(cedure.) 36 b(This) 26 b(ho) n(w) n(ev) n (er) e(do) i(not) g(a\013ect) g(the) -150 3171 y(main) d(aim) f(of) h (\014nding) g(the) g(v) n(ectors) e(and) i(it) g(is) g(to) f(sho) n(w) g (that) h(our) -150 3267 y(approac) n(h) 38 b(w) n(orks.) 71 b(a=) p Fg(f) p Fr(608683911,) 38 b(17315878,) h(-22061625,) -150 3362 y(-111556858,) g(20961326) p Fg(g) p Fr(,) g(b=) p Fg(f) p Fr(3,) j(68,) g(-123,) g(52,) g(4) p Fg(g) p Fr(,) h(c=) p Fg(f) p Fr(1,) -150 3458 y(3,) e(5,) g(7,) g(11) p Fg(g) p Fr(,) f(d=) p Fg(f) p Fr(11,) g(-11,) g(11,) g(-11,) h(4) p Fg(g) p Fr(,) f(e=) p Fg(f) p Fr(1788,) f(-8663,) -150 3553 y(-1348,) 63 b(8223,) g(-2420) p Fg(g) p Fr(,) f(f=) p Fg(f) p Fr(5791304343,) d(-304905182408,) -150 3649 y(-1387655556967) o (,) 19 b(1686769435032) o(,) g(7600253389432) p Fg -1 w(g) p Fr(,) f(g=) p Fg(f) p Fr(1,) -150 3744 y(1,) 36 b(1,) g(1,) g(0) p Fg(g) p Fr(.) 56 b(The) 35 b(reader) e(can) h(in) n (tro) r(duce) g(the) h(v) n(ectors) e(in) n(to,) -150 3840 y(e.g.,) 21 b(W) -7 b(olfram's) p Fi 18 w(Mathematic) l(a) p Fr 31 w(and) 19 b(con) n(vince) f(her-) h(or) f(himself) -150 3935 y(that) 28 b(they) h(really) e(are) g(orthogonal) f(and,) i(b) n (y) g(simple) h(com) n(bina-) -150 4031 y(torics,) i(that) h(one) e (cannot) h(ascrib) r(e) f(0) h(or) f(1) h(to) g(all) g(of) g(them) h (\(in) -150 4126 y(eac) n(h) 23 b(triple) h(one) g(elemen) n(t) g(m) n (ust) g(b) r(e) h(1) e(and) h(the) h(other) e(t) n(w) n(o) h(zero) -150 4222 y(and) j(this) h(is) g(imp) r(ossible\).) -67 4317 y(Cab) r(ello) h([38) o(]) g(related) f(his) h(Ko) r(c) n(hen-Sp) r(ec) n(k) n(er) e(set) i(of) g(18) f(v) n(ec-) -150 4413 y(tors) c(in) g(9) g (blo) r(c) n(ks) g(with) h(his) g(exp) r(erimen) n(tal) e(prop) r(osal) g(in) i(a) f(four-) -150 4508 y(dimensional) c(Hilb) r(ert) h(space) e ([26) o(]) i(and) f(he) g(deals) g(with) h(4-tuples.) -150 4604 y(W) -7 b(e) 27 b(deal) e(with) i(triples) f(and) g(lea) n(v) n(e) f(a) g(problem) h(of) g(\014nding) h(a) e(re-) -150 4699 y(lated) 36 b(exp) r(erimen) n(t) g(op) r(en.) 61 b(This) 36 b(is) g(b) r(ecause) f(w) n(e) h(are) f(\014rst) g(of) -150 4795 y(all) f(in) n(terested) f(in) i(\014nding) f(a) f(general) g (algorithm) g(for) g(\014nd) i(all) -150 4890 y(orthogonalities) i (that) i(do) f(not) h(allo) n(w) f(0) 14 b(-1) 37 b(states.) 70 b(So,) 42 b(e.g.,) -150 4985 y(Cab) r(ello's) 25 b(18) f(v) n(ectors) g (in) i(9) f(blo) r(c) n(ks) g(can) g(form) g(ab) r(out) g(cca) g(7000) -150 5081 y(MMP) j(diagrams) f(that) i(do) g(not) f(allo) n(w) g(0) 14 b(-1) 27 b(states.) 40 b(but) 29 b(do) f(al-) -150 5176 y(lo) n(w) 22 b(quan) n(tum) h(states.) 35 b(Still,) 25 b(none) d(of) h(these) g(examples) g(\(there-) 2042 -83 y(fore) 36 b(not) g(ev) n(en) g(the) h(one) f(elab) r(orated) g(in) g (Ref.) h([38]\)) f(b) n(y) h(itself) 2042 12 y(corresp) r(ond) c(to) i (a) g(Hilb) r(ert) h(space) e(b) r(ecause) h(their) g(MMP) g(dia-) 2042 108 y(grams) 30 b(do) h(not) h(corresp) r(ond) d(to) j(lattices:) 44 b(the) 32 b(smallest) f(triple) 2042 203 y(lattice) 38 b(whic) n(h) g(do) g(not) g(allo) n(w) f(states) g(are) g(t) n(w) n(o) h (lattices) g(with) 2042 299 y(19) 25 b(atoms) g(and) g(13) g(blo) r(c) n (ks) g(and) h(a) f(quadruple) g(lattice) h(can) g(only) 2042 394 y(ha) n(v) n(e) 18 b(more) h(atoms) g(and/or) e(blo) r(c) n(ks.) 34 b(Other) 19 b(smaller) f(cases) h(with) 2042 490 y(18) 24 b(v) n(ectors) g(whic) n(h) h(do) g(not) g(allo) n(w) g(0) 14 b(-1) 23 b(states) i(are:) 35 b(4) 25 b(diagrams) 2042 585 y(with) j(8) g(\(quadruple) g(blo) r(c) n(ks\)) f(blo) r(c) n(ks.) 37 b(The) 29 b(lo) n(w) n(est) d(n) n(um) n(b) r(er) i(of) 2042 681 y(quadruple) d(blo) r(c) n(ks) f(and) h(v) n(ectors) f(are:) 35 b(1) 25 b(4-blo) r(c) n(k) f(case) g(with) i(10) 2042 776 y(v) n(ectors.) 2125 884 y(Let) c(us) f(b) r(e) h(more) f(sp) r (eci\014c:) 34 b(one) 21 b(of) g(the) h(obtained) g(18-9) d(MMP) 2042 979 y(diagrams) 33 b(is:) p Fh 51 w(abcd) p Fr(,) p Fh 37 w(de) -14 b(f) -5 b(g) p Fr 3 w(,) p Fh 36 w(g) s(hij) p Fr 5 w(,) p Fh 36 w(j) 5 b(k) s(l) r(m) p Fr(,) p Fh 35 w(mnop) p Fr(,) p Fh 37 w(pq) s(r) r(a) p Fr(,) p Fh 37 w(bik) s(r) p Fr 2 w(,) p Fh 2042 1075 a(cel) r(n) p Fr(,) p Fh 13 w(f) -5 b(hoq) p Fr 3 w(.) 38 b(And) 28 b(with) p Fh 29 w(a) p Fr(=100) 3062 1057 y(\026) 3062 1075 y(1,) p Fh 25 w(b) p Fr(=0110,) p Fh 26 w(c) p Fr(=11) 3655 1057 y(\026) 3655 1075 y(11,) p Fh 25 w(d) p Fr(=1) 3937 1057 y(\026) 3937 1075 y(111,) p Fh 2042 1170 a(e) p Fr(=111) 2272 1152 y(\026) 2272 1170 y(1,) p Fh 52 w(f) p Fr 9 w(=0101,) p Fh 52 w(g) p Fr 3 w(=10) 2939 1152 y(\026) 2939 1170 y(10,) p Fh 52 w(h) p Fr(=010) 3337 1152 y(\026) 3337 1170 y(1,) p Fh 52 w(i) p Fr(=1) 3590 1152 y(\026) 3590 1170 y(11) 3674 1152 y(\026) 3674 1170 y(1,) p Fh 53 w(j) p Fr 5 w(=1111,) p Fh 2042 1266 a(k) p Fr 3 w(=11) 2237 1248 y(\026) 2237 1266 y(1) 2279 1248 y(\026) 2279 1266 y(1,) p Fh 46 w(l) p Fr 2 w(=1) 2524 1248 y(\026) 2524 1266 y(100,) p Fh 46 w(m) p Fr(=001) 2983 1248 y(\026) 2983 1266 y(1,) p Fh 46 w(n) p Fr(=0011,) p Fh 46 w(o) p Fr(=1000,) p Fh 47 w(p) p Fr(=0100,) p Fh 2042 1361 a(q) p Fr 3 w(=0010,) p Fh 24 w(r) p Fr 2 w(=1001,) d(this) h(is) g(nothing) g(but) h(Cab) r (ello's) f(18-9) e(case.) 2042 1457 y(Graphically) 40 b(it) i(means) f(a) h(hexagram) d(with) j(3) g(ellipses) f(con-) 2042 1552 y(tained) f(in) g(it.) 73 b(The) 40 b(smallest) f(4-blo) r(c) n(k) g(10-5) f(case) g(is:) p Fh 61 w(abcd) p Fr(,) p Fh 2042 1647 a(de) -14 b(f) -5 b(g) p Fr 3 w(,) p Fh 35 w(g) s(hia) p Fr(,) p Fh 35 w(bf) 9 b(ij) p Fr 5 w(,) p Fh 35 w(cehj) p Fr 5 w(.) 57 b(Graphically) 33 b(it) i(means) f(a) g(triangle) 2042 1743 y(with) c(2) f(ellipses) g(con) n(tained) g(in) g(it) h(\(with) g (one) f(common) g(v) n(ertex) 2042 1838 y(not) 35 b(con) n(tained) f (in) h(the) g(triangle\).) 58 b(Ho) n(w) n(ev) n(er,) 35 b(it) g(migh) n(t) g(turn) 2042 1934 y(out) 28 b(\(w) n(e) g(still) h (ha) n(v) n(e) e(not) h(c) n(hec) n(k) n(edt\)) g(that) g(so) g(small) g (a) g(diagram) 2042 2029 y(cannot) 22 b(b) r(e) h(ascrib) r(ed) f(real) g(v) n(ectors) f(in) i(a) f(4-dim) g(space) g(and) h(that) 2042 2125 y(w) n(e) d(should) g(go) g(to) g(higher) g(dimensions) g(to) h (\014nd) g(real) e(v) n(ector) g(sets.) p Fj 2672 2464 a(V.) 88 b(CONCLUSION) p Fr 2125 2690 a(W) -7 b(e) 24 b(ha) n(v) n(e) f(sho) n(wn) g(that) i(one) e(can) h(build) h(an) f (algebra) e(underly-) 2042 2785 y(ing) 29 b(Hilb) r(ert) g(space) g (whic) n(h) g(could) g(b) r(e) g(a) g(univ) n(ersal) e(algebra) h(for) 2042 2881 y(quan) n(tum) 21 b(computers) f(in) i(the) f(same) g(w) n(a) n(y) f(the) h(Bo) r(olean) f(algebra) 2042 2976 y(is) 34 b(for) f(classical) f(computers.) 56 b(In) 34 b(our) f(approac) n(h) f (the) i(algebra) 2042 3072 y(is) k(based) f(on) h(p) r(olynomial) g (series) f(of) h(relations) f(b) r(et) n(w) n(een) h(one) 2042 3167 y(dimensional) 22 b(subspaces) g(of) i(a) e(Hilb) r(ert) i(space) e (and) h(linearly) f(de-) 2042 3263 y(\014ned) i(orthogonalit) n(y) e (relations) h(b) r(et) n(w) n(een) i(either) f(subspaces) f(or) 2042 3358 y(v) n(ectors) j(of) h(a) h(Hilb) r(ert) g(space.) 2125 3466 y(Linear) j(orthogonalit) n(y) f(de\014ned) i(through) g(MMP) g (diagrams) 2042 3561 y(p) r(ossibly) 27 b(op) r(ens) g(a) g(w) n(a) n (y) f(to) h(substitute) i(a) e(linear) f(for) h(nonlinear) 2042 3657 y(coupling) h(b) r(et) n(w) n(een) g(qubits) h(presen) n(tly) f (required) g(for) g(univ) n(ersal) 2042 3752 y(quan) n(tum) 20 b(computation.) 35 b(On) 20 b(the) h(other) f(hand) h(suc) n(h) f (linear) g(or-) 2042 3848 y(thogonalit) n(y) 31 b(de\014ned) i(through) f(MMP) h(diagrams) e(already) g(on) 2042 3943 y(our) 24 b(classical) g(computers) g(enabled) h(sp) r(eeding) g(up) g (calculations) 2042 4039 y(for) 36 b(more) h(than) g(5) g(order) f(of) h (magnitude) g(on) g(the) h(CPU) f(time) 2042 4134 y(scale) f(and) h (enabled) g(us) f(to) h(\014nd) h(p) r(olynomial) e(expressions) f(of) 2042 4229 y(the) p Fh 38 w(n) p Fr(-th) j(order) f(represen) n(ting) f (an) n(y) i(Hilb) r(ert) g(space) f(and) h(un-) 2042 4325 y(kno) n(wn) d(so) h(far.) 62 b(It) 37 b(also) e(enabled) h(us) g (to) g(\014nd) h(a) f(general) f(ap-) 2042 4420 y(proac) n(h) 22 b(to) h(\014nding) h(Ko) r(c) n(hen-Sp) r(ec) n(k) n(er) d(v) n (ectors,) i(some) g(of) g(whic) n(h) 2042 4516 y(w) n(e) k(presen) n (ted) g(ab) r(o) n(v) n(e.) p Fj 2716 4855 a(Ac) n(kno) n(wledgmen) n (ts) p Fr 2208 5081 a(The) d(author) g(ac) n(kno) n(wledges) e(supp) r (ort) j(of) f(the) h(Ministry) f(of) 2042 5176 y(Science) j(of) h (Croatia.) p 90 rotate dyy eop %%Page: 5 5 5 4 bop Fr 4043 -299 a(5) p 948 -81 2039 5 v 1203 -80 1529 7 v 1458 -79 1020 9 v 1712 -78 510 11 v Fo -112 167 a([1]) 35 b(C.) k(J.) g(Isham,) p Fq 74 w(L) l(e) l(ctur) l(es) j (of) c(Quantum) i(The) l(ory) p Fo(,) 77 b(Imp) r(erial) 3 254 y(College) 28 b(Press,) e(London,) g(1995.) -112 341 y([2]) 35 b(P) -6 b(.) 20 b(R.) f(Halmos,) p Fq 26 w(Intr) l(o) l(duction) 24 b(to) e(Hilb) l(ert) h(Sp) l(ac) l(e) g(and) g(the) g(Sp) l(e) l(c-) 3 429 y(tr) l(al) 30 b(The) l(ory) g(of) f(Sp) l (e) l(ctr) l(al) i(Multiplicity) p Fo(,) 39 b(Chelsea,) 30 b(New) e(Y) -6 b(ork,) 3 516 y(1957.) -112 603 y([3]) 35 b(G.) 30 b(Kalm) n(bac) n(h,) p Fq 46 w(Orthomo) l(dular) i(L) l(attic) l(es) p Fo(,) 49 b(Academic) 29 b(Press,) 3 690 y(London,) c(1983.) -112 777 y([4]) 35 b(P) -6 b(.) 43 b(Mittelstaedt,) p Fq 94 w(Quantum) i(L) l(o) l(gic) p Fo(,) 93 b(Syn) n(these) 42 b(Library;) 3 864 y(V) -6 b(ol.) 26 b(18,) g(Reidel,) g(London,) g (1978.) -112 952 y([5]) 35 b(F.) 27 b(Maeda) g(and) g(S.) f(Maeda,) p Fq 39 w(The) l(ory) k(of) e(Symmetric) h(L) l(attic) l(es) p Fo(,) 3 1039 y(Springer-Verlag,) d(New) g(Y) -6 b(ork,) 25 b(1970.) -112 1126 y([6]) 35 b(P) -6 b(.-A.) 27 b(Iv) n(ert) f(and) i (T.) g(Sj\177) -38 b(odin,) 40 b(On) 27 b(the) g(Imp) r(ossibilit) n(y) g(of) h(a) g(Fi-) 3 1213 y(nite) 20 b(Prop) r(ositional) i(Lattice) f (for) g(Quan) n(tum) d(Mec) n(hanics,) 27 b(Helv.) 3 1300 y(Ph) n(ys.) e(Acta) p Fj 26 w(51) p Fo(,) h(635{636) j(\(1978\).) -112 1388 y([7]) 35 b(S.) 29 b(Llo) n(yd,) 46 b(Univ) n(ersal) 29 b(Quan) n(tum) e(Sim) n(ulators,) 46 b(Science) p Fj 29 w(273) p Fo(,) 3 1475 y(1073{1078) 29 b(\(1996\).) -112 1562 y([8]) 35 b(S.) h(Llo) n(yd,) 67 b(Univ) n(ersal) 36 b(Quan) n(tum) e(Sim) n(ulators:) 55 b(Correction,) 3 1649 y(Science) p Fj 26 w(279) p Fo(,) 26 b(1117) h(\(1998\).) -112 1736 y([9]) 35 b(S.) 49 b(Llo) n(yd) g(and) g(S.) g(L.) g(Braunstein,) 112 b(Quan) n(tum) 47 b(Compu-) 3 1823 y(tation) g(o) n(v) n(er) h(Con) n(tin) n(uous) f(V) -6 b(ariables,) 105 b(Ph) n(ys.) 47 b(Rev.) g(Lett.) p Fj 3 1911 a(82) p Fo(,) 41 b(1784{1787) g(\(1999\),) 73 b(h) n(ttp://xxx.lanl.go) n(v/abs/quan) n(t-) 3 1998 y(ph/9810082.) -150 2085 y([10]) 35 b(B.) 21 b(B.) g(Boghosian) h(and) f (W.) f(T) -6 b(a) n(ylor) 21 b(IV,) 26 b(Sim) n(ulating) 19 b(Quan) n(tum) 3 2172 y(Mec) n(hanics) h(on) g(a) f(Quan) n(tum) f (Computer,) 25 b(Ph) n(ysica) 19 b(D) p Fj 20 w(120) p Fo(,) i(30{) 3 2259 y(42) 26 b(\(1998\),) 36 b(h) n(ttp://xxx.lanl.go) n (v/abs/quan) n(t-ph/9701019.) -150 2346 y([11]) f(B.) c(B.) h (Boghosian) h(and) e(W.) g(T) -6 b(a) n(ylor) 32 b(IV,) 50 b(Quan) n(tum) 29 b(Lattice-) 3 2434 y(Gas) h(Mo) r(dels) h(for) f(the) f(Man) n(y-Bo) r(dy) g(Sc) n(hr\177) -38 b(odinger) 30 b(Equations,) 3 2521 y(\(1997\),) 35 b(h) n(ttp://xxx.lanl.go) n (v/abs/quan) n(t-ph/9701016.) -150 2608 y([12]) g(B.) 43 b(D.) f(McKa) n(y) -6 b(,) 46 b(N.) d(D.) f(Megill,) 49 b(and) 42 b(M.) h(P) n(a) n(vi) n(\024) -36 b(ci) n(\023) g(c,) 89 b(Algo-) 3 2695 y(rithms) 43 b(for) j(Greec) n(hie) f(Diagrams,) 96 b(In) n(t.) 44 b(J.) i(Theor.) f(Ph) n(ys.) p Fj 3 2782 a(39) p Fo(,) c(2393{2417) g(\(2000\),) 73 b(h) n(ttp://xxx.lanl.go) n (v/abs/quan) n(t-) 3 2869 y(ph/0009039.) -150 2957 y([13]) 35 b(K.) 29 b(Sv) n(ozil) h(and) f(J.) h(Tk) l(adlec,) 47 b(Greec) n(hie) 31 b(Diagrams,) g(Nonexis-) 3 3044 y(tence) g(of) i (Measures) f(and) g(Ko) r(c) n(hen-Sp) r(ec) n(k) n(er-T) n(yp) r(e) d (Construc-) 3 3131 y(tions,) 35 b(J.) 26 b(Math.) g(Ph) n(ys.) p Fj 26 w(37) p Fo(,) g(5380{5401) j(\(1996\).) -150 3218 y([14]) 35 b(R.) d(Go) r(do) n(wski,) 56 b(V) -6 b(arieties) 33 b(of) g(Orthomo) r(dular) e(Lattices) i(with) 3 3305 y(a) f(Strongly) f(F) -6 b(ull) 32 b(Set) f(of) i(States,) 54 b(Demonstratio) 31 b(Math.) p Fj 33 w(14) p Fo(,) 3 3392 y(725{733) d(\(1981\).) -150 3480 y([15]) 35 b(N.) i(D.) h(Megill) h (and) e(M.) i(P) n(a) n(vi) n(\024) -36 b(ci) n(\023) g(c,) 73 b(Equations,) 42 b(States,) f(and) 3 3567 y(Lattices) c(of) g (In\014nite-Dimensional) e(Hilb) r(ert) h(Space,) 69 b(In) n(t.) 36 b(J.) 3 3654 y(Theor.) 28 b(Ph) n(ys.) p Fj 27 w(39) p Fo(,) g(2349{2391) j(\(2000\),) 41 b(h) n (ttp://xxx.lanl.go) n(v) 3 3741 y(/abs/quan) n(t-ph/0009038.) -150 3828 y([16]) 35 b(M.) 27 b(P) n(a) n(vi) n(\024) -36 b(ci) n(\023) g(c,) 40 b(Quan) n(tum) 24 b(Loic) k(for) g(Gen) n(uine) e (Quan) n(tum) f(Sim) n(u-) 3 3916 y(lators,) 31 b(in) p Fq 22 w(Quantum) 25 b(Computing) p Fo(,) f(edited) e(b) n(y) g(E.) h (Donk) n(or) f(and) 3 4003 y(A.) j(R.) g(Piric) n(h,) h(v) n(olume) e (4047) j(of) p Fq 26 w(Pr) l(o) l(c) l(e) l(e) l(dings) i(of) e(SPIE) p Fo(,) e(pages) 3 4090 y(90{96,) j(SPIE,) d(Bellingham,) i(W) -6 b(ashington,) 26 b(2000.) -150 4177 y([17]) 35 b(N.) 23 b(D.) h(Megill) i(and) e(M.) g(P) n(a) n(vi) n(\024) -36 b(ci) n(\023) g(c,) 33 b(Quan) n(tum) 22 b(State) i(Equations,) 3 4264 y([to) i(app) r(ear]) 52 b(\(2002\).) -150 4351 y([18]) 35 b(M.) f(P) n(a) n(vi) n(\024) -36 b(ci) n(\023) g(c) 35 b(and) f(J.) h(Summhammer,) 57 b(In) n(terferometry) 33 b(with) 3 4439 y(Tw) n(o) 26 b(P) n(airs) g(of) h(Spin) d(Correlated) j (Photons,) 34 b(Ph) n(ys.) 25 b(Rev.) g(Lett.) p Fj 3 4526 a(73) p Fo(,) h(3191{3194) j(\(1994\).) -150 4613 y([19]) 35 b(M.) d(P) n(a) n(vi) n(\024) -36 b(ci) n(\023) g(c,) 56 b(Spin-Correlated) 32 b(In) n(terferometry) f(with) h(Beam) 3 4700 y(Splitters:) 56 b(Preselection) 39 b(of) e(Spin-Correlated) g (Photons,) 71 b(J.) 3 4787 y(Opt.) 25 b(So) r(c.) h(Am.) f(B) p Fj 26 w(12) p Fo(,) h(821{828) i(\(1995\).) -150 4874 y([20]) 35 b(C.) 30 b(H.) f(Bennett,) h(G.) g(Brassard,) i(C.) d(Cr) n (\023) -36 b(ep) r(eau,) 32 b(R.) d(Jozsa,) j(and) 3 4962 y(W.) 40 b(K.) g(W) -6 b(o) r(otters,) 81 b(T) -6 b(elep) r(orting) 42 b(an) e(Unkno) n(wn) e(Quan) n(tum) 3 5049 y(State) 25 b(via) g(Dual) g(Classical) j(Einstein-P) n(o) r (dolsky-Rosen) e(Chan-) 3 5136 y(nels,) 34 b(Ph) n(ys.) 26 b(Rev.) f(Lett.) p Fj 26 w(70) p Fo(,) h(1895{1898) k(\(1993\).) -150 5232 y([21]) 35 b(M.) 126 5213 y(_) 113 5232 y(Zuk) n(o) n(wski,) 20 b(A.) f(Zeilinger,) i(M.) e(A.) g(Horne,) h(and) e(A.) h(K.) f(Ek) n (ert,) 3 5319 y(\\Ev) n(en) n(t-Ready-Detectors") j(Bell) i(Exp) r (erimen) n(t) e(via) h(En) n(tangle-) 3 5406 y(men) n(t) i(Sw) n (apping,) 34 b(Ph) n(ys.) 26 b(Rev.) f(Lett.) p Fj 25 w(71) p Fo(,) i(4287{4290) i(\(1993\).) 2042 167 y([22]) 34 b(J.) 20 b(P) n(an,) h(D.) d(Bou) n(wmeester,) j(H.) e(W) -6 b(einfurter,) 20 b(and) f(A.) f(Zeilinger,) 2194 254 y(Exp) r(erimen) n(tal) 33 b(En) n(tanglemen) n(t) e(Sw) n(apping:) 49 b(En) n(tagling) 34 b(Pho-) 2194 341 y(tons) c(that) e(Nev) n(er) h(In) n(teracted,) 44 b(Ph) n(ys.) 29 b(Rev.) g(Lett.) p Fj 29 w(80) p Fo(,) h(3891{) 2194 429 y(4894) e(\(1998\).) 2042 516 y([23]) 34 b(D.) d(Bou) n(wmeester,) h(J.) f(P) n(an,) h(K.) e (Mattle,) j(M.) e(Eibl,) i(H.) d(W) -6 b(ein-) 2194 603 y(furter,) 37 b(and) c(A.) h(Zeilinger,) 61 b(Exp) r(erimen) n(tal) 33 b(Quan) n(tum) f(T) -6 b(ele-) 2194 690 y(p) r(ortation,) 36 b(Nature) p Fj 25 w(390) p Fo(,) 26 b(575{579) j(\(1997\).) 2042 777 y([24]) 34 b(D.) 47 b(Bou) n(wmeester,) 53 b(K.) 46 b(Mattle,) 54 b(J.) 47 b(P) n(an,) 52 b(H.) 47 b(W) -6 b(einfurter,) 2194 873 y(A.) 28 b(Zeilinger,) i(and) e(M.) 2925 854 y(_) 2912 873 y(Zuk) n(o) n(wski,) 42 b(Exp) r(erimen) n(tal) 27 b(Quan) n(tum) 2194 960 y(T) -6 b(elep) r(ortation) 27 b(of) g(Arbitrary) e(Quan) n(tum) e(States,) 34 b(Appl.) 26 b(Ph) n(ys) p Fj 2194 1048 a(B) k(67) p Fo(,) c(749{752) i(\(1998\).) 2042 1135 y([25]) 34 b(M.) 24 b(P) n(a) n(vi) n(\024) -36 b(ci) n(\023) g(c,) 32 b(Spin) 22 b(Correlated) j(In) n(terferometry) d (for) i(P) n(olarized) 2194 1222 y(and) c(Unp) r(olarized) g(Photons) h (on) f(a) g(Beam) g(Splitter,) 26 b(Ph) n(ys.) 20 b(Rev.) 2194 1309 y(A) p Fj 26 w(50) p Fo(,) 26 b(3486{3491) j(\(1994\).) 2042 1396 y([26]) 34 b(A.) 55 b(Cab) r(ello) i(and) e(G.) g(Garc) -9 b(\023) -30 b(\020a-Alcaine,) 132 b(Prop) r(osed) 56 b(Ex-) 2194 1483 y(p) r(erimen) n(tal) k(T) -6 b(ests) 62 b(of) f(the) f(Bell-Ko) r(c) n(hen-Sp) r(ec) n(k) n(er) g(Theo-) 2194 1571 y(rem,) 65 b(Ph) n(ys.) 35 b(Rev.) g(Lett.) p Fj 35 w(80) p Fo(,) k(1797{1799) f(\(1998\),) 67 b(h) n(ttp://) 2194 1658 y(xxx.lanl.go) n(v/abs/quan) n(t-ph/9709047.) 2042 1754 y([27]) 34 b(C.) 24 b(Simon,) e(H.) h(W) -6 b(einfurter,) 23 b(M.) 3181 1735 y(_) 3168 1754 y(Zuk) n(o) n(wski,) h(and) e(A.) g (Zeilinger,) 2194 1841 y(F) -6 b(easible) 32 b(Ko) r(c) n(hen-Sp) r(ec) n(k) n(er) d(Exp) r(erimen) n(t) h(with) h(Single) g(P) n(arti-) 2194 1928 y(cles,) 69 b(Ph) n(ys.) 36 b(Rev.) f(Lett.) p Fj 36 w(85) p Fo(,) 40 b(1783{1786) f(\(2000\),) 69 b(h) n(ttp://) 2194 2015 y(xxx.lanl.go) n(v/abs/quan) n(t-ph/0009074.) 2042 2102 y([28]) 34 b(D.) 50 b(A.) g(Mey) n(er,) 112 b(Finite) 50 b(Precision) h(Measuremen) n(t) e(Nulli-) 2194 2189 y(\014es) d(the) e (Ko) r(c) n(hen-Sp) r(ec) n(k) n(er) g(Theorem,) 96 b(Ph) n(ys.) 45 b(Rev.) g(Lett.) p Fj 2194 2277 a(83) p Fo(,) d(3751{3754) e(\(1999\),) 74 b(h) n(ttp://xxx.lanl.go) n(v/abs/quan) n(t-) 2194 2364 y(ph/9905080.) 2042 2451 y([29]) 34 b(A.) 29 b(Ken) n(t,) 43 b(Noncon) n(textual) 27 b(Hidden) h(V) -6 b(ariables) 29 b(and) f(Ph) n(ysical) 2194 2538 y(Measuremen) n(ts,) 49 b(Ph) n(ys.) 30 b(Rev.) f(Lett.) p Fj 30 w(83) p Fo(,) j(3755{3757) h (\(1999\),) 2194 2625 y(h) n(ttp://xxx.lanl.go) n(v/abs/quan) n (t-ph/9906006.) 2042 2712 y([30]) h(N.) 43 b(D.) g(Mermin,) 90 b(A) 43 b(Ko) r(c) n(hen-Sp) r(ec) n(k) n(er) e(Theorem) i(for) h(Im-) 2194 2800 y(precisely) j(Sp) r(eci\014ed) e(Measuremen) n(t,) 99 b(h) n(ttp://xxx.lanl.go) n(v) 2194 2887 y(/abs/quan) n(t-ph/9912081) 28 b(\(1999\).) 2042 2974 y([31]) 34 b(C.) 22 b(Simon,) 2552 2955 y(\024) 2544 2974 y(C.) f(Brukner,) g(and) f(A.) h(Zeilinger,) 28 b(Hidden-V) -6 b(ariable) 2194 3061 y(Theorems) 52 b(for) g(Real) g (Exp) r(erimen) n(ts,) 117 b(Ph) n(ys.) 51 b(Rev.) g(Lett.) p Fj 2194 3148 a(86) p Fo(,) 42 b(4427{4430) e(\(2001\),) 74 b(h) n(ttp://xxx.lanl.go) n(v/abs/quan) n(t-) 2194 3236 y(ph/0006043.) 2042 3323 y([32]) 34 b(A.) 20 b(Cab) r(ello,) 29 b(Finite) 20 b(Precision) i(Measuremen) n(t) d(Do) r(es) i(Not) f(Nul-) 2194 3410 y(lify) 38 b(the) e(Ko) r(c) n(hen-Sp) r(ec) n(k) n(er) f (Theorem,) 69 b(h) n(ttp://xxx.lanl.go) n(v) 2194 3497 y(/abs/quan) n(t-ph/0104024) 28 b(\(2001\).) 2042 3584 y([33]) 34 b(S.) h(Ko) r(c) n(hen) f(and) g(E.) h(P) -6 b(.) 35 b(Sp) r(ec) n(k) n(er,) 62 b(The) 35 b(problem) f(of) h(hidden) 2194 3671 y(v) l(ariables) 23 b(in) f(quan) n(tum) e(mec) n(hanics,) 29 b(J.) 23 b(Math.) g(Mec) n(h.) p Fj 22 w(17) p Fo(,) h(59{) 2194 3759 y(87) j(\(1967\).) 2042 3846 y([34]) 34 b(A.) 23 b(P) n(eres,) 30 b(Tw) n(o) 24 b(Simple) d(Pro) r(ofs) k(of) e(the) f (Bell-Ko) r(c) n(hen-Sp) r(ec) n(k) n(er) 2194 3933 y(Theorem,) 35 b(J.) 26 b(Ph) n(ys.) g(A) p Fj 25 w(24) p Fo(,) g(L175{L178) j (\(1991\).) 2042 4020 y([35]) 34 b(A.) 43 b(P) n(eres,) p Fq 88 w(Quantum) h(The) l(ory:) 66 b(Conc) l(epts) 45 b(and) e(Metho) l(ds) p Fo(,) 2194 4107 y(Klu) n(w) n(er,) 26 b(Dordrec) n(h) n(t,) g(1993.) 2042 4194 y([36]) 34 b(A.) j(Cab) r (ello,) 42 b(J.) c(M.) g(Estebaranz,) j(and) 36 b(G.) i(Garc) -9 b(\023) -30 b(\020a-Alcaine,) 2194 4282 y(Bell-Ko) r(c) n(hen-Sp) r(ec) n(k) n(er) 40 b(Theorem:) 64 b(A) 40 b(Pro) r(of) i(with) f(18) g(V) -6 b(ec-) 2194 4369 y(tors,) 101 b(Ph) n(ys.) 46 b(Lett.) h(A) p Fj 45 w(212) p Fo(,) 52 b(183{187) d(\(1996\),) 102 b(h) n(ttp://) 2194 4456 y(xxx.lanl.go) n(v/abs/quan) n(t-ph/9706009.) 2042 4543 y([37]) 34 b(A.) 73 b(Cab) r(ello,) 88 b(J.) 74 b(M.) f(Estebaranz,) 86 b(and) 73 b(G.) h(Garc) -9 b(\023) -30 b(\020a-) 2194 4630 y(Alcaine,) 40 b(New) 28 b(V) -6 b(arian) n(ts) 27 b(of) h(the) f(Bell-Ko) r(c) n(hen-Sp) r(ec) n(k) n (er) f(The-) 2194 4717 y(orem,) 90 b(Ph) n(ys.) 43 b(Lett.) g(A) p Fj 42 w(218) p Fo(,) 48 b(115{118) e(\(1996\),) 91 b(h) n(ttp://) 2194 4805 y(xxx.lanl.go) n(v/abs/quan) n(t-ph/9706009.) 2042 4892 y([38]) 34 b(A.) f(Cab) r(ello,) 60 b(Ko) r(c) n(hen-Sp) r(ec) n (k) n(er) 32 b(Theorem) h(and) g(Exp) r(erimen-) 2194 4979 y(tal) 39 b(T) -6 b(ests) 39 b(on) f(Hidden) f(V) -6 b(ariables,) 75 b(In) n(t.) 37 b(J.) i(Mo) r(d.) g(Ph) n(ys.) f(A) p Fj 2194 5066 a(15) p Fo(,) k(2813{2820) e(\(2000\),) 74 b(h) n(ttp://xxx.lanl.go) n(v/abs/quan) n(t-) 2194 5153 y(ph/9911022.) 2042 5241 y([39]) 34 b(M.) d(Kernaghan) f(and) f(A.) h (P) n(eres,) 48 b(Ko) r(c) n(hen-Sp) r(ec) n(k) n(er) 28 b(Theorem) 2194 5328 y(for) 38 b(Eigh) n(t-Dimensional) e(Space,) 69 b(Ph) n(ys.) 36 b(Lett.) g(A) p Fj 36 w(198) p Fo(,) k(1{5) 2194 5415 y(\(1995\),) c(h) n(ttp://xxx.lanl.go) n(v/abs/quan) n (t-ph/9412006.) p 90 rotate dyy eop %%Page: 6 6 6 5 bop Fr 4043 -299 a(6) p Fo -150 -83 a([40]) 35 b(J.) 25 b(Zim) n(ba) f(and) g(R.) h(P) n(enrose,) 33 b(On) 24 b(Bell) i(Non-Lo) r(calit) n(y) f(without) 3 4 y(Probabilities:) 36 b(More) 26 b(Curious) f(Geometry) -6 b(,) 32 b(Stud.) 24 b(Hist.) h(Phil.) 2194 -83 y(Sci.) p Fj 27 w(24) p Fo(,) h(697{720) i (\(1993\).) p 90 rotate dyy eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF