8.06 Spring 2016 Lecture Notes 3. Entanglement, density matrices and decoherence

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1 8.06 Spring 016 Lecture Notes 3. Entangleent, ensity atrices an ecoherence Ara W. Harrow Last upate: May 18, 016 Contents 1 Axios of Quantu Mechanics One syste Two systes The proble of partial easureent Density operators 5.1 Introuction an efinition Exaples The general rule for ensity operators Positive seiefinite atrices Proof of the ensity-atrix conitions Application to spin-1/ particles: the Bloch ball Dynaics of ensity atrices Schröinger equation Measureent Decoherence Exaples of ecoherence Looking insie a Mach-Zehner interferoeter Spin rotations in NMR Spontaneous eission Multipartite ensity atrices Prouct states Partial easureent an partial trace Purifications Axios of Quantu Mechanics We begin with a (very) quick review of soe concepts fro 8.04 an

2 1.1 One syste States are given by unit vectors ψ) V for soe vector space V. ˆ Observables are Heritian operators A L(V ). Measureents Suppose  = i=1 λ i v i )(v i for { v 1 )..., v )} an orthonoral basis of eigenvalues an (for ˆ siplicity) each λ i istinct. If we easure observable A on state ψ) then the outcoes are istribute accoring to Pr[λ i ] = (ψ v i ). Tie evolution is given by Schröinger s equation: i ψ) = H ψ) where H is the Hailtonian. t Heisenberg picture We can instea evolve operators in tie using i A ˆ H = [A ˆ H, H H ]. (1) t Tie-inepenent solution If the Hailtonian oes not change in tie, then the tie evolution operator for tie t is the unitary operator U = e iht -. The state evolves accoring to ψ(t)) = U ψ(0)) in the Schröinger picture or the operator evolves accoring to  H (t) = U A H (0)U in the Heisenberg picture. Systes are escribe by a pair (V, H). 1. Two systes Let s see how things change when we have two quantu systes: (V 1, H 1 ) an (V, H ). States are given by unit vectors ψ) V where V = V 1 V span{ ψ 1 ) ψ ) : ψ 1 ) V 1, ψ ) V }. A special case are the prouct states of the for ψ 1 ) ψ ). States that are not prouct are calle entangle. Observables are still Heritian operators A ˆ L(V ). A general observable ay involve interactions between the two systes. Local observables are of the for A ˆ I, I B ˆ or ore generally Aˆ I +I B, ˆ an correspon to properties that can be easure without interacting the two systes. Measureents The usual easureent rule still hols for collective easureents. But when only one syste is easure, we nee a way to explain what happens to the other syste. Suppose we easure the first syste using the orthonoral basis { v 1 ),..., v )}. (Equivalently, we easure an operator with istinct eigenvalues an with eigenvectors v 1 ),..., v ).) If the overall syste is in state ψ), then the first step is to write ψ) as ψ ) = p i v i ) w i ), i=1 for soe unit vectors w 1 ),..., w ) (not necessarily orthogonal) an soe p 1,..., p such that p i 0 an i=1 p i = 1. Then the probability of outcoe i is p i an the resiual state in this case is v i ) w i ).

3 Tie evolution is still given by Schröinger s equation, but now the joint Hailtonian of two non-interacting systes is H = H 1 I + I H. () q 1 q r 1 r Interactions can a ore ters, such as the Coulob interaction, which generally cannot be written in this way. Note that a Hailtonian ter of the for Aˆ 1 Aˆ oes represent an interaction; e.g. σ z σ z has energy ±1 epening on whether the two spins have Z coponents pointing in the sae or opposite irections. Tie-inepenent solution For a Hailtonian of the for in (), the tie evolution operator is iht ih1t iht U = e - = e - e -. You shoul convince your that this secon equality is true. Of course if the Hailtonian contains interactions then U will generally not be of this for. These principles are actually profounly ifferent fro anything we have seen before. For exaple, consier the nuber of egrees of freeo. One -level syste nees coplex nubers to escribe (neglecting noralization an the overall phase abiguity) but N -level systes nee N coplex nubers to escribe, instea of N. This exponential extravagance is behin the power of quantu coputers, which will be iscusse briefly at the en of the course, if tie perits. It also seee intuitively wrong to any physicists in the early 0th century, ost notably incluing Einstein. The objections of EPR [A. Einstein, B. Poelsky an N. Rosen, Physical Review, (1935)] le to Bell s theore, which we saw in 8.05 an will review on pset 6. Here, though, we will consier a sipler proble. 1.3 The proble of partial easureent Let us revisit the scenario where we easure part of an entangle state. Suppose that Alice an Bob each have a spin-1/ particle in the singlet state +) ) ) +) ψ ) = (3) (The singlet is an arbitrary but nice choice. The arguent woul be essentially the sae for any entangle state.) Iagine that Alice an Bob are far apart so that they shoul not be able to quickly sen essages to one another. Now suppose that Alice ecies to easure her state in the { +), )} basis. Using the above rules we fin that the outcoes are as escribe in Table 1. Alice s outcoe joint state Bob s state Pr[+] = 1 +) ) ) Pr[ ] = 1 ) +) +) Table 1: Outcoes when Alice easures her half of the singlet state (3) in the { +), )} basis. 3

