10 Interpreting Quantum Mechanics

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1 10 Interpreting Quantum Mechanics In this final chapter, I want to explore the process of measurement in quantum mechanics The Density Operator To get started, we must first come clean about an unrealistic feature of our treatment so far. Up to this point, we ve assumed that we know the precise quantum state our system is in. For macroscopic objects this assumption is totally implausible; we can t possibly hope to discover the quantum state of the 10 3 C atoms in a diamond, or the 10 5 atoms in a protein molecule. In fact, even in purely classical systems there s always some uncertainty in our knowledge of the system. In reality, we cannot know the momentum of a single particle with infinite precision as all our measurements will be subject to some experimental error. To handle this situation in quantum mechanics, we need a way to describe systems even when we re not sure which state they re in. Suppose we know only that our system is in one of the states { ni}, and that the probability it is in state ni is p n. It s important to be clear that we re not saying that system is in state i = X n p pn ni, (10.1) because i itself is a well defined quantum state. Rather, we re admitting that we don t know the true state of the system, which could be any of the states { ni}. Indeed, these states do not need to form a complete set, and do not even need to be orthogonal. Given our incomplete knowledge, the expectation value of measuring some observable Q will be Q = X p n hn Q ni. (10.) n This expression combines the quantum uncertainty in Q inherent in any state ni (which may not be an eigenstate of Q), together with our lack of knowledge of the system s state, represented by the p n s. When our knowledge of the state is incomplete, we say that the system is in an impure state, and correspondingly we refer to a regular quantum state i as pure. This terminology is somewhat misleading, because it is really just our knowledge that is incomplete the system itself is presumably in some perfectly well defined quantum state, it s just that we don t know which one. We define the density operator := X n p n nihn (10.3) where the p n are the probabilities introduced above. Thus projects onto state ni with probability p n. The density operator has the following three properties: First, since probabilities are real =, so the density operator is Hermitian. Second, tr H = 1 (10.) 16

2 since the probabilities sum to one, and third h i 0 for all ih (10.5) since probabilities are non negative. We often write this property as 0 for short. 70 If we re definitely in some state i, sothesystemispure,then = ih. For such pure states it is easy to see that n = using the fact that i is normalized. The definition (10.3) of the density operator is reminiscent of the definition Q = X j q j q j ihq j (10.6) of an arbitrary operator in terms of its eigenstates. However, should not be considered as corresponding to an observable, because the p n are subjective rather than objective: they have nothing to do with the state of the system itself, rather they just quantity our knowledge or ignorance of that state. For example, if all our records of the results of measurements become lost, our values of the p n will change, but the physical system itself will not. By contrast, the spectrum {q j } of Q is determined by the Laws of Nature, independent of our knowledge of them. To see the point of the density operator, we write Q= X n,j p n q j nihn q j ihq j. (10.7) Taking the trace of this equation over H we have tr H ( Q)= X p n q j hq k nihn q j ihq j q k i = X n,j,k n,j = X p n hn Q ni = Q. n p n q j hq j ni (10.8) Thus, given the density operator, we can obtain the expectation value of any observable of our system, accounting for both the quantum and classical uncertainties. In the Schrödinger picture the time evolution of is given by i~ d dt =i~ ni p n hn n = X n p n (H nihn nihn H) =[H, ]. (10.9) This is the quantum analogue of Liouville s equation d /dt = {H, } in Classical Dynamics, which governs the time evolution of a probability density on phase space. To obtain the time evolution of an arbitrary expectation value Q =tr H ( Q)(where the Schrödinger picture operator Q has no explicit time dependence) we expand the trace in terms of a time-independent basis { ai} and use (10.9) to find i~ dq dt =i~ X a ha d dt Q ai = X a ha (H H)Q ai =tr H ( [Q, H]) (10.10) 70 In fact, these three properties can be taken to be the defining properties of a density operator, in the sense that any operator obeying these three properties is the density operator for some system. 17

3 where the last equality uses the cyclicity of the trace. We know from before that the rate of change of the expectation value of Q in any pure quantum state is given by the expectation value of [Q, H]/i~. Equation (10.10) states that even when our knowledge of the quantum state is imprecise the expected rate of change of Q is the appropriately weighted average of the rates of change of Q for each of the possible states of the system. The density operator and the operators for the Hamiltonian and other observables encapsulate a complete, self contained theory of quantum dynamics. If we have incomplete knowledge of the system s quantum state, then use of this formalism is mandatory. If we do happen to know that our system is initially in the precise quantum state i, we can still use this apparatus by setting = ih, rather than using the TDSE, though in this case use of the density operator is optional Reduced density operators If our system comprises two (or more) identifiable subsystems, then we learned that the full Hilbert space H could be written as a tensor produce H A H B of the two subsystems. Let s suppose A describes the system we re really interested in, whilst B is the environment. Thus B describes the quantum state of everything in the Universe, except our immediate object of study A. Of course, we cannot hope to know the precise quantum state of B. We now show that, even if A starts in a pure state, once it has entangled itself with B, it will be in an impure state. Let { A; ii} denote a complete set of orthonormal states for A and { B; ri} a complete orthonormal set for B. We can write the density operator for the entire system as AB = X A; ii B; ri ir,js ha; j hb; s. (10.11) ij rs Now let Q be an observable property of subsystem A. The expectation value of Q is Q =tr H ( Q) 0 1 = X ha; k hb; X A; ii B; ri ir js ha; j hb; s A Q A; ki B; ti kt ij,rs = X ha; j Q A; ii X! ir,jr, ij r (10.1) where in going to the last line we have used the orthonormality of the states, together with the fact that since Q is a property just of A, it acts on H = H A H B as Q = Q I. We now define the reduced density operator of the subsystem A to be taking the trace only over subsystem B. Thus A := tr HB ( AB ) (10.13) A = X r hb; r AB B; ri = X ij A; ii! X ir,jr ha; j, (10.1) r 18

4 where the second equality uses our expression (10.11) for AB. We can write our expression (10.1) for Q in terms of the reduced density operator as Q = X i ha; i A Q A; ii =tr HA ( A Q). (10.15) That is, for any operator of the form Q I we have tr H ( AB Q I) =tr HA ( A Q) (10.16) because the trace over H B is trivial. The reduced density operator enables us to obtain expectation values of subsystem A s observables without bothering about the states of B Decoherence Suppose both A and B start in pure quantum states (0)i and (0)i, respectively. Initially then, AB (0) = (0)ih (0) =( (0)i (0)i)( (0)i (0)i) (10.17) and the whole system will evolve unitarily in time via the operator U AB (t) built from the Hamiltonian of the full system. If the full Hamiltonian does not couple A and B, so that H = H A I + I H B,then U AB (t) =e iht/~ =e i(h A+H B )t/~ =e ih At/~ e ih Bt/~ = U A (t) U B (t), (10.18) using the fact that [H A,H B ] = 0 since they act on di erent Hilbert spaces. Thus, in this case (t)i = U AB (t) (0)i = U A (t) (0)i U B (t) (0)i = (t)i (t)i (10.19) and the density operator at time t will be AB (t) =U AB (t) (0)ih (0) U 1 AB (t) =( (t)i (t)i)(h (t) h (t) ). (10.0) Consequently, tracing over H B we have that the reduced density operator A (t) =tr HB AB (t) = (t)ih (t) (10.1) at time t. In other words, if A starts in a pure state and it does not interact with the environment then it will remain in a pure state. In every realistic case, subsystems are coupled to eachother however weak, there is some term H AB in the Hamiltonian that is not diagonal with respect to the splitting H A H B. In the presence of an interaction term H AB, the time evolution operator is not generically a product of the time evolution operators of the two subsystems, and states of A and B will typically become entangled. Let s demonstrate this assertion in the simplest case that A and B each have only two possible states. Suppose that at time t they have evolved into the entangled state (t)i = 1 p ( A;0i B;0i + A;1i B;1i). (10.) 19

5 In this (t)i, the states of A and B are correlated. Before making any measurements, there is a probability 1/ that A is in either of its two states. However, if we discover (by performing some measurement) that B is in state B;0i then A must certainly be in state A;0i, whilst if we learn that B is in state B;1i, A must be in state A;1i. However, suppose we don t keep track of the state of B. Then we should trace over the states in H B, giving A = 1 X hb; a ( A;0i B;0i + A;1i B;1i)(hA;0 hb;0 + ha;1 hb;1 ) B; ai a=1, = 1 ( A;0ihA;0 + A;1ihA;1 ), (10.3) which is the density operator of an impure state. In tracing over H B we ve lost the information carried by the correlations between the two subsystems, so A now we only know that A is equally likely to be in either state. Thus, even when the whole system is in a pure state, if we only measure part of it, the state can appear to be impure. As another example, system A could represent a hydrogen atom which we monitor closely, whilst system B could describe an electromagnetic field to which the atom couples, but which we do not measure precisely. If our atom starts in its first excited state and the electromagnetic field in its ground state then initially, the atom, field and whole system are all in pure states. After some time, coupling between the atom and field means the whole system will evolve into a pure but entangled state (t)i = a 0 (t) A;0i F ;1i + a 1 (t) A;1i F ;0i (10.) where A; ni represents the n th excited state of the atom, while F ; ni is the state of the field when it contains n photons of frequency associated with transitions between the atom s ground and first excited state. If we fail to monitor the electromagnetic field, then we must describe the atom by its reduced density operator A = a 0 (t) A;0ihA;0 + a 1 (t) A;1ihA;1. (10.5) This density operator shows that the atom is now in an impure state. In general, interactions between an experimental system and the wider environment mean that the state of the whole Universe rapidly becomes entangled. Since we don t keep track of all the details of the environment, sooner or later we re obliged to describe our experimental system by its reduced density operator, which will be impure. The tendency for subsystems to evolve from pure quantum states to impure states through interactions with the environment is known as quantum decoherence. Trying to isolate a system from the environment so as to prevent it from becoming impure is one of the main challenges to be overcome in building a practical quantum computer. 10. Entropy If our density operator has p m = for some state mi, with the remaining probability spread in some way among other states, then although our knowledge of the 130

6 system s state is imperfect, the e ects of the impurity are likely to be negligible. On the other hand, if the density operator has p n apple 10 0 for all states ni so that a tremendous number of states are equally probable, then our knowledge of the true quantum state of a system is very poor indeed. We would like to have a way to quantify how much knowledge, or information, about a state we have once the probability distribution {p i } has been specified. To achieve this, define the von Neumann entropy S( ) associated to a density operator by S( ) = tr H ( ln ). (10.6) Recall that for any operator Q with eigenstates { q j i} and corresponding eigenvalues {q j } we define the function f(q) byf(q) = P j f(q j) q j ihq j provided f(q j )exists. Thuswe have tr H ( ln ) = X! 0 1 X hn p i X ln(p j ) jihj A ni n i j (10.7) = X p n ln(p n ) n in terms of the probabilities appearing in the density operator. In this form, it s easy to see that S( ) 0withS( ) =0i describes a pure state, where only one of the p n s is non-zero (and hence equal to 1) as we have complete certainty about which state our system is in. Also easy to see is that, if dim(h) = N then the maximum value of S( ) islnn, which is attained i = I/N where I is the identity operator on H. Toseethis,usethemethod of Lagrange multipliers to impose the constraint tr H = 1 and vary S( ) tr H ( ) with respect to the probabilities and Lagrange multiplier. At an extremum, 0= tr ln = tr. (10.8) In the first line, we ve used the fact that tr( 1 )=tr( 1 ) inside the trace, so the order of the variation in the logarithm doesn t matter. These equations must hold for arbitrary variations and, so the first tells us that =e 1 I (10.9) so that in particular, at an extremum is proportional to the identity matrix. Taking the trace, the second equation fixes the constant of proportionality to be = I/N for an N-state system, so the entropy is maximised when all states are equally likely meaning we have no idea about which state our system is actually in. One can also easily show that S(U U)=S( ) for any unitary transformation U : H!H, which means that the entropy is invariant under changes in basis of the Hilbert space this is important, because as we ve seen there can be many di erent ways to present the same density operator. Perhaps the most important properties of entropy come from considering it s behaviour on subsystems. If H = H A H B then S( AB ) apple S( A )+S( B ) (10.30) 131

7 Figure 18: Strong subadditivity of the von Neumann entropy. where A and B are the reduced density matrices for the two subsystems. The equality is saturated if and only if the two subsystems are uncorrelated (unentangled) so that AB = A B. This property is known as subadditivity and you ll explore it further in the final problem sheet. It tells us that the entropy of a whole is no greater than the sum of entropies of its parts. It follows from subadditivity that if H = H A H B H C then S( ABC ) apple S( A )+S( BC ) apple S( A )+S( B )+S( C ) (10.31) but in fact something stronger is true: we have S( ABC )+S( B ) apple S( AB )+S( BC ) (10.3) which is known as strong subadditivity and was proved in 1973 by Elliott Lieb and Mary Beth Ruskai. To interpret it, we consider AB and BC are each subsystems of ABC, with AB \ BC = B. Strong subadditivity states that the entropy of the whole is no greater than the entropies of the overlapping subsystems AB and BC, even when the entropy of the overlap B is removed. One of the main uses of the density operator and von Neumann entropy is in Statistical Quantum Mechanics. As an example, suppose we wish to extremize the entropy subject to both tr = 1 and tr H = U, saying that we know our system has a fixed average energy U. Then using two Lagrange multipliers and, at an extremum we have 0= [S( ) (tr( ) 1) (tr( H) U)] = tr (ln +1+ H + ) (tr( ) 1) + (tr( H) U) (10.33) for all variations, and. Setting the coe cients of and to zero just enforces our constraints, while considering the terms involving gives =e Since tr = 1 we can fix e +1 = Z( )where H e 1. (10.3) Z( ):=tr H (e H ). (10.35) 13

8 defines the partition function of our system. Thus, in a state of maximum entropy for fixed average energy, the density operator takes the form = 1 Z( ) e H = 1 Z( ) X e n E n nihn, (10.36) This form of density operator is known as the Gibbs distribution, It plays a fundamental role in quantum statistical mechanics, as you ll see next term. is usually denoted 1/k B T where T is called the temperature and k B the Boltzmann constant. For fixed average energy U of the system, the temperature is determined by the constraint tr H ( H) =U. In other words, the temperature T is determined by the average energy of the system. You ll work much more with the density operator and entropy if you take the Part II Statistical Mechanics course next term Measurement in Quantum Mechanics The EPR Gedankenexperiment Einstein was never happy with the probabilistic nature of quantum mechanics. In the EPR thought experiment, an electron and a positron 71 are produced in a state with net spin 0, perhaps by the decay of some nucleus from a spin-0 excited state to a lower spin-0 state. The electron and positron travel in opposite directions, each carrying the same amount of momentum. At some distance from the decaying nucleus Alice detects the electron and measures the component of its spin in a direction a of her choice. Since electrons have spin 1, Alice inevitably discovers either +~/ or ~/. Meanwhile Bob, who is sitting at a similar distance on the opposite side of the nucleus, detects the positron and measures its spin in some direction b of his choice. We are free to choose the z-axis to be aligned with Alice s direction a. Since the electron positron system has combined spin zero, it must be in the state i = 1 p ( "i #i #i "i) (10.37) that entangles the separate spins of the electron and positron. According to the Copenhagen interpretation, when Alice measures +~/ for the electron spin, the system collapses into the state 0 i = "i #i. (10.38) Thus, whilst before Alice s measurement the amplitude for the positron to have spin +~/ along a was 1/, after she has measured the electron spin, there is no chance that the positron also has spin up along the same axis. The state of the positron corresponding to definitely having spin +~/ along the b-axis is " b i = cos e i / "i+sin e i / #i (10.39) 71 The positron is the antiparticle of the electron, predicted in the relativistic theory by Dirac s equation. It has the same mass and spin as an electron, but opposite sign electric charge. 133

9 as we found in the second problem sheet, where = cos 1 (a b). Given that after Alice s measurement the positron is certainly in state #i, it follows from (10.39) that the probability Bob measures spin up along b is h " b #i =sin. (10.0) In particular, there is only a small probability he would find spin-up along a direction closely aligned with Alice s choice a. We ve supposed that Alice measures first, but if the electron and positron are far apart when the measurements are made, a light signal sent to Bob by Alice when she makes her measurement would not have arrived at Bob by the time he makes his measurement, and vice versa. In these circumstances, relativity tells us that the order in which the measurements appear to be made depend on the velocity of the observer who is judging the matter. Consequently, for consistency the predictions of quantum mechanics must be independent of who is supposed to make the first measurement and thus collapse the state. This condition is satisfied by the above discussion, since the final probability depends only on a b and is thus symmetric between Alice and Bob. What bothered EPR is that after Alice s measurement there is a direction (a) along which Bob can never find +~/ for the positron s spin, and this direction depends on what exactly Alice chooses to measure. This fact seems to imply that the positron somehow knows what Alice measured for the electron, and the collapse of the entangled wavefunction 1 p ( "i #i #i "i)! " i # i apparently confirms this suspicion. Since relativity forbids news of Alice s work on the electron from influencing the positron at the time of Bob s measurement, EPR argued that the required information must have travelled out in the form of a hidden variable which was correlated at the time of the nuclear decay with a matching hidden variable in the electron. These hidden variables would then explain the probabilistic nature of quantum mechanics QM would contain no uncertainties once replaced by a better theory taking into account these hidden variables Bell s Inequality Remarkably, Bell was able to show that the predictions of any theory of hidden variables are in conflict with the predictions of quantum mechanics. Suppose we assume that the result of measuring the electron s spin in the a-direction is completely determined by the values taken by hidden variables in addition to a. That is, if we knew these variables, we could predict with certainty the result of measuring the component of the electron s spin along any direction a; Alice is only uncertain of the outcome because she does not yet know the values of the hidden variables. We take these variables to be the components of some n-dimensional vector v. Thus the result of measuring the electron s spin along a is a function e (a, v) that takes values ±~/ only. Similarly, the result of measuring the positron s spin along b is some function p (b, v). 13

10 We have e(a, v)+ p (a, v) = 0 (10.1) by conservation of angular momentum. Let s suppose that v has a probability distribution (v), such that the probability dp that v lies in the infinitesimal volume d n v is dp = (v) d n v. (10.) Then the expectation value of interest is Z h e (a, v) p (b, v)i = = (v) e (a, v) p (b, v) d n v Z (v) e (a, v) e (b, v) d n v. (10.3) Now suppose Bob sometimes measures the spin of the positron parallel to b 0 rather than b. The fact that p(b, v) = ~ allows us to write Z h e (a, v) p (b, v)i h e (a, v) p (b 0, v)i = (v) e (a, v)[ p(b, v) p(b 0, v)] d n v = Z apple (v) e (a, v) e (b, v) 1 ~ e(b, v) e (b 0, v) d n v. (10.) The expression [1 e (b, v) e (b 0, v)] is non negative, while the product e(a, v) e (b, v) fluctuates between ± ~. Hence we obtain the bound Z apple h e (a, v) p (b, v)i h e (a, v) p (b 0, v)i apple ~ (v) 1 ~ e(b, v) e (b 0, v) d n v = ~ h e (b, v) p (b 0, v)i (10.5) This is Bell s inequality. It must hold for any three unit vectors a, b and b 0 if the probabilistic nature of QM really comes from some underlying hidden variables. Bell s inequality can be tested experimentally as follows: for a large number of trials, Alice measures the electron s spine along a while Bob measures the positonr s spin along b in half these trials, and along b 0 in the other half. From the results of these trials, the lhs of (10.5) can be estimated. The rhs is estimated from a new series of trials in which Alice measures the electron s spin along b and Bob measures the positron s spin along b 0. If hidden variables exist, the results will obey Bell s inequality. We now show that quantum mechanics itself violates Bell s inequality. Since the random variables e(a) and p (b) can each take only the values ±~/ there are just four cases to consider. We find h e (a) p (b)i = ~ [P A(" a )P B (" b # a ) P A (" a )P B (# b # a ) P A (# a )P B (" b " a )+P A (# a )P B (# b " a )], (10.6) 135

11 where P B (" b " a ) is the probability Bob measures the positron s spin to be +~/ along b, given that Alice it s spin must be down along a as a result of the collapse of wavefunction after Alice s measurement. Since nothing is known about the orientation of the electron before Alice makes her measurement, quantum mechanics gives P A (" a )=P A (# a )= 1 (10.7) Above, we calculated that P B (" b " a )=sin and P B (# b " a ) = cos (10.8) and similarly for P B (# b # a ) and P B (" b # a ). Thus we have h e (a) p (b)i = ~ sin cos = ~ cos = ~ a b (10.9) according to quantum mechanics. Using this correlation in either side of Bell s inequality we find LHS = ~ a (b b0 ) whereas RHS = ~ (1 b b0 ) (10.50) In particular, if a b = 0 and b 0 = b cos + a sin we find LHS = ~ ~ sin whereas RHS = (1 cos ) (10.51) and it is easy to see that Bell s inequality is violated for all 6= 0, /. The predictions of quantum mechanics are thus inconsistent with the existence of hidden variables. Experiments 7 have been carried out to determine whether Nature herself obeys Bell s inequalities or violates them and quantum mechanics triumphs. 7 Freedman, S. & Clauser, J., Experimental Test of Local Hidden Variable Theories, Phys.Rev.Lett. 8, 938 (197). Freedman & Clauser s experiment used a modified version of Bell s inequalities using the polarization states of photons, rather than the spins of electrons. See also Aspect, A. & Roger, G. Experimental Realization of the Einstein Podolski Rosen Bohm Gedankenexperiment: A New Violation of Bell s Inequalities, Phys.Rev.Lett.9, 91(198). 136