C191 - Lecture - Quatum states ad observables I ENTANGLED STATES We saw last time that quatum mechaics allows for systems to be i superpositios of basis states May of these superpositios possess a uiquely quatum feature kow as etaglemet As a example, cosider the two-qubit state ψ 1 00 + 1 11 Because this is a two qubit state, I ca separate the two qubits, givig oe to Alice ad the other to Bob If Alice measures her qubit, she will have a 50% chace of measurig 0 ad 50% chace of measurig 1 But if Alice measures a 1, the ecessarily Bob will measure a 1, eve if the qubits are removed from causal cotact from oe aother (may light secods away, for istace Cotrast this with the state phi 1 00 + 1 01 Now Alice will measure 0 with 100% probability, but Bob s measuremet will still be 0 or 1 with 50% probability each This state is called separable because it may be writte as φ 1 00 + 1 0 1 1 0 0 + 1 1 0 0 0 ( 0 + 1 1 Etagled states are those which ca ever be writte this way We will see later that etaglemet is a extremely importat resource for quatum computatio II DENSITY MATRICES Quatum states are iheretly probabilistic, but how do we hadle situatios where there is additioal classical ucertaity? This could arise, for istace, if we were give a quatum state that, with probability p 1 was prepared i some state ψ, ad with probability p was prepared i a state φ The expectatio value of a operator, A over some state, ψ, is defied as, A ψ ψ A ψ But recall what a expectatio value is: it the sum of all possible measuremet outcomes weighted by their probabilities So if we have some classical ucertaity, we ca just build that ito the defiitio of the expectatio value, A p 1 A ψ +p A φ p 1 ψ A ψ +p φ A φ At this poit we itroduce a mathematical object kow as the resolutio of the idetity We ca take a complete set of states, { } ad take their outer products, I This is equal to the idetity We ca see this i a simple example of a two level system: multiply ay two level state o the left by 0 0 + 1 1 ad you will see that you get the same state back I matrix form, this ca be writte as: 1 0 1 0 1 0 + 0 1, 0 1 0 1 which is the idetity matrix Isertig the resolutio of the idetity ito the above expressio, we have A p 1 ψ A ψ +p ψ A φ p 1 A ψ ψ +p A φ ψ A(p 1 ψ ψ +p φ φ
We call the quatity i the paretheses the desity matrix ρ p 1 ψ ψ +p φ φ ad we ca iterpret B as the trace of the matrix B This ca be see agai by cosiderig a two level system: B ( 1 0 b 00 b 01 1 + ( 0 1 b 00 b 01 0 0 1 So we have b 00 +b 11 TrB A TrρA We ll see desity matrices frequetly whe we begi to talk about errors i quatum states These errors occur whe the cotrols we use to create states are imperfect ad itroduce ucertaity ito the system III HERMITIAN OPERATORS Oe of the axioms of quatum mechaics is that every observable correspods to a Hermitia operator Let s review some properties of these objects 1 By defiitio, a Hermitia operator is equal to its cojugate traspose A A T A Hermitia operators have real eigevalues For some eigestate, A u a u Multiply o the left by u ad we have Now take the cojugate-traspose of both sides u A u a u u a u A u a u A u a u A u a Which implies that a a, so the eigevalue must be real 3 Eigevectors are orthogoal if they possess differet eigevalues Now additioally assume we have aother eigevector with a differet eigevalue, A v b v The with A actig to the right, we have but with A actig to the left we have v A u a v u v A u b v u this meas that a v u b v u If a b, the v u 0, so the vectors are orthogoal 4 Hermitia operators may be expressed i terms of their spectral represetatio: A a Where A a This is called a spectral represetatio because the collectio of eigevalues is ofte referred to as the spectrum of a operator 5 Commutig observables possess a simultaeous eigebasis This proof has two parts: i Two operators that share a eigebasis must commute, ad ii two operators that commute must share a eigebasis I ll do the first oe here, the secod oe is worth workig out yourselves Assume we have two operators which share a eigebasis, A a ad B b The AB A(b b A a b BA B(a a B a b So if AB BA o every state, the AB BA, ad we say that the operators commute
3 A Commutators Thecommutatoroftwohermitia operators,a,b isdefied as[a,b] AB BA Herearesomeoftheirproperties, 1 The are liear i both elemets [c 1 A 1 +c A,B] c 1 [A 1,B]+c [A,B] [A,c 1 B 1 +c B ] c 1 [A,B 1 ]+c [A,B ] The are atisymmetric 3 They satisfy the Jacboi idetity 4 They satisfy the Leibitz idetity [A,B] [B,A] [A,[B,C]]+[B,[C,A]]+[C,[A,B]] 0 [AB,C] A[B,C]+[A,C]B [A,BC] B[A,C]+[A,B]C IV TENSOR PRODUCTS OF OPERATORS Whe we combie quatum systems, the state vectors live i a Hilbert space that is the tesor product of the two costituet Hilbert spaces If each qubit is i a particular state, the the combied state is the tesor product of the idividual states Ad if each qubit is operated o by a particular operator, the the combied operator is the tesor product of the idividual operators 1 Tesor products are liear (a 1 A 1 +a A B a 1 A 1 B +a A B Each term i a tesor product acts o its ow compoet (A B( m (A B( m A m B 3 Multiplicatio of operators (A B(C D AC BD 4 The matrix represetatio of tesor products is, for states: a0 b0 a 1 b 1 For operators, a00 01 a 10 a 11 b0 a 0 b 1 a 1 ( b0 b 1 a 0 b 0 a 0 b 1 a 1 b 0 a 1 b 1 00 10 01 11 a 00 b 00 a 00 b 01 b 00 b 01 a 00 b 10 a 00 b 11 b 10 b 11 a 10 b 00 a 10 b 01 b 00 a 11 b 01 a 10 b 10 a 10 b 11 b 10 a 11 b 11
4 A Complete set of commutig observables For ay Hilbert space, we ca specify a (ot uique! set of commutig observables, {A,B,} If each eigestate is associated with a uique set of eigevalues over this set, the set is called a complete set of commutig observables or CSCO Takig a qubit as a example, we ca measure the Pauli Z operator, Z σ z σ 3 ( 1 0 0 1 This is a somewhat trivial example, because there is oly oe operator i the set, but we ca oetheless associate each eigestate, 0 ad 1 with its uique eigevalue, +1 ad 1, respectively For multiple qubits, we ca take tesor products of the Z with the idetity operator, 1 0 I σ i σ 0 0 1 For example, take a six qubit system ad the state b 1 b b 3 b 4 b 5 b 6, where b i is a biary digit We will cosider the expectatio values of the followig operators: Z 1 ZIIIII Z IZIIII Z 3 IIZIII All of these operators commute, which ca be prove by direct computatio: Ad their expectatio values are [Z 1,Z ] Z 1 Z Z Z 1 (ZIIIII(IZIIII (IZIIII(ZIIIII (ZI IZ I I I I (IZ ZI I I I I (Z Z I I I I (Z Z I I I I 0 Z 1 b 1 b b 3 b 4 b 5 b 6 ZIIIII b 1 b b 3 b 4 b 5 b 6 b 1 Z b 1 b b b 3 b 3 b 4 b 4 b 5 b 5 b 6 b 6 ( 1 b1 1 1 1 1 1 ( 1 b1 Followigthetred,weseethatayeigestate, b 1 b b 3 b 4 b 5 b 6 correspodstoauiquesetofeiegvalues,{( 1 bi } i [16] with respect to the CSCO listed above V FUNCTIONS OF OPERATORS AND THE SCHRODINGER EQUATION The Schrodiger equatio is i ψ(t H(t ψt i This looks very familiar to the ODE, for which the solutio is d y(t a(ty(t dt y(t exp t 0 a(sdsy(0
5 Similarly, the formal solutio to the Schrodiger equatio is ( t ψ(t T exp i 0 H(sds/ ψ(0 The time orderig operator T is icluded to hadle the fact the H(t might ot commute with H(t But for ow we will assume that H(t H is costat, so the time orderig complicatio goes away ad we are left with: ψ(t exp( iht/ ψ(0 But ow we must iterpret this expoetial of a operator, exp( iht/ I geeral, fuctios of operators ca be iterpreted i terms of the spectral represetatio of the operators Recall that a operator A ca be writte i terms of its eigevalues ad eigevectors as A a A fuctio f of this operator is the iterpreted as f(a f(a So the solutio to the Schrodiger equatio may be writte as ψ(t exp( ie t/ ψ(0 Where the eigevaluesof the Hamiltoia operator, H aregive the special symbol E ad areiterpreted as eergies