4 What can we say about Bob s state after such a easureent? It is not a eterinistic object, but rather an enseble of states, each with an associate probability. For this we use the notation {(p 1, ψ 1 )),..., (p, ψ ))} (4) to inicate that state ψ i ) occurs with probability p i. The nubers p 1,..., p shoul for a probability istribution, eaning that they are nonnegative reals that su to one. The states ψ i ) shoul be unit vectors but o not have to be orthogonal. In fact, the nuber coul be uch larger than the iension, an coul even be infinite; e.g. we coul iagine a state with soe coefficients that are given by a Gaussian istribution. We generally consier to be finite because it keeps the notation siple an oesn t sacrifice any iportant generality. In the exaple where Alice easures in the { +), )} basis, Bob is left with the enseble 1 1, +),, ). (5) What if Alice chooses a ifferent basis? Recall fro 8.05 that if n R 3 is a unit vector then a spin-1/ particle pointing in that irection has state θ θ n) n; +) = cos +) + sin e iφ ). (6) Here (1, θ, φ) are the polar coorinates for n; i.e. n x = sin θ cos φ, n y = sin θ sin φ, an n y = cos θ. The notation n; +) was what we use in 8.05 an n) will be the notation use in 8.06 in contexts where it is clear that we are talking about spin states. The orthonoral basis { n; +), n; )} in our new notation is enote { n), n)}. Suppose that Alice easures in the { n), n)} basis. It can be shown (see 8.05 notes or Griffiths 1.) that for any n, n) n ) n ) n) ψ ) =. (7) Thus, for any choice of n, the two outcoes are equally likely an in each case Bob is left with a spin pointing in the opposite irection, as escribe in Table. Alice s outcoe joint state Bob s state Pr[n] = 1 n) n) n) Pr[ n] = 1 n) n) n) Table : Outcoes when Alice easures her half of the singlet state (3) in the { n), n)} basis. This leaves Bob with the enseble 1 1, n),, n n). (8) 4

5 Uh-oh! At this point, our elegant theories of quantu echanics have run into a nuber of probles. Theory isn t close. When we cobine two systes with tensor prouct we get a new syste, eaning a new vector space an a new Hailtonian. It still fits the efinition of a quantu syste. But when we look at the state of a subsyste, we o not get a single quantu state, we get an enseble. Thus, if we start with states being represente by unit vectors, we are inevitably force into having to use ensebles of vectors instea. Ensebles aren t unique. Any choice of n will give Bob a ifferent enseble. We expect our physical theories to give us unique answers, but here we cannot uniquely eterine which enseble is the right one for Bob. Note that other choices of easureent can leave Bob with ifferent ensebles as well; e.g. if Alice flips a coin an uses that to choose between two easureents settings, then Bob will have a istribution over four states, each occurring with probability 1/4. Tie travel?! If Bob coul istinguish between these ifferent ensebles (incluing the case in which Alice oes nothing an he still hols half of an entangle state), then Alice coul instantaneously counicate to Bob with her choice of easureent basis (or perhaps her choice of whether to easure at all or not). Accoring to special relativity, there is a ifferent inertial frae in which this process looks like Alice sening a essage backwars in tie. This rapily leas to trouble... Fortunately ensity operators solve all three probles! As a bonus, they are far ore elegant than ensebles. Density operators.1 Introuction an efinition We woul like to evelop a theory of states that cobines ranoness an quantu echanics. So it is worth reviewing how both ranoness an quantu echanics can be viewe as two ifferent ways of generalizing classical states. For siplicity, consier a classical syste which can be in ifferent states labelle 1,,.... The quantu echanical generalization of this woul be to consier coplex -iensional unit vectors while the probabilistic generalization woul be nonnegative real -iensional vectors whose entries su to one. These can be thought of as two incoparable generalizations of the classical picture. We are intereste in consiering both generalizations at once so that we consier state spaces that are both probabilistic an quantu. We suarize these ifferent choices of state spaces in Table 3. classical quantu eterinistic {1,..., } ψ) C s.t. (ψ ψ) = 1 probabilistic p 1,..., p 0 ensebles? s.t. p p = 1 ensity operators? Table 3: Different theories yiel ifferent state spaces. 5

6 What o we put in the fourth box (probabilistic quantu) of Table 3? One possibility is to put ensebles of quantu states, as efine in (4). Besies the rawbacks entione in the previous section, these also have the flaw that of involving an unboune nuber of egrees of freeo. For exaple, let s take a spin-1/ particle (i.e. = ), so our quantu states are of the for c + +) + c ). Then one such probability istribution is +) with probability 1/3, ) with + ) i ) probability 1/ an with probability 1/6. Another istribution is cos(θ) +) + sin(θ) ) where θ is istribute accoring to a Gaussian with ean 0 an variance σ. This works, but there are an infinite nuber of egrees of freeo, even if we start with a single lousy electron spin! Surely nature woul not be so cruel. Another rawback with this approach is that ifferent istributions can give the sae easureent statistics for all possible easureents. As a result, any of these infinite egrees of freeo turn out to be siply reunant. To see how this works, suppose that we have a iscrete istribution where state ψ a ) occurs with probability p a, for a = 1,...,. Consier an observable A. ˆ The expectation of A ˆ with respect to this enseble is: p a (ψ a A ψ ˆ a ) = p a tr (ψ a A ψ ˆ a ) since tr has no effect on 1 1 atrices a=1 a=1 ˆ = p a tr A ψ a )(ψ a cyclic property of the trace a=1 = tr Aˆ p a ψ a )(ψ a linearity of the trace = tr[aρ ˆ ] a=1 D DA C ensity atrix ρ nice siple forula = ( A, ˆ ρ) alternate interpretation We see that the easureent statistics are a function of the istribution only via the ensity operator ρ = ψ a=1 p a a )(ψ a. This has about egrees of freeo, by contrast with O() egrees of freeo for known quantu states an with the egrees of freeo associate with ensebles. This alreay solves one of the probles of ensebles, naely their use of unliite aounts of reunant inforation. This arguent use only iscrete istributions over C but the extension to continuous istributions an/or infinite-iensional states is straightforwar. Facts about traces: Let X be a atrix of iension n an Y a atrix of iension n. In general these will not be square, but XY an Y X both are, so their traces are well-efine. In fact, they are equal! A quick calculation shows n tr[xy ] = X i,j Y j,i = tr[y X]. (9) i=1 j=1 This is calle the cyclic property of the trace because it is often applie to traces of long strings of atrices. For exaple, we can repeately apply (9) (using curly braces to inicate which blocks of atrices we are calling X an Y ) to obtain tr[ab D DA C DDAC D ] = tr[d D DA AB C DDAC C ] = tr[cd DA DA C DDAC B ] = tr[bcda] (10) X Y X Y X Y 6

7 The trace can also be use to efine an inner prouct on operators. Define (X, Y ) tr[x Y ] = X i,j Y i,j. (11) Fro this last expression we see that (X, Y ) is equivalent to turning X an Y into vectors in the natural way (just listing all the eleents in orer) an taking the conventional inner prouct between those vectors.. Exaples..1 Pure states If we know the state is ψ), then the ensity atrix is ψ)(ψ. Observe that there is no phase abiguity ( ψ) e iφ ψ) leaves the ensity atrix unchange) an each ψ) gives rise to a istinct ensity atrix. Such ensity atrices are calle pure states, an soeties this terinology is also use when talking about wavefunctions, to justify not using the ensity atrix foralis. By contrast, all other ensity atrices are calle ixe states... Spin-1/ pure states Let us consier the special case of pure states when =, corresponing to a spin-1/ particle. If n) = cos( θ ) +) + sin( θ )e iφ ) then i,j θ θ θ cos ( ) cos( ) sin( )e iφ n )( n = cos( θ ) sin( θ iφ )e sin ( θ ) 1 0 cos ( θ ) 1 0 iφ θ θ 0 e = + + cos sin 1 θ sin ( ) e iφ 0 I cos(θ) sin(θ) = + σ z + (cos(φ)σ x + sin(φ)σ y ) I + n σ = This result is beautiful enough to frae. I + n nσ n)(n = (1) With this in han we can return to the exaple of Alice easuring half of a singlet state. Whatever her choice of n, Bob s ensity atrix is I + n σ 1 I n σ I n)(n + n)( n = + =. (13) This rules out their earlier attepts at instantaneous signaling (an later we will prove this in ore generality). Bob s ensity atrix fully eterines the results of any easureent he akes, an it is inepenent of Alice s choice of n. 7

8 ..3 The axially ixe state If { v 1 ),..., v )} are an orthonoral basis, an each occurs with probability 1/, then the resulting ensity atrix is 1 I ρ = v i )(v i =, (14) Now consier the istribution 1 ψ 1 ) +) + ) with probability 1/ ψ ) +) ) with probability 1/ 3 3 i=1 inepenent of the choice of basis. This is calle the axially ixe state. The previous exaple was the = case of this: a 1/ probability of spin-up an 1/ probability of spin-own results in the sae ensity atrix, no atter which irection up refers to. The continuous istribution over all unit vectors in C also yiels the sae ensity atrix, although this is a harer calculation...4 Multiple ecopositions Consier the istribution where +) occurs with probability /3 an ) with probability 1/3. The ensity atrix is )( + + )( = 3. (15) The ensity atrix is now ψ + )(ψ + + ψ )(ψ = + =. (16) One lesson is that we shouln t take the probabilities p a too seriously; i.e. they are not uniquely eterine by the ensity atrix. Neither is the property of the states in the enseble being orthogonal...5 Theral states Introuce a Hailtonian H = i=1 E i i)(i. We can think of this classically, as saying that state i has energy E βe i i. In this case, the Boltzann istribution at teperature T is p i = e /Z, where Z = e βe i, β = 1/k 3 i=1 B T an k B = J/K is Boltzann s constant. In the quantu setting, state i is replace by i). The resulting ensity atrix is e βe i i)(i e βh e βh i=1 i i Z Z tre βh i=1 ρ = p )(i = = = (17) This is known as the Gibbs state or the theral state. It escribes the state of a quantu syste at theral equilibriu. 8

9 One specific exaple coes fro NMR. Consier a proton spin in a agnetic fiel, say a Tesla fiel in the ẑ irection. At this fiel strength, the proton spin will experience the Hailtonian H = ω 0 σ z where ω MHz. (In fact, if you buy a 11.74T superconucting agnet, the venor will probably call it a 500 MHz agnet for this reason. It coul also reasonably be calle a 500K agnet because of its price.) The theral state is βh β ω 0 β ω 0 e e + )(+ + e )( ρ = = tre βh e β ω 0 + e β ω 0 (1 + β ω 0 ) +)(+ + (1 β ω 0 ) )( I β ω 0 = + σ z. Let T = 300K, which is close to roo teperature. Then 1/β 6.5THz an so β ω This is the polarization of the state. In an NMR experient we can think of a fraction of spins aligning with the external fiel an a fraction of the spins anti-aligning with the external fiel at theral equilibriu. This eans that observables are effectively attenuate by a factor of (in this case) The general rule for ensity operators We know what akes a vector a vali probability istribution or a vali quantu wave-vector. What akes an operator a vali ensity operator? One answer is ρ is a vali ensity operator if there exists p 1,..., p, ψ 1 ),..., ψ ) such that ρ = a p a ψ a )(ψ a, p 1,..., p 0, p p = 1 an (ψ a =)1 for each i. This is soewhat unsatisfactory if we want to buil a theory where the ensity operators are the funaental objects. Fortunately, there is a siple answer. Theore 1. If a atrix ρ is a ensity atrix for soe enseble of quantu states then 1. trρ = 1.. ρ c 0. Conversely, for any atrix ρ satisfying these two conitions, there exists an enseble {p a, ψ a )} 1 a such that ρ = a=1 p a ψ a )(ψ a. Here can be taken to be the rank of ρ. The inequality ρ c 0 eans that ρ is positive seiefinite, which is efine to ean that (ψ ρ ψ) 0 for all ψ). It is the atrix analogue of being nonnegative. 3.1 Positive seiefinite atrices We say that a square atrix A is positive seiefinite if (ψ A ψ) 0 for all ψ). Physically, A ight be an observable that takes on only nonnegative values. Or it ight be a ensity atrix. If furtherore A is Heritian, then there are three equivalent ways to characterise the conition of being positive seiefinite. Theore. If A = A then TFAE (the following are equivalent) 1. For all ψ), (ψ A ψ) 0.. All eigenvalues of A are nonnegative. 9

10 3. There exists a atrix B such that A = B B. (This is calle a Cholesky factorization.) To get intuition for this last conition, observe that for 1 1 atrices, it is the stateent that a real nuber x 0 iff x = z z for soe coplex z. Proof of Theore. Since A is Heritian, we can write A = basis { e 1 ),..., e )} an soe real λ 1,..., λ. i=1 λ i e i )(e i for soe orthonoral (1 ): Take ψ) = e i ). Then 0 (ψ A ψ) = (e i A e i ) = λ i. ( 3): Let B = i=1 λ i e i )(e i. As an asie, one can show that B satisfies A = B B if an only if B = i=1 λ i e i )(f i for soe orthonoral basis { f 1 ),..., f )}. Soeties we say that B = A, by analogy to the scalar case. (3 1): For any ψ), let ϕ) = B ψ). Then (ψ A ψ) = (ψ B B ψ) = (ϕ ϕ) Proof of the ensity-atrix conitions Here we prove Theore 1. Start with an enseble {p i, ψ i )} 1 i. Define ρ = i ψ i )(ψ i. i=1 p i=1 p i Then ρ = ψ i )(ψ i = ρ, since p i = p i. Thus ρ is Heritian. Next, efine B = p i ψ i )(i. Then ρ = B B, iplying that ρ c 0. i=1 Finally, trρ = i=1 p itr ψ i )(ψ i = i=1 p i = 1. To prove the other irection, suppose that trρ = 1 an ρ c 0. By Theore, ρ = i=1 λ i e i )(e i for { e 1 ),..., e )} an orthonoral basis an each λ i 0. Aitionally trρ = i=1 λ i = 1. Thus we can take p i = λ i an now ρ is the ensity atrix corresponing to the enseble {p i, e i )} 1 i. If rank ρ <, then the su only nees rank ρ ters. 3.3 Application to spin-1/ particles: the Bloch ball The geoetry of the set of ensity atrices is unfortunately not quite as siple as the state spaces we have encountere previously. Pure quantu states for a ball (oulo the phase abiguity) an probability istributions for a siplex. Intersections of planes (such as tr[ρ] = 1) with the set of positive seiefinite atrices are calle spectrahera, an apart fro the wonerful nae, I will not explore their general properties here. However, the case of = is inee siple an elegant. Given a Heritian atrix A, when is it a vali ensity atrix? First, if it is Heritian then we can express it as a 0 I + a 1 σ 1 + a σ + a 3 σ 3 A = for soe real nubers a 0, a 1, a, a 3. The factor of in the enoinator is arbitrary, but we will see I+ra rσ later that it siplifies things. If A is a ensity atrix, then 1 = tr[a] = a 0. Thus, A = with 1± ra na (a 1, a, a 3 ). We saw in 8.05 that eig(na nσ) = ± na. Thus eig(a) = both nonnegative, which is true iff na 1. This proves that the set of two-iensional ensity atrices is precisely equal to the set. A is ps iff these are I + na nσ : na 1. (18) 10

11 Geoetrically this looks like the unit ball in R 3. The pure states for the surface of the ball, corresponing to the case na = 1. The axially ixe state I/ correspons to na = 0. In general, na can be thought of as the purity of a state. This set is calle the Bloch ball. The unit vectors at the surface are calle the Bloch sphere. These have nothing to o with Bloch states or Bloch s theore (which arise in the solution of perioic potentials) except for the nae of the inventor. Beware also that for >, the set of ensity atrices is no longer a ball an there is no longer a canonical way to quantify purity. However, notions of entropy o exist an are use in fiels such as quantu statistical echanics. 4 Dynaics of ensity atrices 4.1 Schröinger equation The Schröinger equation states that i ) ψ) = H ψ t (19a) i (ψ = ( ψ H t (19b) ψ)(ψ H) = ψ)(ψ ] t i ψ)(ψ = (H ψ)(ψ i [H, (19c) i p a ψ a )(ψ a = [H, p a ψ a )(ψ a ] (19) t a=1 a=1 We conclue that the ensity atrix evolves in tie accoring to i ρ = [H, ρ]. (0) t This is reiniscent of the Heisenberg equation of otion for operators, but with the opposite sign i A ˆ H = [A ˆ H, H H ]. (1) t One way to explain the ifferent signs is that states an observables are ual to each other, in the sense that they appear in the expectation value as ( Â, ρ). Another way to talk about quantu ynaics is in ters of unitary transforations. If a syste unergoes Hailtonian evolution for a finite tie then this evolution can be escribe by a unitary operator U, so that state ψ) gets appe to U ψ). In this case ψ)(ψ is appe to U ψ)(ψ U. By linearity, a general ensity atrix ρ is then appe to UρU. 4. Measureent A siilar arguent shows that if we easure ρ in the orthonoral basis { v 1 ),..., v )}, then the probability of outcoe j is (v j ρ v j ) an the post-easureent state is v j )(v j. The fastest way to see this is to consier the observable v j )(v j which has eigenvalue 1 (corresponing to obtaining outcoe v j )) an eigenvalue 0 repeate 1 ties (corresponing to the orthogonal outcoes). Then we use the fact that (A) ˆ = tr[aρ] ˆ an set A ˆ = v j )(v j. 11

12 An alternate erivation is to ecopose ρ = Pr[j] = p a Pr[j a] a=1 = p a (v j ψ a ) a =1 = p a ( v j ψ a )( ψ a v j ) a=1 a=1 p a = (v j p a ψ a )(ψ a v j ) a=1 = (v j ρ v j ) ψ a )(ψ a. Then Pr[j a] = (v j ψ a ) an It shoul be reassuring that, even though we use the enseble ecoposition in this erivation, the final probability we obtaine epens only on ρ. What if we forget the easureent outcoe, or never knew it (e.g. soeone else easures the state while our back is turne)? Then ρ is appe to (v j ρ v j ) v j )( v j = v j )( v j ρ v j )( v j. () j=1 j=1 Here it is iportant to note that ensity atrices, like probability istributions, represent not only objective states of the worl but also subjective states; in other wors, they escribe our knowlege about a state. So subjective uncertainty (i.e. the state really is soething efinite but we on t know what it is) will have iplications for the ensity atrix. If we now write ρ fro () as a atrix in the v 1 ),..., v ) basis, this looks like ρ 1, ρ, ρ, Can we unify easureent an unitary evolution the way that we have unifie the probabilistic an quantu pictures of states? For exaple, how shoul we oel an ato in an excite state unergoing fluorescence? We will return to this topic later when we iscuss open quantu systes an quantu operations. However, alreay we are equippe to hanle the phenoenon of ecoherence, which is the onster lurking in the closet of every quantu echanical experient. 4.3 Decoherence Unitary operators correspon to reversible operations: if U is a vali unitary tie evolution then so is U. In ters of Hailtonians, evolution accoring to H will reverse evolution accoring to H. But other quantu processes cause an irreversible loss of inforation. Irreversible quantu processes are generally calle ecoherence. This soewhat iprecise ter refers to the fact that this inforation loss is always associate with a loss of coherence an with quantu systes 1

13 becoing ore like classical systes. In what follows we will illustrate it via a series of exaples, but will not give a general efinition. Let s war up with the concept of a ixture. If state ψ a ) occurs with probability p a, then the ensity atrix is a p a ψ a )(ψ a. But what if we have an enseble of ensity atrices? e.g. {(p 1, ρ 1 ),..., (p, ρ )} Then the average ensity atrix is ρ = p a ρ a. (3) a=1 We can use this to oel rano unitary evolution. Suppose that our state experiences a rano Hailtonian. Moel this by saying that unitary U a occurs with probability p a for a = 1,...,. This correspons to the ap ρ p a U a ρu. (4) a a=1 Let s see how this can explain how coherence is lost in siple quantu systes. Suppose we start with the ensity atrix ρ +,+ ρ +, ρ = ρ,+ ρ, an choose a rano unitary to perfor as follows: with probability 1 p we o nothing an with probability p we perfor a unitary transforation equal to σ z. This correspons to the enseble of unitary transforations {(1 p, I), (p, σ z )}. The ensity atrix is then appe to ρ 1 (1 p)iρi + pσ z ρσ z ρ +,+ = (1 p) ρ +, ρ +,+ + p ρ +, ρ,+ ρ, ρ,+ ρ, = ρ +,+ (1 p)ρ +, (1 p)ρ,+ ρ, If p = 0 then this of course correspons to oing nothing, an if p = 1, we siply have ρ 1 = σ z ρσ z. In between we see that the iagonal ters reain the sae, but the off-iagonal ters are reuce in absolute value. The iagonal ters correspon to the probability of outcoes we woul observe if we easure in the ẑ basis, an so it is not surprising that a ẑ rotation woul not affect these. However, the off-iagonal ters reuce just as we woul expect for a vector that is average with a rotate version of itself. If p = 1/, then the off-iagonal ters are copletely eliinate, eaning that all polarization in the xˆ an ŷ irections has been eliinate. One way to see this is that the xˆ an ŷ polarization of σ z ρσ z is opposite to that of ρ. Thus averaging ρ an σ z ρσ z leaves zero polarization in the xˆ-ŷ plane. With a series of exaples, I will illustrate that: Decoherence can be achieve in several ways that look ifferent but have the sae results. Decoherence estroys soe quantu/wave-like effects, such as interference. This also involves the loss of inforation, often of phase inforation. 13

14 5 Exaples of ecoherence 5.1 Looking insie a Mach-Zehner interferoeter This exaple is physically unrealistic (in one place) but akes the ecoherence phenoenon clearest to see. A Mach-Zehner interferoeter is epicte in Fig. 1. Bea-Splitter Colliate Bea RB Source SB Saple Detector 1 Mirror Detector Iage by MIT OpenCourseWare, aapte fro the wikipeia article on Mach-Zehner interferoeters. Figure 1: Mach-Zehner interferoeter. Iage taken fro the wikipeia article with this nae. At each point the photon can take one of two possible paths, which we enote by the states 1) an ). Technically 1) eans photon nuber in one oe an zero in the other oes, an siilarly for ). Also, we use 1), ) to first enote the two inputs to the first bea splitter, then the two possible paths through the interferoeter, an finally the two outputs of the secon bea splitter leaing to the etectors. Each bea splitter can be oele as a unitary operator. If they are bea splitters, then this operator is U bs =. 1 1 Thus, a photon entering in state 1) will go through the first bea splitter an be transfore into )+ ) the state 1, corresponing to an even superposition of both paths. Assuing the paths have the sae length an refractive inex, it will have the sae state when it reaches the secon bea splitter. At this point the state will be appe to 1) + ) 1) + ) + 1) ) U bs = = 1) an the first etector will click with probability 1. This is very ifferent fro what we observe if a particle entering a bea splitter chose ranoly which path to take. In that case, both etectors woul click half the tie. The usual reason to buil a Mach-Zehner experient, though, is not only to eonstrate the wave nature of light, but to easure soething. Suppose we put soe object in one of the paths so that light passing through it experiences a phase shift of θ. This correspons to the unitary transforation eiθ 0 U ph. (5)

15 Our oifie experient now correspons to the sequence U bs U ph U bs, which aps 1) to 1) + ) e iθ 1) + ) e iθ 1 ) + e iθ ) + 1 ) ) U bs U ph U bs 1) = U bs U ph = U bs =. The probability of the first etector clicking is now 1+eiθ = cos (θ/). Now a ecoherence. Suppose you fin a way to look at which branch the photon is in without estroying the photon. (This part is a bit unrealistic, but if we use larger objects, then it becoes ore reasonable. See the reaings for a escription of a two-slit experient conucte with C 60 olecules.) If we observe it then we will fin that regarless of the phase shift θ: the photon is equally likely to be in each path; an each etector is equally likely to click. Our easureent has cause ecoherence that has estroye the phase inforation in θ. 5. Spin rotations in NMR Start with a spin-1/ particle in the +) state. Apply H = S y for tie t = π/, so that iht π I iσ y i σ U = e - = e 4 y = = )+ ) Applying U once yiels U +) = an applying U a secon tie yiels ) (calculation oitte). Suppose that we easure in the { +), )} basis after applying the first U. Then each outcoe occurs with probability 1/ an the resulting ensity atrix is 1 1 I + )( + + )( =. Applying U again leaves the ensity atrix unchange. Decoherence has estroye the polarization of the spin. In actual NMR experients, we have a test tube with 10 0 water olecules at roo teperature an we are not going to easure their iniviual spins. Instea, suppose that two nuclear spins get close to each other an interact briefly. Suppose that the first spin is in state ρ an the secon spin is axially ixe (i.e. ensity atrix I/). Suppose that they interact for a tie T accoring to the Hailtonian H = λs z S z. n S n () 3 (Why not S (1) i=1 S i S i? This is a consequence of perturbation theory: if there is a large S z I + I S z ter in the Hailtonian, then the S x S x an S y S y ters are suppresse but the S z S z ter is not.) This is equivalent to the first spin experiencing a Hailtonian λs z if the secon spin is in a +) state. an experiencing λs z if the secon spin is in a ) state. 15

16 Averaging over these, the first spin is appe to the state 1 i 1 i tλs tλs z 1 i tλs i tλs ρ = e - z ρe - + e - z ρe - z iλt/ iλt/ iλt/ iλt/ 1 e 0 e 0 1 e 0 e 0 = ρ + ρ 0 eiλt/ 0 e iλt/ 0 e iλt/ 0 eiλt/ = ρ ++ cos(λt/)ρ + cos(λt/)ρ + ρ This oesn t coplete estroy the off-iagonal ters, but attenuates the. Here we shoul think of λt as usually sall. If we average over any such interactions, then this ight (skipping any steps, which you will explore on the pset) result in a process that looks like 1 0 ρ + ρ =, (6) T ρ + 0 where T is the ecoherence tie, soetie also calle the ephasing tie for this kin of ecoherence. If this is T, then is there also a T 1? Yes, T 1 refers to a ifferent kin of ecoherence. In NMR, there is typically a static agnetic fiel in the ẑ irection, which gives rise to a Hailtonian of the for H = γbs z. Fro this (together with the teperature) we obtain a theral state ρ theral escribe in Section..5. The process of theralization is challenging to rigorously erive fro the Schröinger equation but it is usually sufficient to oel it phenoenologically. Suppose that accoring to a Poisson process with rate 1/T 1, the spin is iscare an replace with a fresh spin in the state ρ theral. Then we woul obtain the ifferential equation 1 ρ = (ρ ρ theral ). (7) T 1 Of course, there is another source of ynaics, which is the natural tie evolution fro the Schröinger equation: ρ = i [H, ρ]. Putting this together, we obtain the Bloch equation: i ρ + ρ = [H, ρ] (ρ ρ theral ). (8) T 1 T ρ + 0 If we write ρ = I+ra r σ, then (8) becoes na = Mna ˆ + nb, (9) t ˆ n for M, b to be eterine on a pset. (Why o I keep talking about NMR, an not ESR (electron spin resonance)? The electron s gyroagnetic ratio is about 657 ties higher than the proton s so its roo-teperature polarization is larger by about this aount, an signals fro it are easier to etect. However, it also interacts ore proiscuously an thus often ecoheres quickly, with T on the orer of icrosecons or worse in ost cases. So when you get a knee injury, your iagnosis will be ae via your nuclei an not your electrons.) 16

17 5.3 Spontaneous eission Consier an ato with states g) an e), corresponing to groun an excite. We will also consier a photon oe, i.e. a haronic oscillator. Suppose the initial state of the syste is ψ) ato 0) photon with ψ) = c 1 g) + c e). These will interact via the Jaynes-Cuings Hailtonian H = Ω( g)(e â + e)(g â). (30) (For siplicity we have left out soe ters that are usually in this Hailtonian. This Hailtonian can be erive using perturbation theory, as we iscusse on a pset.) Suppose that the ato an photon fiel interact via this Hailtonian for a tie t. Assue that δ Ωt is sall an expan the state of the syste in powers of δ: iht δ 3 e - ψ) 0) = (c1 g) + c e)) 0) iδc g) 1) c e) 0) + O(δ ). (31) Now easure an we see that with probability c δ the photon nuber is 1 an the ato is in the state g). In this case, we observe an eitte photon an can conclue that the ato ust currently be in the state g). (It is tepting to conclue that we know it was previously in the state e). This sort of reasoning about the past can be angerous. In fact, all we can conclue is that c ust have been nonzero.) With probability c 1 + (1 δ ) c = 1 c δ we observe 0 photons an the state is c 1 g) + (1 δ )c e), 1 c δ again, up to O(δ 3 ) corrections. If we repeat this for long enough then we also en up in the state g). This is because if we watch an ato for a long tie an it never eits a photon we can conclue that it s probably in the groun state. 6 Multipartite ensity atrices In Section 1.3 I coplaine that pure-state quantu echanics is not close uner iscaring subsystes. Here we will see that ensity atrices solve this proble. More generally we will exten the foralis of ensity atrices to hanle coposite quantu systes. 6.1 Prouct states Suppose that we have two systes, calle A an B, with ensity atrices ρ an σ respectively. I clai that their joint state is ρ σ. One way to see this is by explicitly ecoposing ρ = i p i α i )(α i, σ = j q j β j )(β j an consiering these as inepenent ensebles {(p i, α i ))} an {(q i, β i ))}. Accoring to the rule for inepenent probability istributions the probability of fining syste A in state α i ) an syste B in state β j ) is p i q j. In this case the joint state is α i ) β j ). This correspons to the enseble {(p i q j, α i ) β j ))} which has ensity atrix p i q j ( α i ) β j ))((α i (β j ) = p i q j α i )(α i β j )(β j = ρ σ. (3) i,j Therefore the prouct rule for ensity atrices can be inferre fro the prouct rule for pure states. i,j 17

18 We can also erive it fro observables. If we easure observable A ˆ on the first syste then this correspons to the observable A ˆ I on the coposite syste; likewise B ˆ on the secon syste correspons to I Bˆ on the joint syste. Their prouct is  Bˆ. This arises for exaple when the ipole oents of two spins are couple an the Hailtonian gets a ter proportional to ns 1 S n = S x S x + S y S y + S z S z. Let ω be the joint state of a syste where the first particle is in state ρ an the secon is in state σ. The expectation of A ˆ B ˆ with respect to ω shoul be ˆ ˆ ˆ ˆ tr[ρa]tr[σb] = tr[(ρ σ)(a B)]. (33) ˆ ˆ Since this is equal to tr[ω(â Bˆ)] for all choices of A, B, we ust have that ω = ρ σ. 6. Partial easureent an partial trace Density atrices were introuce by the fact that easuring one part of a larger syste leaves the rest in a rano state. As a result, pure-state quantu echanics is not a close theory; if the state of the joint syste AB is pure, then it is possible that the states of A an B are not theselves pure. However, if there is a ensity atrix ρ AB escribing the joint state of systes A an B then we shoul be able to efine ensity atrices for the iniviual systes. Inee any observable X on syste A shoul have a well-efine expectation value, an these can be use to efine a reuce ensity atrix for syste A. Matheatically we enote the ensity atrix of syste A by ρ A an efine it to be the unique ensity atrix satisfying tr[ρ A X] = tr[ρ AB (X I)], (34) for all observables X. (It is an instructive exercise to verify that there is always a solution to (34) an that it is unique.) Siilarly we can efine the state of syste B to be ρ B satisfying tr[ρ B X] = tr[ρ AB (I X)]. Expaning (34) in ters of atrix eleents yiels AB AB ρ A ' X ' a,a a,a = ρ ' b ' X a,a ' δ b,b ' = '. (35) ab,a ρ ab,a ' b X a,a a,a ' a,b,a ',b ' a,a ',b Since this ust hol for any X, we have ρ A AB ' = (tr B [ρ]) ' a,a a,a = ρ ab,a ' b. (36) b This looks like taking a trace over the B subsyste (i.e. suing over the b = b 1 entries) while leaving the A syste alone. For this reason we call the ap fro ρ AB ρ A the partial trace an enote it tr B ; i.e. ρ A = tr B [ρ AB ]. The partial trace is the quantu analogue of the rule for arginals of probability istributions: p X (x) = y p XY (x, y). A siilar equation hols for ρ B tr A [ρ AB ] which can be expresse in ters of atrix eleents as ρ AB ρ B ' = (tr A [ρ]) = '. (37) b,b b,b ' ab,ab a If A an B have iensions A an B respectively an M enotes the set of atrices, then tr A : M A B M B an tr B : M A B M A are linear aps efine by tr A [ α)(β γ)(δ ] = (α β) γ)(δ tr B [ α)(β γ)(δ ] = (γ δ) α)(β 18

19 If { a)} an { b)} are orthonoral bases then tr A [ a)(a 1 b)(b 1 ] = δ a,a ' b)(b 1 an tr B [ a)(a 1 b)(b 1 ] = δ b,b ' a)(a 1. Let s illustrate this by revisiting the exaple of spontaneous eission fro Section 5.3. Suppose iht/ g ) + e) ψ) = e 0), where H = Ω( g)(e a + e)(g a). H g, 0) = 0 an H acts on the { e, 0), g, 1)} subspace as a rotation. Thus 1 1 ψ) = g, 0) + (cos(θ) e, 0) i sin(θ) g, 1 )). The corresponing ensity atrix is (g, 0 (e, 0 (g, 1 (e, i g, 0) cos(θ) sin(θ) 0 1 e, 0) 1 i cos(θ) cos (θ) cos(θ) sin(θ) 0 ψ)(ψ = i i 1 (38) g, 1) sin(θ) sin(θ) cos(θ) sin (θ) 0 e, 1) The reuce state of the ato is an the reuce state of the photon is (g (e 1 1 g) + sin 1 (θ) cos(θ) tr photon ψ )(ψ = 1 1 (39) e) cos(θ) cos (θ) (0 ( ) + cos i (θ) sin(θ) tr ato ψ)(ψ = i 1. (40) 1) sin(θ) sin (θ) (The ecorations surrouning the above atrices are eant as reiners of which basis eleents the rows an coluns correspon to.) 6.3 Purifications One way ensity atrices can arise is via subjective uncertainty; i.e. we on t know what the state is, but it really is pure. If so, we ight iagine that ensity atrices woul be useful for a quantu theory of statistics or inforation, but are not essential to quantu physics. However, ensity atrices also arise in settings where the overall state is known exactly. We saw this earlier where Bob coul not istinguish his half of a singlet fro a uniforly rano state. Conversely, a uniforly rano state cannot be istinguishe fro half of a singlet, with the other half in an unknown location. This is in fact only a representative exaple of the general rule that any ensity atrix coul arise by being part of an entangle state. A B First, let ψ) = i=1 j=1 α i,j i) j). If Bob easures his syste, he obtains outcoe j with probability p j α i,j an the resiual state for Alice is α i,j i)/ p j. Her ensity atrix is i B A A A A α B i,j i) i α ' i ',j (i1 i=1 =1 1 p j = α i,j α p i ',j p )( j=1 j j i=1 i ' =1 j=1 i D DA C =(αα ) i,i ' i i = αα. 19

20 What if Alice easures? Working this out is a goo exercise. The answer is (α α) T. By Theore any ensity atrix ρ can be written as αα for soe atrix α. It reains only to check the noralization to ensure that ψ) is a vali state: 1 = trρ = trαα = α i,j. This eans that if we prouce ρ in the lab, we can never know whether the state is ixe because of uncertainty about which pure state it is, or because it is entangle with a particle that is out of our control. i,j 0

21 MIT OpenCourseWare Quantu Physics III Spring 016 For inforation about citing these aterials or our Ters of Use, visit: