%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: lipt-jan.dvi %%Pages: 11 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: A4 %%EndComments %DVIPSCommandLine: dvips lipt-jan.dvi %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX output 1999.05.18:1912 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N 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b(describ)s(e)h(the)f(system)h(b)m(y)-91 2214 y(a)c(unit)g(v)m(ector)i(in)d(the)i(Hilb)s(ert)e(space)j Fp(H)1438 2178 y Fi(2)1508 2214 y Fu(o)m(v)m(er)g(the)f(\014eld)f(of)g (complex)g(n)m(um)m(b)s(ers.)44 b(W)-8 b(e)31 b(denote)h(the)f(t)m(w)m (o)-91 2338 y(orthogonal)21 b(states)j(b)m(y)f Fp(j)p Fu(0)p Fp(i)k Fu(=)h(\(1)p Fn(;)17 b Fu(0\))22 b(and)h Fp(j)p Fu(1)p Fp(i)k Fu(=)g(\(0)p Fn(;)17 b Fu(1\).)40 b(The)24 b(states)f(mak)m(e)g(an)g(orthogonal)e(basis)i(for)g Fp(H)3740 2302 y Fi(2)3779 2338 y Fu(.)-91 2463 y(In)33 b(a)f(quan)m(tum)h(computer)g(w)m(e)h(deal)e(with)h(a)f(big)g(n)m(um)m (b)s(er)h Fn(n)g Fu(of)g(qubits)g(whic)m(h)g(build)f(up)h(a)f(comp)s (osite)-91 2588 y(Hilb)s(ert)38 b(space)j Fp(H)h Fu(=)e Fp(H)847 2551 y Fi(2)914 2588 y Fp(\012)27 b Fn(:)17 b(:)g(:)27 b Fp(\012)h(H)1350 2551 y Fi(2)1389 2588 y Fu(.)66 b(The)41 b(computational)d(basis,)k(i.e.,)f(the)g(basis)f(of)f (this)h(space,)-91 2712 y(consists)g(of)f(the)i(follo)m(wing)c(2)1037 2676 y Fj(n)1123 2712 y 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(computing|whic)m(h)e(is)i(their)f(most)g(attractiv)m(e)-91 4679 y(feature|b)m(y)40 b(using)g(sup)s(erp)s(osition)f(whic)m(h)h (puts)h Fn(n)f Fu(quan)m(tum)g(states)h(in)e(a)g(sup)s(erp)s(osition)g (of)h(all)e(2)3760 4643 y Fj(n)-91 4804 y Fu(basis)33 b(states)i(in)e(one)g(step.)48 b(Again,)33 b(for)g(a)g(parallel)e (computation)h(a)h(classical)f(computer)i(w)m(ould)f(need)-91 4928 y(either)f(an)h(exp)s(onen)m(tially)e(gro)m(wing)h(hardw)m(are)h (or)g(an)f(exp)s(onen)m(tially)g(rising)f(time.)75 5053 y(Consider)42 b(for)f(example)h(the)g(follo)m(wing)d(state)j(of)f(2)h (particles,)h(kno)m(wn)g(as)f(the)h Fo(entangle)-5 b(d)51 b Fu(state)-91 5178 y(\(P)m(a)m(vi)m(\024)-46 b(ci)m(\023)g(c)31 b(&)h(Summhammer,)e(1994;)h(P)m(a)m(vi)m(\024)-46 b(ci)m(\023)g(c,)32 b(1995\))f(of)g(the)h(particles)f(whic)m(h)h(can)g(then)g(also)f(b)s(e) h(used)-91 5302 y(for)44 b(a)g(telep)s(ortation)f(of)h(states)i(or)e (Bell)g(exp)s(erimen)m(ts)h(or)f(quan)m(tum)h(cryptograph)m(y)h(\(w)m (e)g(omit)d(the)-91 5427 y(normalization)28 b(factors)33 b(throughout\):)1633 5551 y Fp(j)p Fu(00)p Fp(i)22 b Fu(+)g Fp(j)p Fu(11)p Fp(i)1599 b Fu(\(6\))1833 5881 y(2)p eop %%Page: 3 3 3 2 bop -91 299 a Fu(Here)38 b(no)f(one)g(of)g(the)g(t)m(w)m(o)h (qubits)g(has)f(a)g(de\014nite)g(state:)54 b(the)37 b(state)h(of)e(the) i(system)g(is)f(not)g(a)g(tensor)-91 423 y(pro)s(duct)32 b(of)h(the)g(states,)g(and)g(w)m(e)g(cannot)g(\014nd)g Fn(a)1757 438 y Fi(1)1797 423 y Fn(;)17 b(a)1892 438 y Fi(2)1931 423 y Fn(;)g(b)2016 438 y Fi(1)2056 423 y Fn(;)g(b)2141 438 y Fi(2)2213 423 y Fu(suc)m(h)34 b(that)909 635 y(\()p Fn(a)998 650 y Fi(1)1038 635 y Fp(j)p Fu(0)p Fp(i)21 b Fu(+)h Fn(b)1314 650 y Fi(1)1354 635 y Fp(j)p Fu(1)p Fp(i)p Fu(\))f Fp(\012)i Fu(\()p Fn(a)1718 650 y Fi(2)1757 635 y Fp(j)p Fu(0)p Fp(i)f Fu(+)g Fn(b)2034 650 y Fi(2)2074 635 y Fp(j)p Fu(1)p Fp(i)p Fu(\))k(=)i Fp(j)p Fu(00)p Fp(i)21 b Fu(+)h Fp(j)p Fu(11)p Fp(i)-91 847 y Fu(since)282 1060 y(\()p Fn(a)371 1075 y Fi(1)410 1060 y Fp(j)p Fu(0)p Fp(i)g Fu(+)g Fn(b)687 1075 y Fi(1)727 1060 y Fp(j)p Fu(1)p Fp(i)p Fu(\))f Fp(\012)h Fu(\()p Fn(a)1090 1075 y Fi(2)1130 1060 y Fp(j)p Fu(0)p Fp(i)f Fu(+)h Fn(b)1406 1075 y Fi(2)1446 1060 y Fp(j)p Fu(1)p Fp(i)p Fu(\))27 b(=)h Fn(a)1782 1075 y Fi(1)1821 1060 y Fn(a)1872 1075 y Fi(2)1912 1060 y Fp(j)p Fu(00)p Fp(i)21 b Fu(+)h Fn(a)2247 1075 y Fi(1)2287 1060 y Fn(b)2328 1075 y Fi(2)2368 1060 y Fp(j)p Fu(01)p Fp(i)f Fu(+)h Fn(b)2693 1075 y Fi(1)2733 1060 y Fn(a)2784 1075 y Fi(2)2823 1060 y Fp(j)p Fu(10)p Fp(i)f Fu(+)h Fn(b)3148 1075 y Fi(1)3188 1060 y Fn(b)3229 1075 y Fi(2)3269 1060 y Fp(j)p Fu(11)p Fp(i)-91 1272 y Fu(and)30 b Fn(a)147 1287 y Fi(1)187 1272 y Fn(b)228 1287 y Fi(2)295 1272 y Fu(=)e(0)i(implies)e(that)i (either)g Fn(a)1341 1287 y Fi(1)1381 1272 y Fn(a)1432 1287 y Fi(2)1499 1272 y Fu(=)e(0)i(or)g Fn(b)1840 1287 y Fi(1)1880 1272 y Fn(b)1921 1287 y Fi(2)1988 1272 y Fu(=)e(0.)42 b(These)33 b(states)e(represen)m(t)h(situations)d(that)-91 1396 y(ha)m(v)m(e)i(no)e(classical)g(coun)m(terpart.)43 b(These)32 b(are)d(also)g(the)h(states)h(that)f(pro)m(vide)g(the)g(exp) s(onen)m(tial)f(gro)m(wth)-91 1521 y(of)j(quan)m(tum)h(state)g(spaces)h (with)e(the)h(n)m(um)m(b)s(er)g(of)f(particles.)75 1645 y(T)-8 b(o)32 b(see)i(this)e(let)g(us)h(consider)g(the)g(follo)m(wing)d (sup)s(erp)s(osition)i(of)g Fn(n)h Fu(qubits:)1612 1813 y Fi(1)1557 1843 y Fl(X)1463 2053 y Fj(i)1487 2062 y Fg(1)1522 2053 y Fj(i)1546 2062 y Fg(2)1580 2053 y Fj(:::)o(i)1663 2061 y Ff(n)1705 2053 y Fi(=0)1812 1938 y Fp(j)p Fn(i)1873 1953 y Fi(1)1913 1938 y Fn(i)1946 1953 y Fi(2)2002 1938 y Fn(:)17 b(:)g(:)f(i)2166 1953 y Fj(n)2213 1938 y Fp(i)1430 b Fu(\(7\))-91 2243 y(Applying)30 b(the)j(linear)d(unitary)h(op)s (eration)f(whic)m(h)i(computes)g Fn(f)11 b Fu(,)32 b(from)e(Eq.)j (\(4\),)f(to)f(this)g(state,)i(yields:)1531 2416 y Fi(1)1476 2446 y Fl(X)1353 2656 y Fj(i)1377 2665 y Fg(1)1411 2656 y Fj(;i)1455 2665 y Fg(2)1490 2656 y Fj(;:::;i)1614 2664 y Ff(n)1654 2656 y Fi(=0)1761 2541 y Fp(j)p Fn(f)11 b Fu(\()p Fn(i)1919 2556 y Fi(1)1958 2541 y Fn(i)1991 2556 y Fi(2)2047 2541 y Fn(:)17 b(:)g(:)f(i)2211 2556 y Fj(n)2259 2541 y Fu(\))p Fp(i)p Fn(:)1319 b Fu(\(8\))-91 2844 y(Hence,)34 b Fn(U)43 b Fu(computes)33 b Fn(f)43 b Fo(p)-5 b(ar)g(al)5 b(lel)g(ly)41 b Fu(on)32 b(all)f(the)i(2)1772 2808 y Fj(n)1851 2844 y Fu(p)s(ossible)f(inputs)h Fn(i)p Fu(.)75 2968 y(T)-8 b(o)39 b(ac)m(hiev)m(e)h(suc)m(h)h(a)e(parallel)d (computing)i(in)g(an)h(assumed)h(realistic)d(computer,)k(w)m(e)f(start) g(with)-91 3093 y(an)c(initial)d(state)k Fp(j)p Fn(i)p Fp(i)f Fu(whic)m(h)h(corresp)s(onds)h(to)e(an)g(\\input")g(to)g(the)h (computation.)53 b(W)-8 b(e)37 b(then)g(p)s(erform)-91 3217 y(elemen)m(tary)25 b(op)s(erations)g(on)g(the)h(system)g(using)g (the)f(quan)m(tum)h(gates)g(de\014ned)h(ab)s(o)m(v)m(e.)42 b(The)26 b(op)s(erations)-91 3342 y(corresp)s(ond)48 b(to)f(the)h(computational)d(steps)k(in)e(the)h(computation,)i(just)e (lik)m(e)f(logic)e(gates)j(are)g(the)-91 3466 y(elemen)m(tary)41 b(steps)h(in)e(classical)f(computers,)44 b(and)d(are)g(p)s(erformed)f (on)h(an)g(isolated)f(system,)k(so)d(the)-91 3591 y(ev)m(olution)29 b(can)i(alw)m(a)m(ys)g(b)s(e)f(describ)s(ed)h(b)m(y)g(a)f(unitary)g (matrix)f(op)s(erating)g(on)h(the)h(state)g(of)e(the)i(system.)75 3716 y(Therefore)25 b(w)m(e)h(can)e(alw)m(a)m(ys)h(implemen)m(t)e(a)h (unitary)g(op)s(erator)f(whic)m(h)i(is)f(giv)m(en)h(b)m(y)g(the)g (Hamiltonian)-91 3840 y(of)38 b(a)g(giv)m(en)h(pro)s(cess)g(or)f(state) h(of)f(a)h(system)g(as)g(a)f(set)h(of)f(instructions)g(on)h(ho)m(w)g (to)f(transform)g(input)-91 3965 y(states)d(in)f(time.)48 b(But)35 b(the)f(crucial)g(problem)f(are)i(initial)30 b(states)36 b(themselv)m(es.)50 b(Can)35 b(w)m(e)h(write)e(do)m(wn)i(a) -91 4089 y(general)c(input)g(state:)1514 4214 y Fp(j)p Fu(\011)1618 4229 y Fi(0)1657 4214 y Fp(i)c Fu(=)f Fp(j)p Fn(i)1888 4229 y Fi(1)1928 4214 y Fn(i)1961 4229 y Fi(2)2000 4214 y Fn(:::i)2114 4229 y Fj(n)2162 4214 y Fp(i)1481 b Fu(\(9\))-91 4387 y(b)m(y)43 b(means)g(of)g(quan)m(tum)g(gate)f(op)s (erations)g(o)m(v)m(er)i(elemen)m(tary)f(input)g(prop)s(ositions)e(so)i (as)g(to)f(corre-)-91 4512 y(sp)s(ond)32 b(to)g(a)g(general)f(w)m(a)m (v)m(e)j(function)e(of)f(the)i(Hilb)s(ert)d(space)k(whic)m(h)e(describ) s(es)h(the)g(input)f(state?)43 b(The)-91 4636 y(answ)m(er)c(is)e (curren)m(tly)i(in)e(the)h(negativ)m(e.)59 b(There)39 b(is)e(no)h(kno)m(wn)h(\014nite)f(and)g(de\014nite)g(receipt)g(for)f (suc)m(h)-91 4761 y(a)i(corresp)s(ondence.)66 b(But)40 b(in)f(the)h(recen)m(t)h(y)m(ears)g(a)e(lot)f(has)i(b)s(een)g(ac)m (hiev)m(ed)h(to)e(narro)m(w)h(the)g(gap)g(b)s(e-)-91 4885 y(t)m(w)m(een)34 b(an)f(algebra)e(of)h(elemen)m(tary)h(prop)s (ositions)e(\(corresp)s(onding)i(to)f(pure)h(states\))h(and)f(the)g (Hilb)s(ert)-91 5010 y(space)39 b(description.)60 b(Let)38 b(us)h(consider)f(some)g(most)g(imp)s(ortan)m(t)e(details)h(in)h(a)g(p) s(ossible)f(construction)-91 5134 y(of)c(a)h(quan)m(tum)g(mac)m(hine)f (language)g(whic)m(h)i(could)e(mimic)e(quan)m(tum)j(system)h(and)f (therefore)h(directly)-91 5259 y(corresp)s(ond)e(to)f(its)g(Hilb)s(ert) f(space)j(represen)m(tation.)75 5383 y(In)28 b(Sec.)h(3)f(w)m(e)h(sho)m (w)h(\(using)d(only)h(lattice)e(theory\))j(that)f(b)s(oth)g(a)g(prop)s (er)g(quan)m(tum)g(logic)f(of)g(prop)s(o-)-91 5508 y(sitions)f(and)h(a) f(prop)s(er)h(classical)f(logic)f(of)h(prop)s(ositions)g(ha)m(v)m(e)i (mo)s(dels)e(that)h(are)g(not)f(ev)m(en)j(orthomo)s(du-)-91 5632 y(lar.)40 b(Therefore)28 b(in)e(b)s(oth)g(classical)g(and)g(quan)m (tum)i(computers)f(one)g(m)m(ust)g(use)g(algebras)g(instead.)41 b(Suc)m(h)1833 5881 y(3)p eop %%Page: 4 4 4 3 bop -91 299 a Fu(an)33 b(algebra|orthomo)s(dular)c(lattice|whic)m (h)j(is)h(usually)g(considered)h(to)f(b)s(e)g(an)h(algebra)e (underlying)-91 423 y(quan)m(tum)40 b(measuremen)m(t)h(and)f(a)g(Hilb)s (ert)f(space)i(represen)m(tation)g(w)m(e)g(analyze)f(in)g(Sec.)h(2)f (and)g(sho)m(w)-91 548 y(that)j(all)e(its)i(binary)g(op)s(erations)f (are)i(am)m(biguous)e(and)h(that)g(bare)h(orthomo)s(dular)d(lattices)h (cannot)-91 672 y(b)s(e)c(emplo)m(y)m(ed)g(in)f(quan)m(tum)i (computers.)60 b(In)38 b(Sec.)h(4)f(w)m(e)h(sho)m(w)g(ho)m(w)f(one)g (can)h(construct)g(Hilb)s(ertian)-91 797 y(lattices)d(whic)m(h)i (enable)f(a)g(direct)g(represen)m(tation)h(in)f(a)g(Hilb)s(ert)e(space) k(of)d(quan)m(tum)i(computational)-91 922 y(sim)m(ulation)j(and)k(pro)m (vide)f(a)g(lattice)f(for)h(the)g(purp)s(ose,)k(whic)m(h)d(at)f(the)h (same)f(time)f(eliminates)f(the)-91 1046 y(so-called)c(4-dim)g(p)s (ostulate.)66 b(W)-8 b(e)41 b(also)e(sho)m(w)i(that)f(quan)m(tum)h (theory)f(is)g(at)g(least)g(so)g(incompatible)-91 1171 y(with)33 b(the)g(strong)h(form)e(of)g(the)i(Ch)m(urc)m(h-T)-8 b(uring)34 b(principle)e(as)h(an)m(y)h(classical)e(theory)i(con)m (trary)g(to)f(the)-91 1295 y(Deutsc)m(h's)h(claim.)41 b(\(Deutsc)m(h,)34 b(1985\))-91 1594 y FA(2)161 b Fq(All)45 b(Op)t(erations)h(in)e(Orthomo)t(dular)i(Lattices)g(Are)f(Am)l(biguous) -91 1818 y Fu(The)33 b(Birkho\013-v)m(on)f(Neumann)h(requiremen)m(t)g (\(Kalm)m(bac)m(h,)f(1983\):)780 2040 y Fn(a)c Fp(!)959 2055 y Fj(i)1015 2040 y Fn(b)223 b Fp(\))f Fn(a)28 b Fp(\024)g Fn(b;)408 b(i)28 b Fu(=)f(1)p Fn(;)17 b(:)g(:)g(:)f(;)h Fu(5)p Fn(;)864 b Fu(\(10\))-91 2283 y(where)29 b Fn(a)f Fp(!)365 2298 y Fi(1)432 2283 y Fn(b)502 2227 y Fi(def)513 2283 y Fu(=)39 b Fn(a)679 2247 y Fv(0)716 2283 y Fp([)14 b Fu(\()p Fn(a)g Fp(\\)g Fn(b)p Fu(\),)30 b Fn(a)d Fp(!)1293 2298 y Fi(2)1360 2283 y Fn(b)1430 2227 y Fi(def)1441 2283 y Fu(=)39 b Fn(b)1597 2247 y Fv(0)1648 2283 y Fp(!)1748 2298 y Fi(1)1815 2283 y Fn(a)1866 2247 y Fv(0)1889 2283 y Fu(,)30 b Fn(a)e Fp(!)2125 2298 y Fi(3)2191 2283 y Fn(b)2261 2227 y Fi(def)2272 2283 y Fu(=)39 b(\()p Fn(a)2476 2247 y Fv(0)2513 2283 y Fp(\\)14 b Fn(b)p Fu(\))g Fp([)g Fu(\()p Fn(a)2855 2247 y Fv(0)2893 2283 y Fp(\\)g Fn(b)3014 2247 y Fv(0)3037 2283 y Fu(\))g Fp([)g Fu(\()p 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b(&)h(Megill,)d(1998b\))352 3024 y Fn(a)22 b Fp([)h Fn(b)28 b Fu(=)g(\()p Fn(a)f Fp(!)903 3039 y Fj(i)959 3024 y Fn(b)p Fu(\))h Fp(!)1166 3039 y Fj(i)1221 3024 y Fu(\(\(\()p Fn(a)g Fp(!)1514 3039 y Fj(i)1570 3024 y Fn(b)p Fu(\))g Fp(!)1777 3039 y Fj(i)1832 3024 y Fu(\()p Fn(b)g Fp(!)2039 3039 y Fj(i)2095 3024 y Fn(a)p Fu(\)\))g Fp(!)2350 3039 y Fj(i)2405 3024 y Fn(a)p Fu(\))195 b Fn(i)28 b Fu(=)g(1)p Fn(;)17 b(:)g(:)g(:)e(;)i Fu(5)p Fn(;)436 b Fu(\(11\))-91 3246 y(this)26 b(means)g(that)g(all)f(op)s(erations)g(\(\014v)m(e-fold) h(negation)g(follo)m(ws)f(trivially\))e(in)j(an)g(orthomo)s(dular)e (lattice)-91 3371 y(are)32 b(\014v)m(efold)h(de\014nable.)75 3495 y(A)m(t)22 b(the)h(\014rst)g(sigh)m(t)f(it)f(still)f(seems)j(that) f(one)g(can)h(pro)m(v)m(e)g(a)f(conjecture)i(that)e(the)g(relation)f (of)h(equation)-91 3620 y(in)29 b(the)h(lattice)e(is)h(uniquely)h (de\014nable.)42 b(The)31 b(reasons)f(for)g(suc)m(h)h(a)e(conjecture)i (are)e(the)h(follo)m(wing)d(ones.)-91 3759 y(All)40 b(\014v)m(e)k(quan) m(tum)e(implications)d Fn(a)44 b Fp(!)1441 3774 y Fj(i)1513 3759 y Fn(b)f Fu(collapse)e(to)h(the)h(classical)e(one)h Fn(a)i Fp(!)3045 3774 y Fi(0)3129 3759 y Fn(b)3212 3703 y Fi(def)3223 3759 y Fu(=)53 b Fn(a)3403 3723 y Fv(0)3455 3759 y Fp([)29 b Fn(b)43 b Fu(in)f(a)-91 3884 y(distributiv)m(e)31 b(lattice.)42 b(Ev)m(en)35 b(more)689 4106 y Fn(a)28 b Fp(!)868 4121 y Fj(i)923 4106 y Fn(b)g Fu(=)g Fn(a)g Fp(!)1275 4121 y Fj(j)1338 4106 y Fn(b;)408 b(i)28 b Fp(6)p Fu(=)f Fn(j)6 b Fu(;)212 b Fn(i;)17 b(j)34 b Fu(=)27 b(0)p Fn(;)17 b(:)g(:)g(:)f(;)h Fu(5)p Fn(;)772 b Fu(\(12\))-91 4328 y(mak)m(es)33 b(an)f(ortho-lattice)f(distributiv)m(e.\(P)m(a)m(vi) m(\024)-46 b(ci)m(\023)g(c,)32 b(1987\))f(On)i(the)g(other)f(hand)1232 4551 y Fn(a)c Fp(!)1411 4566 y Fi(0)1478 4551 y Fn(b)223 b Fp(\))f Fn(a)28 b Fp(\024)g Fn(b;)1317 b Fu(\(13\))-91 4773 y(also)26 b(mak)m(es)i(an)f(ortholattice)f(distributiv)m(e.)41 b(\(P)m(a)m(vi)m(\024)-46 b(ci)m(\023)g(c,)28 b(1998\))e(In)i(an)m(y)g 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1220 y Fn(a)28 b Fp(\021)1310 1235 y Fi(0)1377 1220 y Fn(b)h Fu(=)e(1)223 b Fp(,)f Fn(a)28 b Fu(=)f Fn(b;)1239 b Fu(\(16\))-91 1445 y(whic)m(h)33 b(mak)m(es)g(an)f(ortholattice)f(distributiv)m(e)h(\(P)m(a)m(vi)m(\024) -46 b(ci)m(\023)g(c,)33 b(1998\).)75 1569 y(Unfortunately)-8 b(,)44 b(the)f(conjecture)h(do)s(es)f(not)f(hold.)72 b(The)44 b(reason)f(is)f(simple.)71 b(In)43 b(a)f(distributiv)m(e)-91 1694 y(lattice)k Fn(a)55 b Fp(!)428 1709 y Fj(i)509 1694 y Fn(b;)66 b(i)54 b Fu(=)g(1)p Fn(;)17 b(:)g(:)g(:)e(;)i Fu(5)48 b(all)e(merge)i(to)f Fn(a)55 b Fp(!)2018 1709 y Fi(0)2111 1694 y Fn(b)48 b Fu(and)g(therefore)h(\()p Fn(a)54 b Fp(!)3073 1709 y Fj(i)3155 1694 y Fn(b)p Fu(\))33 b Fp(\\)g Fu(\()p Fn(b)55 b Fp(!)3600 1709 y Fj(j)3690 1694 y Fn(a)p Fu(\),)-91 1842 y Fn(i)46 b Fp(6)p Fu(=)g Fn(j)6 b Fu(,)46 b Fn(i;)17 b(j)51 b Fu(=)46 b(1)p Fn(;)17 b(:)g(:)g(:)f(;)h Fu(5)43 b(m)m(ust)g(merge)g(to)g Fn(a)j Fp(\021)1733 1857 y Fi(0)1819 1842 y Fn(b)1904 1786 y Fi(def)1914 1842 y Fu(=)54 b(\()p 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y(the)c(follo)m(wing)d (theorem.)43 b(\(P)m(a)m(vi)m(\024)-46 b(ci)m(\023)g(c)32 b(&)h(Megill,)d(1999\))-91 4824 y Fd(Theorem)37 b(2.1.)49 b Fo(A)n(n)34 b(ortholattic)-5 b(e)35 b(in)g(which)708 5048 y Fn(a)27 b Fp(\021)863 5063 y Fj(i)920 5048 y Fn(b)h Fu(=)f(1)227 b Fp(\))g Fn(a)28 b Fu(=)f Fn(b;)416 b(i)28 b Fu(=)f(2)p Fn(;)17 b(:)g(:)g(:)f(;)h Fu(5)791 b(\(17\))-91 5273 y Fo(holds)34 b(is)g(an)h(orthomo)-5 b(dular)34 b(lattic)-5 b(e)35 b(and)f(vic)-5 b(e)35 b(versa.)75 5480 y Fu(Hence,)k(putting)d(together)i(Eq.)g(\(15\))e(and)h(Eq.)h (\(17\))e(w)m(e)i(ha)m(v)m(e)g(an)f(indication)e(that)i(the)g(relation) -91 5605 y(of)d(equiv)-5 b(alence)34 b(whic)m(h)h(establishes)g(a)f (connection)h(b)s(et)m(w)m(een)i(quan)m(tum)d(logic)f(and)h(its)g(mo)s (dels)g(migh)m(t)1833 5881 y(5)p eop %%Page: 6 6 6 5 bop -91 299 a Fu(turn)27 b(out)h(to)f(b)s(e)g(based)h(on)g(sev)m (eral)g(di\013eren)m(t)f(op)s(erations)g(of)g(iden)m(tit)m(y)g(at)g (the)h(same)f(time)f(th)m(us)j(making)-91 423 y(a)39 b(direct)h(ev)-5 b(aluation)37 b(of)j(elemen)m(tary)f(logical)e(prop)s (ositions)h(imp)s(ossible.)62 b(Ho)m(w)m(ev)m(er,)44 b(as)c(w)m(e)h(will)c(see)-91 548 y(in)45 b(the)i(next)h(section,)i (there)d(is)f(an)h(ev)m(en)h(more)e(imp)s(ortan)m(t)f(reason)i(wh)m(y)g (w)m(e)h(cannot)f(use)g(prop)s(er)-91 672 y(quan)m(tum)41 b(logic)d(to)j(ev)-5 b(aluate)40 b(quan)m(tum)h(prop)s(ositions)e(and)h (this)h(is)f(that)g(a)h(prop)s(er)f(quan)m(tum)h(logic)-91 797 y(is)34 b(not)h(orthomo)s(dular.)49 b(As)36 b(ev)m(en)h(bigger)d (surprise)i(comes)g(the)f(result)g(that)g(ev)m(en)i(standard)f (classical)-91 922 y(logic)30 b(need)k(not)e(b)s(e)h(orthomo)s(dular.) -91 1221 y FA(3)161 b Fq(Non-Orthomo)t(dular)62 b(Mo)t(dels)h(for)f (Both)g(Quan)l(tum)g(and)g(Classical)151 1371 y(Logics)-91 1594 y Fu(A)35 b(crucial)e(di\013erence)j(b)s(et)m(w)m(een)h(logics)c (and)j(lattices)d(as)j(their)e(mo)s(dels)g(is)g(that)h(prop)s(erties)g (that)f(pla)m(y)-91 1718 y(a)42 b(decisiv)m(e)i(role)e(in)h(lattices)f 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b(mac)m(hine)e(in)h(classical)f(ph)m(ysics.)83 b(Owing)45 b(to)g(con)m(tin)m(uit)m(y)h(of)-91 5444 y(classical)38 b(dynamics,)j(the)f(p)s(ossible)f(states)h(of)g(a)f(classical)f(system) j(necessarily)f(form)e(a)h(con)m(tin)m(uum.)p -91 5532 1560 4 v 21 5593 a Fk(2)58 5623 y Fa(ftp://ftp.shore.)o(net)o(/m)o(emb) o(er)o(s/n)o(dm)o(/q)o(uan)o(tu)o(m-l)o(og)o(ic)1833 5881 y Fu(9)p eop %%Page: 10 10 10 9 bop -91 299 a Fu(Y)-8 b(et)36 b(there)g(are)g(only)f(coun)m(tably) h(man)m(y)f(w)m(a)m(ys)j(of)d(preparing)g(a)h(\014nite)f(input)g(for)g ([a)h(quan)m(tum)g(T)-8 b(uring)-91 423 y(mac)m(hine].)43 b(Consequen)m(tly)35 b([it])c(cannot)i(p)s(erfectly)g(sim)m(ulate)e(an) m(y)i(classical)e(system.")75 548 y(This)44 b(distinction)e(do)s(es)j (not)f(hold,)i(b)s(ecause,)j(as)44 b(w)m(e)h(ha)m(v)m(e)h(seen,)i(a)c (con)m(tin)m(uum)g(app)s(ears)h(with)-91 672 y(quan)m(tum)i(spin)f (systems)i(as)f(w)m(ell)f(and)h(on)f(the)i(other)e(hand)h(preparing)f (a)h(\014nite)f(input)g(is)h(not)f(in)-91 797 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(quaternionic\))f(\014eld)g(for)h(our)f(Hilb)s(ert)g(space|it)g Fo(fol)5 b(lows)57 b Fu(from)48 b(the)h(in\014nit)m(y)f(of)h(the)-91 1669 y(lattice.)20 b(\(Holland,)31 b(JR.,)i(1995\))-91 1964 y FA(5)161 b Fq(Conclusion)-91 2187 y Fu(States)28 b(of)g(a)g(general)g(purp)s(ose)g(quan)m(tum)h(computer)f(m)m (ust|apart)g(from)e(conditions)i(imp)s(osed)f(b)m(y)i(the)-91 2311 y(standard)j(quan)m(tum)h(logic)d(gates|satisfy)i(additional)d (conditions)j(giv)m(en)g(in)f(Sec.)j(4)e(and)g(required)h(to)-91 2436 y(yield)38 b(a)g(general)h(algebra)e(of)i(the)g(states.)64 b(One)39 b(of)f(the)i(conditions)e(is)g(a)h(4-v)-5 b(ariable)36 b(orthoarguesian)-91 2560 y(la)m(w)d(giv)m(en)g(b)m(y)i(Eq.)f(\(25\))f (whic)m(h)h(giv)m(es)g(all)e(kno)m(wn)j(orthoarguesian)d(equations)i (\(including)e(6-v)-5 b(ariable)-91 2685 y(ones\))46 b(and)g(whic)m(h)g(eliminates)e(the)i(so-called)f(4-dim)e(lattice)i(p)s (ostulate.)82 b(The)47 b(obtained)e(algebra,)-91 2809 y(whic)m(h)31 b(is)e(a)i(Hilb)s(ertian)c(lattice)i(is)h(then)h (isomorphic)e(to)h(the)g(subspaces)j(of)d(the)h(Hilb)s(ert)e(spaces)j (whic)m(h)-91 2934 y(c)m(haracterizes)24 b(general)e(computation)g (algorithms.)37 b(Prop)s(ositions)22 b Fn(a;)40 b(b;)g(c)26 b Fu(of)d(the)g(lattice)f(are)h(therefore)-91 3058 y(connected)33 b(to)f(probabilistic)c(outcomes)k(of)g(a)f(calculation)e(of)j(an)f (observ)-5 b(able)32 b Fn(A)g Fu(b)m(y)h(means)f(of)f Fn(\026)p Fu(\()p Fn(a)p Fu(\))c(=)-91 3183 y Fp(h)p Fu(\011)p Fp(j)p Fn(P)115 3198 y Fj(A)171 3183 y Fp(j)p Fu(\011)p Fp(i)g Fu(where)h Fn(P)680 3198 y Fj(A;E)840 3183 y Fu(\()p Fn(E)33 b Fu(is)27 b(a)g(Borel)g(set\))h(is)f(a)g(pro)5 b(jector)28 b(of)f Fn(A)p Fu(,)i Fn(\026)e Fu(is)g(the)h(pure)g(full)e (\()p Fn(\026)p Fu(\()p Fn(a)p Fu(\))h Fp(\024)h Fn(\026)p Fu(\()p Fn(b)p Fu(\))40 b Fp(\))-91 3307 y Fn(a)c Fp(\024)h Fn(b)p Fu(\))h(strongly)f(con)m(v)m(ex)j(\()p Fn(\026)1031 3322 y Fj(j)1103 3307 y Fp(2)d(S)c Fu(&)1417 3233 y Fl(P)1539 3307 y Fn(c)1581 3322 y Fj(j)1654 3307 y Fu(=)j(1)62 b Fp(\))2038 3233 y Fl(P)2160 3307 y Fn(c)2202 3322 y Fj(j)2239 3307 y Fn(\026)2298 3322 y Fj(j)2370 3307 y Fp(2)37 b(S)7 b Fu(\))38 b(probabilit)m(y)e(measure)i(on)f(HL:)-91 3432 y Fn(\026)47 b Fu(:)i(HL)f Fp(7!)f Fu([0)p Fn(;)17 b Fu(1])44 b(and)h(\011)f(w)m(e)i(obtain)d(from)g(the)i(Gleason)f (theorem:)67 b(if)44 b Fn(\026)g Fu(is)g(a)g(pure)h(probabilit)m(y)-91 3557 y(measure,)34 b(then)f(there)i(exists)f(a)f(v)m(ector)h(\011)29 b Fp(2)g(H)q Fu(,)34 b(whic)m(h)f(satis\014es)i(the)e(ab)s(o)m(v)m(e)h Fn(\026)p Fu(\()p Fn(a)p Fu(\).)46 b(The)34 b(mean)f(v)-5 b(alue)-91 3681 y(of)32 b(the)h(op)s(erator)f Fn(A)g Fu(is)g(then)i(giv)m(en)e(b)m(y)i(the)f(sp)s(ectral)f(theorem.)75 3806 y(In)44 b(Sec.)g(3)f(w)m(e)i(ha)m(v)m(e)g(sho)m(wn)g(that)e(no)h (calculation)d(can)j(b)s(e)f(carried)g(out)h(within)e(prop)s(ositional) -91 3930 y(quan)m(tum)28 b(logic)d(since)j(the)g(latter)f(can)h(b)s(e)f (mo)s(deled)g(with)g(a)g(non-orthomo)s(dular)e(mo)s(del.)40 b(\(In)28 b(addition)-91 4055 y(w)m(e)42 b(sho)m(w)f(that)g(the)g (standard)g(classical)f(logic)e(has)k(a)e(non-orthomo)s(dular)e(mo)s (del)h(to)s(o)h(and)h(explain)-91 4179 y(wh)m(y)34 b(this)e(is)g(of)g (no)h(consequence)i(for)d(classical)f(computers.\))75 4304 y(In)k(Sec.)h(2)f(w)m(e)h(ha)m(v)m(e)h(sho)m(wn)f(that)f(an)g (orthomo)s(dular)e(lattice)g(cannot)j(b)s(e)f(used)h(as)g(a)e (satisfactory)-91 4428 y(algebra)43 b(of)i(states)g(b)s(ecause)h(all)d (op)s(erations)h(in)g(the)h(lattice)e(are)i(\014v)m(e-fold)g (de\014ned,)k(including)43 b(the)-91 4553 y(iden)m(tit)m(y)32 b(and)h(the)g(relation)d(of)j(equiv)-5 b(alence.)-91 4962 y FA(References)-91 5185 y Fu(Boghosian,)45 b(B.)e(M.,)k(&)c(T)-8 b(a)m(ylor,)46 b(W.)75 b(\(1998\).)f(Sim)m(ulating)41 b(quan)m(tum)i(mec)m(hanics)h(on)f(a)g(quan)m(tum)153 5310 y(computer.)g Fo(Physic)-5 b(a)34 b(D)p Fu(,)f Fd(120)p Fu(,)f(30-42.)-91 5508 y(Christianson,)c(B.,)h(Knigh)m(t,)g(P)-8 b(.)28 b(L.,)h(&)f(Beth,)i(T.)36 b(\(1998\).)f(Implemen)m(tations)27 b(of)g(quan)m(tum)i(logic.)34 b Fo(Phil.)153 5632 y(T)-7 b(r)i(ans.)33 b(R)-5 b(oy.)35 b(So)-5 b(c.)34 b(L)-5 b(ondon)34 b(A)p Fu(,)f Fd(356\(1743\))p Fu(,)f(1823-1838.)1809 5881 y(10)p eop %%Page: 11 11 11 10 bop -91 299 a Fu(Deutsc)m(h,)34 b(D.)43 b(\(1985\).)f(Quan)m(tum) 32 b(theory)-8 b(,)33 b(the)g(Ch)m(urc)m(h-Turing)h(principle)d(and)i (the)g(univ)m(ersal)f(quan-)153 423 y(tum)f(computer.)44 b Fo(Pr)-5 b(o)g(c.)35 b(R.)f(So)-5 b(c.)35 b(L)-5 b(ond.)34 b(A)p Fu(,)f Fd(400)p Fu(,)g(97{117.)-91 616 y(F)-8 b(eynman,)48 b(R.)e(P)-8 b(.)82 b(\(1982\).)f(Sim)m(ulating)42 b(ph)m(ysics)47 b(with)f(computers.)82 b Fo(Int.)47 b(J.)f(The)-5 b(or.)46 b(Phys.)p Fu(,)j Fd(21)p Fu(,)153 741 y(467-488.)-91 933 y(F)-8 b(eynman,)32 b(R.)h(P)-8 b(.)43 b(\(1986\).)f(Quan)m(tum)33 b(mec)m(hanical)e(computers.)44 b Fo(F)-7 b(ound.)33 b(Phys.)p Fu(,)g Fd(16)p Fu(,)g(507-531.)-91 1126 y(Go)s(do)m(wski,)c (R.,)g(&)f(Greec)m(hie,)h(R.)36 b(\(1984\).)g(Some)28 b(equations)g(related)g(to)g(the)g(states)i(on)e(orthomo)s(dular)153 1250 y(lattices,.)42 b Fo(Demonstr)-5 b(atio)34 b(Math.)p Fu(,)f Fo(17)p Fu(,)f(241{250.)-91 1443 y(Gro)m(v)m(er,)e(L.)f(K.)37 b(\(1997\).)f(Quan)m(tum)29 b(computers)g(can)g(searc)m(h)h (arbitrarily)c(large)i(databases)i(b)m(y)g(a)e(single)153 1567 y(query)-8 b(.)44 b Fo(Phys.)35 b(R)-5 b(ev.)35 b(L)-5 b(ett.)p Fu(,)33 b Fd(79)p Fu(,)f(4709{4712.)-91 1760 y(Holland,)h(JR.,)h(S.)h(S.)48 b(\(1995\).)g(Orthomo)s(dularit)m (y)31 b(in)j(in\014nite)f(dimensions;)h(a)g(theorem)g(of)g(M.)h(Sol)m (\022)-46 b(er.)153 1884 y Fo(Bul)5 b(l.)34 b(A)n(m.)g(Math.)h(So)-5 b(c.)p Fu(,)32 b Fd(32)p Fu(,)h(205{234.)-91 2077 y(Iv)m(ert,)40 b(P)-8 b(.-A.,)38 b(&)f(Sj\177)-49 b(odin,)38 b(T.)59 b(\(1978\).)d(On)38 b(the)g(imp)s(ossibilit)m(y)33 b(of)k(a)g(\014nite) h(prop)s(ositional)c(lattice)i(for)153 2201 y(quan)m(tum)c(mec)m (hanics.)44 b Fo(Helv.)35 b(Phys.)g(A)-5 b(cta)p Fu(,)33 b Fo(51)p Fu(,)f(635{636.)-91 2394 y(Kalm)m(bac)m(h,)f(G.)43 b(\(1983\).)g Fo(Orthomo)-5 b(dular)34 b(lattic)-5 b(es.)43 b Fu(London:)g(Academic)32 b(Press.)-91 2587 y(Maczy)s(\023)-51 b(nski,)27 b(M.)f(J.)32 b(\(1972\).)f(Hilb)s(ert)24 b(space)j (formalism)22 b(of)j(quan)m(tum)h(mec)m(hanics)g(without)f(the)h(Hilb)s (ert)153 2711 y(space)33 b(axiom.)42 b Fo(R)-5 b(ep.)34 b(Math.)i(Phys.)p Fu(,)c Fd(3)p Fu(,)h(209{219.)-91 2904 y(P)m(a)m(vi)m(\024)-46 b(ci)m(\023)g(c,)31 b(M.)41 b(\(1987\).)f (Minimal)28 b(quan)m(tum)j(logic)e(with)h(merged)h(implications.)37 b Fo(Int.)c(J.)h(The)-5 b(or.)32 b(Phys.)p Fu(,)153 3028 y Fd(26)p Fu(,)g(845{852.)-91 3221 y(P)m(a)m(vi)m(\024)-46 b(ci)m(\023)g(c,)45 b(M.)74 b(\(1993\).)e(Nonordered)43 b(quan)m(tum)g(logic)e(and)h(its)h(YES{NO)g(represen)m(tation.)74 b Fo(Int.)43 b(J.)153 3345 y(The)-5 b(or.)34 b(Phys.)p Fu(,)e Fd(32)p Fu(,)h(1481{1505.)-91 3538 y(P)m(a)m(vi)m(\024)-46 b(ci)m(\023)g(c,)50 b(M.)84 b(\(1995\).)f(Spin-correlated)45 b(in)m(terferometry)h(with)g(b)s(eam)g(splitters:)70 b(Preselection)47 b(of)153 3662 y(spin-correlated)31 b(photons.)44 b Fo(J.)35 b(Opt.)g(So)-5 b(c.)34 b(A)n(m.)g(B)p Fu(,)f Fd(12)p Fu(,)f(821{828.)-91 3855 y(P)m(a)m(vi)m(\024)-46 b(ci)m(\023)g(c,)37 b(M.)53 b(\(1998\).)g(Iden)m(tit)m(y)37 b(rule)e(for)g(classical)g(and)h(quan)m(tum)g(theories.)54 b Fo(Int.)37 b(J.)h(The)-5 b(or.)37 b(Phys.)p Fu(,)153 3980 y Fd(37)p Fu(,)32 b(2099{2103.)-91 4172 y(P)m(a)m(vi)m(\024)-46 b(ci)m(\023)g(c,)52 b(M.,)h(&)48 b(Megill,)i(N.)f(D.)90 b(\(1998a\).)f(Binary)48 b(orthologic)e(with)i(mo)s(dus)g(p)s(onens)i (is)d(either)153 4297 y(orthomo)s(dular)30 b(or)i(distributiv)m(e.)43 b Fo(Helv.)34 b(Phys.)h(A)-5 b(cta)p Fu(,)33 b Fd(71)p Fu(,)g(610{628.)-91 4489 y(P)m(a)m(vi)m(\024)-46 b(ci)m(\023)g(c,)46 b(M.,)g(&)d(Megill,)g(N.)h(D.)74 b(\(1998b\).)g(Quan)m(tum)42 b(and)i(classical)d(implication)e(algebras)k(with)153 4614 y(primitiv)m(e)30 b(implications.)40 b Fo(Int.)34 b(J.)h(The)-5 b(or.)34 b(Phys.)p Fu(,)f Fd(37)p Fu(,)f(2091{2098.)-91 4806 y(P)m(a)m(vi)m(\024)-46 b(ci)m(\023)g(c,)32 b(M.,)h(&)e(Megill,)f (N.)i(D.)42 b(\(1999\).)g(Non-orthomo)s(dular)29 b(mo)s(dels)h(for)i(b) s(oth)g(standard)g(quan)m(tum)153 4931 y(logic)d(and)i(standard)g (classical)f(logic:)41 b(Rep)s(ercussions)32 b(for)f(quan)m(tum)g (computers.)41 b Fo([Submitte)-5 b(d])p Fu(.)-91 5123 y(P)m(a)m(vi)m(\024)-46 b(ci)m(\023)g(c,)38 b(M.,)h(&)e(Summhammer,)f (J.)57 b(\(1994\).)f(In)m(terferometry)38 b(with)f(t)m(w)m(o)g(pairs)g (of)g(spin)g(correlated)153 5248 y(photons.)43 b Fo(Phys.)35 b(R)-5 b(ev.)35 b(L)-5 b(ett.)p Fu(,)33 b Fd(73)p Fu(,)g(3191{3194.)-91 5440 y(Shor,)k(P)-8 b(.)36 b(W.)54 b(\(1997\).)e(P)m(olynomial-time)31 b(algorithms)j(for)h(prime)g(factorization)f(and)i(discrete)g(loga-)153 5565 y(rithms)31 b(on)h(a)h(quan)m(tum)g(computer.)43 b Fo(SIAM)35 b(J.)g(Comp.)p Fu(,)c Fd(26)p Fu(,)i(1484{1509.)1809 5881 y(11)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF