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A simple model of quantum trajectories

Todd A. Bruna)

Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540

Received 18 June 2001; accepted 6 March 2002

Quantum trajectory theory, developed largely in the quantum optics community to describe open

quantum systems subjected to continuous monitoring, has applications in many areas of quantum

physics. I present a simple model, using two-level quantum systems q-bits, to illustrate the

essential physics of quantum trajectories and how different monitoring schemes correspond to

different unravelings of a mixed state master equation. I also comment briefly on the relationshipof the theory to the consistent histories formalism and to spontaneous collapse models. 2002

American Association of Physics Teachers.

DOI: 10.1119/1.1475328

I. INTRODUCTION

Over the last ten years the theory of quantum trajectorieshas been developed by many authors110 for a variety ofpurposes, including the ability to model continuously moni-tored open systems,1,3,4 improved numerical calculations,2,8

and for insight into the problem of quantum measurement.5 7

However, outside the small communities of quantum optical

theory and quantum foundations, the theory remains littleknown and poorly understood.The association with quantum foundations has convinced

many observers that quantum trajectory theory is differentfrom standard quantum mechanics, and therefore is to beregarded with deep suspicion.11 Although it is true that thetypes of stochastic Schrodinger and master equations treatedin quantum trajectories are sometimes postulated in alterna-tive quantum theories,1215 these same equations arise quitenaturally in describing quantum systems interacting with en-vironmentsopen systems16which are subjected to monitor-ing by measuring devices. In these systems, the stochasticequations arise aseffectiveevolution equations, and are in nosense anything other than standard quantum mechanics.

The fact that this similarity is not widely appreciated, evenin fields that might usefully employ quantum trajectories, is agreat pity. I attribute it to confusion about exactly what thetheory is, and how these equations arise.

To combat this confusion I present a simple model forquantum trajectories, using only two-level quantum systemsquantum bits or q-bits. Because of the recent interest inquantum information, a great deal of attention has been paidto the dynamics and measurement of two-level systems. Iwill draw on this background to build my model and showhow effective stochastic evolution equations can arise from asystem interacting with a continuously monitored environ-ment.

This paper is intended as a tutorial, giving the physical

basis of quantum trajectories without the complications aris-ing from realistic systems and environments. As an aid inunderstanding or teaching the material, I have included anumber of problems. For the most part these involve deriva-tions that arise in developing the model, but that are notcrucial for following the main argument. The solutions areincluded in the Appendix.

A. The model

For simplicity, I will consider the simplest possible quan-tum system: a single two-level atom or q-bit. The environ-ment will also consist of q-bits. While this system is far

easier to analyze than the more complicated systems used inmost actual quantum trajectory simulations, it can illustratealmost all of the phenomena that occur in more realisticcases.

I will assume that the environment bits interact with thesystem bit successively for finite, disjoint intervals. I willfurther assume that these bits are all initially in the samestate, have no correlations with each other, and do not inter-

act among themselves see Fig. 1. Therefore all we need toconsider is a succession of interactions between two q-bits.After each environment bit interacts with the system, we canmeasure it in any way we like, and use the information ob-tained to update our knowledge of the state.

This model of system and environment may seem exces-sively abstract, but in fact a number of experimental systemscan be approximated in this way. For instance, in experi-ments with ions in traps, residual molecules of gas can oc-casionally pass close to the ion, perturbing its internal state.17

In cavity quantum electrodynamics QED experiments, thesystem might be an electromagnetic mode inside a high- Q

cavity. If there is never more than a single photon present,the mode is reasonably approximated as a two-level system.

This mode can be probed by sending Rydberg atoms throughthe cavity one at a time in a superposition of two neighboringelectronic states, and measuring the atoms electronic statesupon their emergence. In this case, the atoms are serving asboth an environment for the cavity mode and a measurementprobe.18 Similarly, in an optical microcavity the externalelectromagnetic field can be thought of as a series of incom-ing waves which are reflected by the cavity walls; photonsleaking out from the cavity can move these modes from thevacuum to the first excited state1 see Fig. 2. In solid statephysics, the spin of a single electron on a quantum dot is aperfect two-level system; it might be probed by coupling to asecond dot, which is monitored by a single-electrontransistor.19 Other systems might also suggest themselves for

this kind of analysis.

B. Plan of this paper

In Sec. II, I describe systems of q-bits, and the possibletwo-bit interactions they can undergo. In Sec. III, I examinepossible kinds of measurements, including positive operatorvalued and weak measurements. I present in Sec. IV themodel of the system and environment, and in Sec. V I showhow it is possible to describe an effective evolution for thesystem alone by tracing out the environment degrees of free-dom, producing a kind of master equation.

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In Sec. VI, the effects on the system state of a measure-ment on the environment are considered, and the stochasticevolution equations for the system state are deduced, condi-tioned on the random outcome of the environment measure-ments. I examine in Secs. VII and VIII different interactionsand measurements, and how they lead to different effectiveevolution equations for pure statesstochastic Schrodingerequations. I show how under some circumstances the systemappears to evolve by large discontinuous jumps, while forother models the state undergoes a slow diffusion in Hilbertspace. I further show that by averaging over the differentmeasurement outcomes we recover the master equation evo-

lution of Sec. V.In Sec. IX, I consider measurement schemes that give only

partial information about the system. These schemes lead tostochastic master equations rather than stochastic Schro-dinger equations. Finally, in Sec. X I relax the assumption ofmeasurement, and discuss how quantum trajectories can beused to yield insight into the nature of measurement itself,using the decoherent consistent histories formulation ofquantum mechanics. I also briefly consider alternative quan-tum theories that make use of stochastic equations. In Sec.XI I summarize the paper and draw conclusions. I present thesolution to the problems in the Appendix.

II. TWO-LEVEL SYSTEMS AND THEIR

INTERACTIONS

The simplest possible quantum mechanical system is atwo-level q-bit, which has a two-dimensional Hilbert space

H2 . There are many physical embodiments of such a system,

including the spin of a spin-1/2 particle, the polarizationstates of a photon, two hyperfine states of a trapped atom orion, two neighboring levels of a Rydberg atom, and the pres-ence or absence of a photon in a microcavity. All of thesehave been proposed in various schemes for quantum infor-mation and quantum computation, and many have been usedin actual experiments. For a good general source on quan-

tum computation and information, see Nielsen and Chuang.20

The references also include papers on experimentalproposals19,21 and results.17,22

By convention, we choose a particular basis and label itsbasis states0and 1, which we define to be the eigenstatesof the Pauli spin matrix z with eigenvalues 1. We simi-

larly define the other Pauli operators x, y ; linear combi-

nations of these, together with the identity 1, are sufficient to

produce any operator on a single q-bit.The most general pure state of a q-bit is

01, 221. 1

A global phase may be assigned arbitrarily, so that all physi-cally distinct pure states of a single q-bit form a two-parameter space. A useful parametrization is in terms of twoangular variables and,

cos/2ei/20sin/2e i/21 , 2

where 0 and 02. These two parameters de-fine a point on the Bloch sphere. The north and south poles

of the sphere represent the eigenstates ofz , and the eigen-

states of x and y lie on the equator. Orthogonal statesalways lie opposite each other on the sphere.

If we allow states to be mixed, we represent a q-bit by adensity matrix ; the most general density matrix can bewritten as

p 1p , 3

where and are two orthogonal pure states, 0. The mixed states of a q-bit form a three parameter fam-ily:

Fig. 1. Schematic diagram of the type of model used in

this paper. The system is a single two-level system, or

q-bit. It interacts briefly with a series of environment

q-bits which pass by with average time intervals oft,

and the environment bits are subsequently measured.

Fig. 2. A particular physical system of which the model of this paper is an idealization. The system is an electromagnetic field mode in a cavity with a partially

silvered mirror. On average, there is less than a single photon in the mode at a time, so only the lowest two occupation levels contribute significantly to the

dynamics. The external electromagnetic field is described in terms of incoming and outgoing wave packets of width t, each of which reflects on the outside

of the partially silvered mirror. The incoming packets are all in the vacuum state, but upon reflection they may absorb a photon emitted from the cavity.

720 720Am. J. Phys., Vol. 70, No. 7, July 2002 Todd A. Brun


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1r2

cos2/21r

2 sin2/2 0 0

1r2

sin2/21r

2 cos2/2 11

rcos/2sin/2 e i01ei10 , 4

whereand are the same angular parameters as before and

0r1. The limit r1 is the set of pure states, param-

etrized as in Eq. 2, while r0 is the completely mixed

state 1/2. Thus we can think of the Bloch sphere as hav-ing pure states on its surface and mixed states in its interior;

and the distance rfrom the center is a measure of the states

purity. It is simply related to the parameter p in Eq. 3: p

(1r)/2, 1p(1r)/2.

For two q-bits, the Hilbert space H2H2 has a tensor-product basis

0A 0B 00AB, 5a

0A 1B 01AB, 5b

1A 0B 10AB, 5c

1A

1B 11AB. 5dSimilarly, for N q -bits we can define a basis

iN1iN2i0 , ik0,1. A useful labeling of these 2N ba-

sis vectors is by the integers 0 j2N whose binary expres-

sions are iN1i 0 :

j iN1i0, jk0

N1

i k2k. 6

All states evolve according to the Schrodinger equation

with Hamiltonian H( t),

d

dt

i

H t . 7

Over a finite time this evolution is equivalent to applying a

unitary operator U to the state ,

UT:exp it0tf

dt H t :, 8where T:: indicates that the integral should be taken in atime-ordered sense, with early to late times being composedfrom right to left. For the models I consider in this paper Iwill treat all time evolution at the level of unitary transfor-mations rather than explicitly solving the Schrodinger equa-tion, so time can be treated as a discrete variable

nU nU n1U 10 . 9

For a mixed state, Schrodinger time evolution is equivalent

to UU . Henceforth, I will also assume 1.

If the unitary operator U n is weak, that is, close to the

identity, one can always find a Hamiltonian operator H n such

that

U nexpiH nt 1 iH nt. 10

Thus, one can easily recover the Schrodinger equation froma description in terms of unitary operators,

nnn1 U n 1n1

iH nn1t. 11

Problem 1. Show that the most general unitary operator ona single q-bit can be written

U 1cosin" sinexpin" , 12

modulo an irrelevant overall phase, where nnx,ny,nz isa three-vector with unit norm and x,y ,z . In the

Bloch sphere picture this transformation is a rotation by anangle about the axis defined by n, with three real param-etersfour if one includes the global phase.

For two q-bits there is unfortunately no such simple visu-alization as the Bloch sphere. However, to specify any uni-tary transformation it suffices to give its effect on a completeset of basis vectors. I will consider only a fairly limited set oftwo-bit transformations, and no transformations involvingmore than two q-bits, but the simple formalism I derivereadily generalizes to higher-dimensional systems.

Let us examine a couple of examples of two-bit transfor-mations. The controlled-NOT gate or CNOT is widely usedin quantum computation; applied to the tensor-product basisvectors it gives

U CNOT0000 , 13a

U CNOT0101 , 13b

U CNOT1011 , 13c

U CNOT1110 . 13d

If the first bit is in state 0, this gate leaves the second bitunchanged; if the first bit is in state 1, the second bit isflipped 01 . Hence the name: whether a NOT gate isperformed on the second bit is controlledby the first bit. In

terms of single-bit operators, U CNOT 0 0 111x .

Another important gate in quantum computation is theSWAP; applied to the tensor-product basis vectors it gives

U SWAP0000 , 14a

U SWAP0110 , 14b

U SWAP1001 , 14c

U SWAP1111 . 14d

As the name suggests, the SWAP gate just exchanges the

states of the two bits: U

SWAP(

)

.CNOT and SWAP are examples of two-bit quantum gates.Such gates are of tremendous importance in the theory ofquantum computation. These two gates in particular have an

additional useful property. Note that the operator U CNOT

U CNOT , that is, it is both unitary and Hermitian, as is also

true of U SWAP . This is only possible if all of the operators

eigenvalues are 1, and implies that these operators are their

own inverses, U CNOT2

U SWAP2

1. Among single-bit opera-

tors, the Pauli matrices x ,y ,z also have this property, as does

any operator n" .



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The property of being both unitary and Hermitian makes itpossible to define one-parameter families of two-bit unitarytransformations:

U CNOTexpiU CNOT 1cos i U CNOT sin ,15

and similarly

U SWAPexp iU SWAP 1cos iU SWAP sin .16

Problem 2. Show that any operator U

that is both unitaryand Hermitian satisfies exp(iU)1cos iU sin .

These families range from the identity 1 for 0 to the

full CNOT or SWAP gate for/2, up to a global phase.

For small values 1, this is a weak interaction, whichleaves the state only slightly altered.

These families do have an undesirable characteristic, how-

ever. Suppose we apply U CNOT() to a two-bit state of the

form (01) . The new state is

U CNOT01)

ei0cos 1)

isin 1 x ). 17The relative phase e i between0 and1 is a complicationin the calculations that will follow. To avoid this problem,

instead ofU CNOT() and U SWAP(), we will use interactions

of the form ZA()U CNOT(), where ZA()(exp(iz/2)

1), and similarly for U SWAP . The single-bit rotation ex-

actly undoes the extra relative phase produced by U CNOT(),while changing nothing else.

III. STRONG AND WEAK MEASUREMENTS

A. Projective measurements

In the standard description of quantum mechanics, observ-

ables are identified with Hermitian operators OO . A mea-

surement returns an eigenvalue on of O and leaves the sys-

tem in the corresponding eigenstate n , Ononn,with probability p nn

2. If a particular eigenvalue is

degenerate, we instead use the projector Pn onto the eigens-

pace with eigenvalue o n; the probability of the measurement

outcome is then p nPn and the system is left in the

state Pn/p n. For a mixed state the probability of out-

comen ispnTrPn, and the state after the measurement

is PnPn/p n.Because two observables with the same eigenspaces are

completely equivalent to each other as far as measurementprobabilities and outcomes are concerned, we will not worry

about the exact choice of Hermitian operator O; instead, we

will choose a complete set of orthogonal projections Pnthat represent the possible measurement outcomes. Thesesatisfy

PnPnP

nnn,

nPn 1. 18

A set of projection operators that obey Eq. 18 is often re-ferred to as an orthogonal decomposition of the identity. Fora single q-bit, the only nontrivial measurements have exactlytwo outcomes, which we label and , with probabilities

p and p and associated projectors of the form

P1n"

2 . 19

Equation 19 is equivalent to choosing an axis n on theBloch sphere and projecting the state onto one of the two

opposite points. All such operators are projections onto purestates. The two projectors sum to the identity operator, P

P 1. The average information obtained from a projec-

tive measurement on a q-bit is just the Shannon entropy forthe two measurement outcomes,

Smeasplog2pplog2p . 20

The maximum information gain is precisely one bit, when

pp1/2, and the minimum is zero bits when either por p is 0. After the measurement, the state is left in an

eigenstate ofP , so repeating the measurement will result

in the same outcome. This repeatability is one of the mostimportant features of projective measurements.

B. Positive operator valued and weak measurements

While projective measurements are the most familiar fromintroductory quantum mechanics, there is a more general no-tion of measurement, the positive operator valued measure-mentPOVM.23 Instead of giving a set of projectors which

sum to the identity, we give a set ofpositive operators Enwhich sum to the identity:

n

En 1. 21

The probability of outcome n is pnEn , or for a

mixed state p nTrEn . Unlike a projective measurement,

knowing the operators En is notsufficient to determine the

state of the system after measurement. One must further

know a set of operators Ank such that

Enk

Ank

Ank. 22

After measurement outcome n the state is

1/p nk

AnkAnk . 23

This measurement will not preserve the purity of states, in

general, unless there is only a single Ankfor each En that is,

EnAnAn, in which caseAn/p n.

Because the positive operators En need not be projectors,

one is not limited to only two possible outcomes; indeed,there can be an unlimited number of possible outcomes.However, if a POVM is repeated, the same result will notnecessarily be obtained the second time. Projective measure-ments are clearly a special case of POVMs in which theresultsare repeatable. Most actual experiments do not corre-spond to projective measurements, but are described by somemore general POVM.



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Conversely, it is easy to show that any POVM can beperformed in principle by allowing the system that is to bemeasured to interact with an additional system, or ancilla,and then doing a projective measurement on the ancilla. Inthis viewpoint, only some of the information obtained comesfrom the system; part can also come from the ancilla, whichinjects extra randomness into the result. This mathematicalidentification should not be pushed too far, however; in prac-tice, trying to interpret a POVM as a projective measurementon an ancilla often adds complexity without improving un-derstanding.

A particularly interesting kind of POVM for the presentpurpose is the weak measurement.24 This is a measurementthat gives very little information about the system on aver-age, but also disturbs the state very little.

Loosely speaking, there are two ways a measurement canbe considered weak. Suppose we have a q-bit in a state ofform 1, and we perform a POVM with the following twooperators:

E00 0 111A02, 24a

E111A12, 24b

A00 0111, 24c

A11 1, 24d

where 1. Clearly E0 and E1 are positive and E0E1

1, so they constitute a POVM. The probability p 0

E012 of outcome 0 is close to 1, while

p 1E12 is very unlikely. Thus, most such

measurements will give outcome 0, and very little informa-tion is obtained about the system.

Problem 3. Show that the average information gain fromthis measurement in bits is

Smeas2 1/ln 2log2

21. 25

The state changes very slightly after a measurement out-come 0, with 0 becoming slightly more likely relative to1. The new state is

0A0/p 0011)/p 0. 26

However, after an outcome of 1, the state can change dra-matically: a measurement outcome of 1 leaves the q-bit inthe state1. We see that this type of measurement is weak inthat it usuallygives little information and has little effect onthe state, but on rare occasions it can give a great deal ofinformation and have a large effect.

By contrast, consider the following positive operators:

E0 12 00 12 1 1A02, 27a

E1 12 001

2 1 1A12, 27b

A012 001

2 11, 27c

A112 001

2 11. 27d

These operators also constitute a POVM. Both E0 andE1 are

close to 1/2, and so are almost equally likely for all states,

p 0p 11/2; the information acquired is approximately onebit. But the state of the system is almost unchanged for both

outcomes, with the new state beingjAj/pj,

01

1 2 2 1011)

12 0 121 , 28a

11

1 2 2 1011)

12 0 121 , 28b

for outcomes 0 and 1, respectively. For this type of weakmeasurement, the measurement outcome is almost random,but includes a tiny amount of information about the state. Byperforming repeated weak measurements and examining thestatistics of the results, one can in effect perform a strongmeasurement; for the particular case considered here, thestate will tend to drift toward either 0 or 1 with overallprobabilities2 and2.

We will see below specific instances of how a POVM canarise by the interaction of the measured system with an an-cilla, and the subsequent projective measurement of the an-cilla. Quantum trajectories are closely related to such indirectmeasurement schemes.

IV. SYSTEM AND ENVIRONMENT

In discussing quantum evolution it is usually assumed thatthe quantum system is very well isolated from the rest of theworld. This idealization is useful, but it is rarely realized inpractice, even in the laboratory. In fact, most systems interactat least weakly with external degrees of freedom.25,26

One way of taking this interaction into account is to in-

clude a model of these external degrees of freedom in ourdescription. Let us assume that in addition to the system in

state in Hilbert space HSwith Hamiltonian H S , there isan external environmentin state EHE with Hamiltonian

HE. The joint Hilbert space of the system and environment

is the tensor product space HSHE, and the full Hamil-

tonian can be written

HH S 1E 1SHEHI, 29

whereHIis a Hamiltonian giving the interaction between the

two. The system and environment evolve according to theSchrodinger equation, Eq. 7, with this full Hamiltonian.

Because of the presence of the HI

term, in general even

initial product states E will evolve into states thatcannot be written as product states:

Ej

j Ej. 30

Such states are called entangled. If the system and environ-ment are entangled, it is no longer possible to attribute aunique pure state to the system or environment alone. Thebest that can be done is to describe the system as being in a

mixed state S, which is the partial trace of the joint state

over the environment degrees of freedom:



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STrenv j,j ,k

jje kEjEjek

j ,j

jjEjEj . 31

The ek are some complete set of orthonormal basis vec-tors on the environment Hilbert space HE.

The entanglement of a state can be quantified. The mostwidely used measure for pure states is the entropy of en-tanglement

. For a system and environment in a joint purestate, this is

SETrSlog2 S , 32

which is the von Neumann entropy of the reduced density

operator S . This entropy is maximized by the density op-

erator S 1S/ Trsys 1S; such a density operator is calledmaximally mixed, and the joint pure state that gives rise to itis called maximally entangled.

Problem 4. Show that the von Neumann entropy is maxi-

mized for 1/Tr 1 .Systems that interact with their environments are said to

be open. Most real physical environments are extremelycomplicated, and the interactions between systems and envi-

ronments are often poorly understood. In analyzing open sys-tems, one often makes the approximation of assuming asimple, analytically solvable form for the environment de-grees of freedom.

For the models considered, I will assume that both thesystem and the environment consist solely of q-bits. I willalso assume a simple form of interaction, namely that thesystem q-bit interacts with one environment q-bit at a time,and after interacting they never come into contact again. Ialso assume that the environment q-bits have no Hamiltonian

of their own, HE0. This environment may seem ridicu-

lously oversimplified, but in fact it suffices to demonstratemost of the physics exhibited by much more realistic de-

scriptions. In fact, the description in terms of q-bits is not abad schematic picture of many environments at low energies.I summarize one such possible environment in Fig. 2.

For this type of model, the Hilbert space of the system is

HSH2 and the Hilbert space of the environment is HEH2H2 . I will in general assume that all the envi-

ronment q-bits start in some pure initial state, usually 0;however, in later elaborations of the model, I will relax thisassumption to include both other pure states and mixed-stateenvironments, which resemble heat baths of finite tempera-ture.

V. THE MASTER EQUATION

A system interacting with an environment usually cannotbe described in terms of a pure state. Similarly, it is notusually possible to give a simple time evolution for the sys-tem alone, without reference to the state of the environment.However, under certain circumstances one can find an effec-tive evolution equation for the system alone, if the interac-tion between the system and environment has a simple formand the initial state of the environment has certain properties.

In cases where such an effective evolution equation exists,it will not be of the form in Eq. 7. Instead, the evolutionequation will be completely positive, which is a weaker con-dition than unitarity. It must take density matrices to density

matrices, but not necessarily pure states to pure states. To betractable, such an equation must also generally be Markov-ian: it must give the evolution of the density operator solelyin terms of its state at the present time, with no retardedterms. The most general such Markovian equation27 is a mas-ter equation in the Lindblad form

d

dt

i

H S,

kLkLk

1/2Lk

Lk

1/2LkLk, 33

where H S is the Hamiltonian for the system alone and the

Lindblad operatorsLk give the effects of interaction withthe environment.

What about the kind of discrete time evolution weve beenassuming? Suppose that a system and environment in an ini-

tial product state undergo some unitary transformation U:

EU E). 34

How does the state of the system alone change? We can write

any operator on HSHEas a sum of product operators

U

j A

j

B

j. 35

The state of the system after the unitary transformation is

STrenvU U

j ,j

TrenvAjAj

BjEEBj

j ,j

AjAj

EBj

BjE . 36

The self-adjoint matrix Mj jEBj

BjE has a set of or-

thonormal eigenvectors kk j with real eigenvalues ksuch that

j

Mj jk jkk jMj jk

kk jk j* . 37

We define new operators O k by

O k kj

k jAj. 38

The expression in Eq. 36 then simplifies to

STrenvU U

kO kO k

. 39

This expression is quite like the outcome of a POVM, butwithout determining a particular measurement result; the sys-tem is left in a mixture of all possible outcomes.

Problem 5. Show that the unitarity ofU implies

k

O k

O k 1S. 40

Suppose now that the system interacts with a second en-vironment q-bit in the same way, with the second bit in the

same initial state E , and that the Hilbert space of the twoenvironment bits is traced out. The system state will become



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S k1, k2

O k2O

k1O k1

O k2

. 41

After interacting successively with n environment bits, thesystem state is

S(n)

k1, . . . ,kn

O knO k1

O k1O kn

. 42

This evolution is a type of discrete master equation. We seeclearly that an initially pure state will in general evolve into

a mixed state. Depending on the O k, it may or may notconverge to a unique final state.

Lets consider the special case when the unitary operator

Uis close to the identity,

Uexpij

AjBj , 43where1, TrAj

AjTrBj

Bj O(1) , and theAj andBj

are Hermitian. By expanding to second order in , we seethat the new density matrix for the system is

STrenvUU

ij

Aj, EBjE

2

2j j

EBjBjE

2AjAjAj

Aj AjAj .

44

Lets make the simplifying assumption that the first-orderterm vanishes:

j

AjEBjE0. 45

Later in the paper we supplement U CNOT() and U SWAP()with one-bit gates on the system to remove this first-orderterm, which simplifies the evolution.

Let the duration of the interaction be t. We can define a

matrix Mj j ,

Mj jEBj

BjE, 46

with orthonormal eigenvectors kk j and real eigenval-ues k just as in Eq. 37, and define operators

Lk2 k

t

jk jAj, 47

similarly to Eq. 38 above. In terms of these operators, Eq.44 can be rewritten as

t

kLkLk

1/2Lk

Lk 1/2Lk

Lk , 48

which has exactly the same form as Eq. 33 with a vanish-ing Hamiltonian. If the first-order term of Eq. 44 does notvanish, Eq. 48 can pick up an effective system Hamil-tonian, though some care must be exercised about the size of

the various terms and their order in t. But for the simple

case we are considering, we can recover a continuous evolu-

tion equation as a limit of successive brief, infinitesimaltransformations, just as we recovered the Schrodinger equa-tion from a succession of weak unitary transformations.

If we wanted to express the operators O k of Eq. 38 in

terms of the Lk for this infinitesimal transformation, then to

second order in , we would find O kLkt, plus one ad-

ditional operator O 0 1(1/2) kL kLkt. If we substitute

these operators in Eq. 42, we reproduce Eq. 48.

VI. ENVIRONMENTAL MEASUREMENTS ANDCONDITIONAL EVOLUTION

Suppose that the system q-bit is in initial state 01 and it interacts with an environment q-bit in state0,so that a CNOT is performed from the system onto the en-vironment bit. The new joint state of the two q-bits is

0S 0E1S 1E. 49

If we trace out the environment bit, the system is in themixed state

S 20 021 1. 50

Now suppose that we make a projective measurement on the

environment bit in the usual z basis. With probability p 02 (p 1

2), we find the bit in state0 1, in whichcase the system bit will alsobe projected into state 0( 1).If the system interacts in the same way with more environ-

ment bits, which are also subsequently measured in the z

basis, the same result will be obtained each time with prob-ability 1. This scheme acts just like a projective measurementon the system.

Similar schemes are not always equivalent to projectivemeasurements. Suppose that instead of a CNOT, a SWAPgate is performed on the two q-bits. In this case the new jointstate after the interaction is

0S 0E1E). 51

A subsequent z measurement on the environment yields 0 or

1 with the same probabilities as before, but now in bothcases the system is left in state 0. Further interactions andmeasurements will always produce the result 0. Clearly, thechoice of measurement on the environment bit makes nodifference to the state of the systemit will be 0 for anymeasurement result, or for no measurement at all. Note that

U CNOT can produce entanglement between two initially un-

entangled q-bits, while U SWAP cannot.In the case of the CNOT interaction, what happens if we

vary our choice of measurement? Suppose that instead of

measuring the environment bit in the z basis, we measure it

in the x basis x( 01)/&? In terms of this basis,we can rewrite the joint state after the interaction as

1

&01) x

1

&01) x .

52

The and results are equally likely; the result leavesthe system state unchanged, while the flips the relativesign between the 0 and 1 terms. Aside from knowingwhether a flip has occurred, this measurement result yieldsno information about the system state. If this state is initiallyunknown, it will remain unknown after the interaction and



8/19

measurement. By changing the choice of measurement, onecan go from learning the exact state of the system to learningnothing about it whatsoever.

One important thing to note is that, whatever measurementis performed and whatever outcome is obtained, the systemand environment bit are afterwards left in a product state. Noentanglement remains. Thus, the environment bit can be fur-ther manipulated in any way, or discarded completely, with-out any further effect on the state of the system. Because ofthis, it is unnecessary to keep track of the environments stateto describe the system. This approximation is somewhat un-

realistic and arises from the assumption of projective mea-surements; in real experiments, some residual entanglementwith the environment almost certainly exists which the ex-perimenter is unable to measure or control. It does, however,enormously simplify the description of the systems evolu-tion. I will later relax this assumption, but for now I makeuse of it.

VII. STOCHASTIC SCHRO DINGER EQUATIONS

WITH JUMPS

Far more complicated dynamics result if we replace the

strong two-bit gates of Sec. VI with weak interactions, suchas U CNOT() or U SWAP() for 1. In this case the mea-surements on the environment bits correspond to weak mea-surementson the system, and the system state evolves unpre-dictably according to the outcome of the measurements.Weak interaction with the environment is the norm for mostlaboratory systems currently studied; indeed, it is only theweakness of this interaction that justifies the division intosystem and environment in the first place, and enables one totreat the system as isolated to the first approximation, withthe effects of the environment as a perturbation.

This problem is simple enough that we can analyze it for ageneral weak interaction. Suppose that the system and envi-

ronment q-bits are initially in the stateS 0

E. Let

the interaction be a two q-bit unitary transformation of theform

Uexpij

AjBj , 53where 1, and TrAj

AjTrBj

BjO(1), so that we

can expand the exponential in powers of. The new state ofthe system and environment is

US 0Eij

AjSBj0E

2

2 j j A

jA

jS

B

jB

j0E

O

3

. 54

Just as in Sec. V, we define the matrix Mj j0BjB

j0 with eigenvectors k and real eigenvalues

k, and use it to find a new set of operators

Lk2 k

t

jk jAj, 55

where t is the time between interactions with successiveenvironment q-bits. Because we can decompose the matrix

Mj j in terms of two vectors

Mj j0Bj

0 0Bj00Bj11Bj0

ujujvj*vj, 56

it has at most two nonvanishing eigenvalues. We can makethe additional simplifying assumption, as in Sec. V, that

j

Aj0Bj0j

Ajuj0. 57

This assumption leaves only a single nonvanishing Lindblad

operator,

L2

t

jAj1Bj0 , 58

and no effective Hamiltonian term. In terms of L, the jointstate of the system and environment is

U 1 1/2LLtS 0E it LS 1E

O3. 59

The time dependence ofL/tmay seem a bit strangeat first. It is chosen so that the master equation, Eq. 48,properly approximates a time derivative. This definition

does, however, imply a somewhat odd scaling. The strengthof the interaction between the system and environment q-bits

can be parametrized by /t. If the two bits interacted foronly half as long, the Lindblad operators would have to be

redefined, LL/&. In the limit t0, the right-hand side

of Eq. 48 would vanish, and the system would not evolve.This limit is an example of the so-called quantum Zenoeffect.28 Most such master equations have some microscopictime scale built in, for instance, a characteristic collisiontime, and are not really valid at shorter time scales. TheMarkov assumption also generally fails for very short times.These equations are used to describe the system at times longcompared to this short scale, which in the case of our model

would be after the system had interacted with a number ofenvironment q-bits.Once the interaction has taken place, we measure the en-

vironment q-bit in the 0, 1 basis. Using the assumptionin Eq. 57, the probabilities p 0,1 of the outcomes are

p 012

j j

AjAj 0Bj1 1B

j0

1LLt1p 1 . 60

The change in the systemstate after a measurement outcome0 on the environment is

12LLLL t. 61

Because the result 0 is highly probable, as the system inter-

acts with successive environment bits in initial state 0E,most of the time the system state will evolve according to thenonlinearand nonunitary continuous equation 61.

Occasionally, however, a measurement result of 1 will beobtained. In this case, the state of the system can changedramatically:

L

LL. 62



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Because these changes are large but rare, they are usuallyreferred to as quantum jumps.

These two different evolutionscontinuous and determin-isticafter a measurement result 0or discontinuous and ran-domafter a measurement result 1can be combined into asingle stochastic Schrodinger equation:

1

2LLLLt

L

L

L

1

N,

63

where N is a stochastic variablewhich is usually 0, but has

a probability p 1LLt of being 1 during a given

time step t. The values ofNobviously represent measure-

ment outcomes; when the environment is measured to be in

state0, the variable is N0; when the measurement out-come is 1, N1. Equation63 combines the two kinds ofsystem evolution into a single equation. Most of the time the

N term vanishes, and the system state evolves according to

the deterministic nonlinear equation 61; however, whenN1, the Nterms completely dominate, and the system

state changes abruptly according to Eq. 62. Generically, Eq.63 can also include a term for Hamiltonian evolution, butthis is eliminated by the assumption in Eq. 57.

We can summarize the behavior of the stochastic variableby giving equations for its ensemble mean,

N2N, MNLLt, 64

where M represents the ensemble mean over all possible

measurement outcomes, and is the quantum expectationvalue in the state .

This analysis is rather abstract. Lets work out an examplewith a particular choice of two-bit interaction. Suppose thatthe system interacts with a succession of environment q-bits,

via the transformation ZS()U CNOT(), where ZS()

exp(iz/2)S 1E and 1; after each interaction the

environment bits are measured in the z basis. If the environ-

ment bits all begin in state0, and the system is in state 1after interacting with the first environment q-bit, the two bitswill be in the joint state

00cos 10isin 11 , 65

modulo an overall phase. This state is entangled, with en-

tropy of entanglement SE22 log2(

22). If the

environment is then measured in the z basis, it will be found

in state 0 with probability p 022 cos2 1

22 and in state1 only with the very small probabilityp 1

22. After a result of 0, the system will then be in the

new state

00cos 1)/p 0

122/20 122/21 , 66

where we have kept terms up to second order in . Theamplitude of0 increases relative to 1. On the other hand,if the outcome 1 occurs, the system will be left in state 1,unaltered by further interactions and measurements. We seethat this combination of interactions and measurements isequivalent to a weak measurement on the system of the firsttype discussed in Sec. III B. This result bears out the fact that

all POVMs are equivalent to an interaction with an ancillarysystem followed by a projective measurement, the well-known Neumarks theorem.23

Suppose that the system interacts successively with n en-

vironment q-bits, each of which is afterwards measured to be

in state 0. The state of the system is nn0n1 ,where

n

n 1

2

2

n

expn2/2

, 67

which, together with the normalization condition n2

n21, implies that

n2

2

2 2 expn2, 68a

n2

2 expn2

2 2 expn2. 68b

We see that n21, which is what one would expect, be-

cause after many measurement outcomes without a single 1result, one would estimate that the1component of the statemust be very small. The probability of measuring 0 at the n thstep, conditioned on having observed 0s at all previous

steps, is p0(n)1 n2

2

1.Problem 6. Show that the probability of getting the result 0

at every step asn , which leads to the system asymptoti-cally evolving into the state0, is

n0

p 0 n 2. 69

The probability of at some time getting a result 1 and

leaving the system in state 1 is therefore 122.The effect, after many weak measurements, is exactly the

same as a single strong measurement in the z basis, withexactly the same probabilities. This evolution is described byEq. 63 with the Lindblad operator

L2/t1 1. 70

In this case, Eq. 63 simplifies to

2

2 1 112

1 11 1 N, 71where MN21222p 1 . Equation 63

may seem unnecessarily complicated. After all, L is Hermit-

ian, so the distinction between L andL is unimportant; also,

L is proportional to the projector 1 1, which simplifiesEq.63considerably. However, many other systems exist in

which L is not so simple, and which still obey Eq. 63.We can readily find such an alternative system. Suppose

that U SWAP() is used instead of U CNOT() in the two-bitinteraction. The system and environment bits after the inter-action are in the state

00cos 10isin 01. 72

Note that unlike the strong swap gate U SWAP , the weak

U SWAP() canproduce entanglement.From Eq.72 we see



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that just as in the CNOT case, a measurement of 0 on theenvironment leaves the system in a slightly altered statein

fact, exactly the same state that is produced by a 0 result inthe CNOT case. The probabilities p0 and p1 for the two

outcomes are identical to those for the CNOT case, p 01

p 11 22. However, unlike the CNOT case, a mea-

surement outcome 1 will put the system into the state 0rather than 1.

This process is quite like spontaneous decay. Suppose that0 is the ground state and1 is the excited state. Initially, thesystem is in a superposition of these two energy levels. Ateach time step, there is a chance for the system to emit aquantum of energy into the environment. If it does, the sys-tem drops immediately into the ground state 0, while the

environment goes into the excited state1; if not, we reviseour estimate of the system state, making it more probable

that the system is unexcited. This model is similar to quan-tum jumps in quantum optics, in which a photodetector out-

side a high-Q cavity clicks if a photon escapes, and gives no

output if it sees no photon.1,3,4 A measurement result of 1 onthe environment corresponds to the photodetector click.

This model obeys exactly the same stochastic Schrodinger

equation 63, but with the Lindblad operator L

2/t01. Because this operator is notHermitian, the

distinction between L and L is important in this case. We

compare these two stochastic evolutions in Figs. 3a and

3b. In both cases, the coefficient 2 decays steadily away;

Fig. 3. Plot of the squared amplitude

2 for the system to be in state1vsthe time for a variety of quantum jump

trajectories for different initial states.

aassumes an interaction U CNOT with

the environment, while b assumes

U SWAP . In both cases, 2 decays to-

ward zero. Some trajectories just de-

cay smoothly to 0, while others un-

dergo a jump. In a this jump is to 1,while in b they jump to 0, as in spon-

taneous emission. Note that the decays

of 2 are not simple exponentialsdue to the nonlinearity of the trajec-

tory equations. As 20, however,the decay approaches an exponential.



11/19

however, in some trajectories the state suddenly jumps eitherto1 or 0, while others continue to decay smoothly toward0.

What happens if we dont actually measure the environ-ment bits? In this case the system q-bit evolves into a mixedstate which is the average over all possible measurement

outcomes. If we denote by the n outcomes after n steps of

the evolution, p() the probability of those outcomes, and

the state conditioned on those outcomes that is, thesolution of the appropriate stochastic Schrodinger equation,

then the system state will be a density operator

(n)

p. 73

If we start with a state , allow it to interact with exactly oneenvironment q-bit, and average over the possible outcomes 0and 1 with their correct probabilities, we will find that the

new state is

k0

1

AkAk , 74

where for U CNOT() ,

A

0

0 0

cos 11, 75aA1sin 11. 75b

In the limit of small 1, this evolution gives an approxi-mate Lindblad master equation

t LL

1

2L L

1

2LL. 76

The Lindblad operatorLis the same as in the jump equation

for the CNOT interaction, Eq. 70. Similarly, for the SWAPinteraction, the evolution equations are identical, but with

A1sin 01 and L2/t01.

We should note that unlike the stochastic trajectory evolu-

tion, the evolution of the density matrix is perfectly deter-ministic, with no sign of jumps or other discontinuities. Thisis generally true; the jumps appear only if measurements areperformed.

Believe it or not, the models analyzed in this section havealready given almost all essential properties of quantum tra-jectories. Quantum trajectory equations arise when a systeminteracts with an environment which is subsequently mea-sured; the equations give the evolution of the system stateconditioned on the measurement outcome. They take theform of stochastic nonlinear differential equations; the sto-chasticity arises due to the randomness of the measurementoutcomes, and the nonlinearity due to the renormalization ofthe state. Averaging over all possible measurement outcomes

recovers the deterministic evolution of the system densityoperator. The next two sections discuss elaborations of thisbasic scheme, and Sec. X discusses how these results fit intothe foundations of quantum mechanics.

VIII. STOCHASTIC SCHRO DINGER EQUATIONS

WITH DIFFUSION

In discussing weak measurements we noted that some usu-ally produced only minor effects, but occasionally caused alarge change in the state, while others changed the state littleregardless of the outcome. The former case is exactly that of

trajectories with jumps: the state usually changes slowly andcontinuously, but occasionally makes a large jump. The lattercase, then, should correspond to states that evolve by smallbut unpredictable changesin other words, by diffusionrather than jumps.

A. Random unitary diffusion

We can easily demonstrate such evolution using the modelweve already constructed. Suppose that the system begins ina state of Eq. 1, and interacts with a succession of environ-

ment bits initially in state 0 by the unitary transformationZS()U CNOT() for 1. However, rather than measuring

the environment bits in thez basis, we measure them in the x

basis,

x 0 1)/&, x 01)/&. 77

In terms of this basis, after an interaction the system andenvironment bit are in the state

1

&0ei1)x

1

&

0ei1)x . 78

We see that the two measurement outcomes and areequally likely, and produce only a small effect on the systemstate: namely, if an outcome occurs, the relative phase

between1and 0is rotated by , and if a occurs, it isrotated by . This measurement scheme, then, has the ef-fect of performing a random unitary transformation on thesystem.

The stochastic Schrodinger equation for this model is

i LW, 79

where L2/t11 is the same as in the earlier case ofCNOT jumps, and W is a stochastic variable taking the

values Wtwith equal probability. We express thisstochastic behavior by giving the ensemble means

MW0, W2t. 80

This particular normalization of the stochastic variable isuseful because it scales properly with time. If, instead ofusing interactions with single environment q-bits in intervals

t, we let the system interact with n bits over an interval

tnt, then Eq. 79 remains completely unchanged, ex-

cept for t being replaced by t in Eqs. 79 and 80.What if we average overthesemeasurement outcomes? In

this case, the density matrix evolution is still given by Eq.74, with operators

A01

& 00ei1 1, 81a

A11

& 00ei1 1. 81b

This transformation looks quite different from the evolutiongiven by Eq. 75, but it is actually identical, as it must be:the same system state must be obtained by tracing out theenvironment bit, regardless of the basis in which we chooseto carry out the trace. At the level of density matrices, the



12/19

nonunitary quantum jumps and the unitary diffusion are ex-actly the same.

Problem 7. Show explicitly that the new density matrix

k

AkAk 82

obtained using the operators defined in Eq. 81 is the sameas that using the operators defined in Eq. 75.

The equivalence of these two evolutions is an example ofa single master equation having different unravelings, that is,

different stochastic evolutions 63 and 79 which give thesame evolution on average. Any set of measurements doneon the environment must give the same averageevolution ofthe system state, and therefore gives an unraveling of thesame master equation. Of course, some unravelings may givesimpler versions of the stochastic system state evolution thanothers. For some systems, it may be much easier to solve thestochastic evolution than the master equation; one can thenaverage over many different stochastic evolutions to approxi-mate the master equation evolution. This numerical methodis called the Monte Carlo wave function technique.2 Even incases where one has no direct access to information about theenvironment, it may be worthwhile to invent a fictitious setof measurements that would give a simple stochastic evolu-

tion, purely as a numerical technique to solve the masterequation. In this case, however, one should not ascribe anyfundamental significance to the individual solutions of thestochastic trajectory equations.

B. Nonunitary diffusion

We can get diffusive behavior from this system in a dif-ferent way. Suppose once again that the interaction is

ZS()U CNOT() and the environment is measured in the zbasis, but this time let the environment q-bits be initially inthe state

y 0i 1)/&. 83

In this case, the joint state after the interaction is

1

&0 cos sin 1) 0

i

&0 cos sin 1) 1 . 84

For small 1, the two measurement outcomes are almost

equally likely,p 0,1(1/2)2, and the conditioned states

of the system are, to second order in ,

0 12342/20 1 2

222/2342/21 , 85a

1 12342/20 1 2

222/2342/21 . 85b

This stochastic evolution is essentially the same as the sec-ond weak measurement scheme in Eq. 27 in Sec. III B. Ifthe system is initially in either state 0 or 1, it is un-changed, but all other states will diffuse along the Blochsphere until they eventually reach either 0 or1. Once thestate is within a small neighborhood of 0 or 1, it is un-likely to diffuse away again. Because of this, this scheme canalso be thought of as an indirect measurement of0 vs 1;

but in this case, the measurement result can only be deter-mined by gathering a very large number of outcomes. If,after the system has interacted with many environment bits,we find that the number of 1 outcomes slightly exceeds thenumber of 0 outcomes, we would conclude that the system isin state1; if not, we conclude that the system is in state 0.

This scheme is somewhat analogous to homodyne mea-surement in quantum optics.1,29 In homodyne detection, theweak signal from the system is coherently mixed with astrong local oscillator in a beam splitter; the two ports are fedinto separate photodetectors. The output from each detector

is almost entirely due to the local oscillator field. Becausethis field is so much stronger than the signal from the system,almost all of the photons which arrive at the photodetectorscome from the local oscillator, and very little informationabout the system is obtained in any one detector click. Togain information about the system, one looks at the smalldifference between the photocurrents from the two detectorsover the course of many clicks. Similarly in our model, onecan deduce information about the system state over thecourse of many events from the small excess of 1 resultsover 0 results. The analogy to homodyne measurement is notas exact as the analogy between the quantum jump equationand spontaneous emission, but it is still suggestive.

For this nonunitary diffusion, the stochastic Schrodingerequation is different than in the earlier quantum jump equa-tion 63. Expanding the solution to second order in , theequation is

12 3L22LLLLt

LLW, 86

where once again L2/t11, and just as in the ran-dom unitary case of Sec. VIII A the stochastic variable W

takes the values t with probabilities p 0,1 . Because inthis case the measurement outcomes are not equally prob-

able, this stochastic variable W does nothave a vanishing

mean:

MW2Lt, W2t. 87

We can simplify Eq. 87 by replacing this stochastic vari-able by one that does have a vanishing mean to second orderin. Define

ZWMW , 88a

MZ0, Z2tOt3/2. 88b

If we use this new stochastic variable, Eq. 87 becomes

LL 12L2

12L

Lt

LLZ. 89

Equation 89 is the quantum state diffusion equation withreal noise.5 Solutions of Eq. 89 are plotted in Fig. 4.

If we average over the measurement outcomes to give amixed-state evolution for the system, the new density matrixis given by Eq. 74 with operators

A01

& 00 cos sin 11, 90a

A11

& 00 cos sin 11. 90b



13/19

Problem 8. Using the operators in Eq. 90, show that the

density operator evolution is exactly the same as that givenby operators 75 and 81, despite the fact that the environ-ments initial state is quite different in this case, and that thequantum trajectory evolution is also very different. We cansee from this equivalence that the system evolution by itselfis not sufficient to uniquely determine the nature of the en-vironment. A particular master equation can result frommany different environments and interactions.

IX. INCOMPLETE INFORMATION AND

STOCHASTIC MASTER EQUATIONS

So far, we have only considered idealized models in whichboth the system and environment are in a pure state, and

measurements are done that leave them pure. In this case, theexperimenter has complete information about the state of thesystem and environment at all times. It is possible to relaxthis assumption in a numbers of ways. We may have onlypartial information about the state of the system, so that it isdescribed by an initial density matrix rather than a pure state.We may have only partial information about the environ-ment; for instance, it might be in an initial thermal state. Orthe measurements we perform on the environment may,themselves, give only partial information. In a realistic ex-periment, of course, all three of these may be true. We willconsider each of them in turn.

A. Trajectories for a mixed system state

A pure state represents the maximum information that onecan possess about a system. If a system is in a pure state,there is some measurement that could be performed on thesystem that would have a definite outcome. If we have onlypartial information about a system, we describe it by a mixedstate. If a system is in a mixed state, there is no measurementthat has a definite outcome. Systems can be mixed becausewe lack information about how they were prepared, or be-cause they are entangled with other systems, or both. Thevon Neumann entropy is the most useful measure of howmixed a state is; it vanishes for a pure state, and is maxi-mized by the maximally mixed state.

For simplicity, lets suppose that we have no information

at all about the state of the system; it is in the maximallymixed state 1/2, and has von Neumann entropy S1. Welet it interact with an environment q-bit in some initial state,and measure the environment bit in some basis afterwards.For these measurement schemes, the state of the system be-comes

iAiAi/p i, 91

with probability

p iTrAiAi . 92

If the interaction is ZS()U CNOT(), as in the first quantum

jump equation of Sec. VII, the environment bits are initiallyin state 0, and are afterwards measured in the z basis, theoperators 91 are given by Eq. 75. Just as for the purestate, an outcome 1 leaves the system in the state 1, whilean outcome 0 leaves the state only slightly changed:

0 1

2 12/200 12/21 1,

93ap 01

2/2

1 1 1, p 12/2. 93b

In either case, the state is no longer maximally mixed.Some information has been acquired about the system. Wecan assess how much information this is by calculating theaverage von Neumann entropy S

of the state after the mea-

surement:

S1S1i

p iTrilog2 i , 94

which in the case of our jump trajectories is S

2/ (2 ln 2). How much information does the measurementgenerate? It is given by the Shannon entropy of Eq. 20 ofthe measurement,

Smeas2/2log2

2/2. 95

Fig. 4. Plot of 2 vs time for thediffusive equation 89, with several

different initial states and realizations

of the stochastic term. We see that

when the trajectories approach 0 or 1

they tend to remain there, while in be-

tween they diffuse freely.



14/19

Note that in Ref. 30, Soklakov and Schack call S the aver-

age conditional entropy and denote it H, and call Smeas the

average preparation information and denote it I.

We see that SmeasS, which of course makes sense; wecan never obtain more information on average about the sys-tem than the measurement produces. If we look at the ratioof the two, though,

Smeas

S 1ln 2/2, 96

we see that as 20, the information produced by the mea-surement is much, much higher than our gain in knowledgeabout the system. Thus, most of the information we receiveis just random noise, telling us not about the state of thesystem, but about the measurement process itself.

If, instead of the jump description 75 given by measur-ing the environment in the z basis, we use the unitary diffu-sive description given by Eq.81which arises from measur-ing the environment in the x basis, the situation is very

different. Because A0 and A1 are in this case proportional to

unitary operators, the initial maximally mixed state 1/2 isleft completely unchanged by this trajectory. Absolutely no

information is gained about the system and S0. This lack

of information is not special to the maximally mixed state.Because the two measurement outcomes are equally likely,

p 0p 11/2, each measurement result represents Smeas1

bit of information, but the result tells one nothing about thestate of the system; it is all random noise.

Consider now the second diffusive case described in Sec.

VIII B, in which the environment bits begin in the stateyand, after interacting with the system, are measured in the z

basis. In this case, with evolution operators given by Eq.90, an initially maximally mixed state is left in one of thestates

0,1 1

2 100 111, 97

with probabilities

p 0,1 1

2 1. 98

Problem 9. Show that, once again, most of the informationin this measurement outcome is random noise, with the ratio

S/Smeas0 as 0.

B. Trajectories for a mixed environment

Suppose that, rather than the system being initially mixed,

the environment bits are in a mixed state E. For simplicity,lets choose the diagonal state

Ew 000w 111, 99

with w 0w 11. Let an initially pure system in state 1

interact with the environment bit via ZS()U CNOT(). For

small1, if we measure the environment in the z basis, astraightforward but tedious calculation shows that the systemwill be left in one of the states

0,1 122w 1,0/w 0,1

22w 1,0/w 0,111, 100

with probabilities

p 0,1w 0,122 w 1w 0 , 101

where

122/20 122/21 . 102

These states 0,1 are both mixed; the uncertainty in the stateof the environment has led to uncertainty in the state of thesystem. The system evolution is perturbed by noise from theenvironment.

This evolution cannot be described by a stochastic Schro-dinger equation; rather, one must use a stochastic master

equation, which gives the evolution of a density matrix con-ditioned on the measurement outcomes. The master equationtakes the form

SS1

2LLLL ,S w 0p0 1F

w 1

p 1F t

LSLLLS w 1p 0 1F

w 0

p1Ft.

103

Here A,BABBA is the anticommutator, and O

TrO . The stochastic variable F takes the value 0 withprobability p 0 and 1 with probability p 1 . We see then that

Eq.103reduces to Eq. 76in the mean, and hence gives apartialunraveling of that equation in terms of mixed states.

When w 10, this master equation exhibits jump-like behav-

ior; when w 1w 0 , it is diffusive. Ifw0w 11/2, then Eq.103 reduces to the mean equation 76, because the mea-surements yield no information whatsoever about the system.Note that this neednt be true for all possible interactions.

For instance, if we use U SWAP() instead ofU CNOT(), it ispossible to obtain information about the system even using amaximally mixed environment, as we see below.

If the system interacts with n environment bits, each of

which is subsequently measured in the z basis giving a mea-surement record , the system will be left in the state

pnn 1p11, 104

where p depends on the exact sequence of measurement

results, andn is given by Eq. 68 and obeysn0 asn . The state will tend to evolve toward either 00 or11 unless w 0w 11/2. So we see that even when astate becomes mixed, under some circumstances it canevolve toward a pure state again.

If instead of U CNOT(), the interaction were U SWAP(),

the noise would affect the system in a different way. In thiscase the system will be left in one of the states

01

p0 w 000w 1

2sin2 1 1 , 105a

11

p1 w 111w 0

2sin2 0 0 , 105b

where



15/19

00cos 1 , 106a

1cos 0e2i1 , 106b

and

p 0w 0cos2 2sin2 1p 1 . 107

In this case, rather than evolving eventually toward one ofthe pure states 0, 1, the system will evolve toward thestate w 00 0w 11 1, that is, it will become identicalwith the mixed state of the environment bits. This evolution

is quite similar to the process of thermalization, by which asystem interacting with a heat bath evolves toward equilib-rium.

C. Trajectories with generalized measurements

Now we will assume that both the system and environ-ment q-bits begin in a pure state, but that the measurementperformed on the environment bit is no longer a projectivemeasurement, but instead a POVM that provides only partialinformation about the state of the environment bit.

Let us consider a system where the system and environ-

ment q-bits interact via ZS()U CNOT(), the environmentbits start initially in the state 0, and the system is in state

01 . After the system and environment interact, theenvironment is measured with a POVM using operators

E0q00 1q 11A02, 108a

E1 1q 0 0q11A12, 108b

A0q001q1 1, 108c

A11q0 0q1 1. 108d

Assume that 1/2q1. A measurement result 0 then favorsthe environment q-bit being in state0, and a result 1 favors1, but neither result is conclusive.

After the system and environment interact, the probabili-

ties of the two outcomes are

p 0q 2 2 cos2 1q 2 sin2 1p 1 .

109

If we assume 1, tracing out the environment leaves thesystem in one of the mixed states

0 1 1q 22/q

1q 22/q 1 1, 110a

1 1q22/ 1q

q22/ 1q 1 1, 110b

with the updated state

1 22/20 122/21 . 111

That is, the system state remains entangled with the environ-ment q-bit.

This environmental interaction and measurement leavesthe system in a state of the form

Sw 1w 11. 112

If a state in that form interacts with another environment biton which the same POVM is performed, the system state will

remain in the same form. After n interactions, the state will

have the form 104, withn0 as n , and the state

tends toward either 00 or 11 at long times. Thus, thesystem evolution looks quite similar to the case when theenvironment is initially in a mixed state and a projectivemeasurement is performed. However, in the case of a mixedenvironment the system has no entanglement with the envi-ronment, and is in a mixed state due to noise; in the case ofincomplete measurements, the system is entangled with theenvironment, and the system and environment as a wholeremain in a pure state.

In this case as well, one can describe the evolution of thesystem by a stochastic master equation, which has exactly

the same form as Eq. 103 with w 0,1 replaced by q and 1q. Unlike the previous case, however, the system state is

not mixed because of classical noise entering from the envi-ronment, but rather because it remains entangled with theenvironment.

The POVM just considered represents a measurement withimperfect discrimination. Lets consider a different way thata measurement may be incomplete. Consider the followingpositive operators:

E0q0 0A02, 113a

E1q1 1A12, 113b

E2 1q 1A22, 113c

A0q00, 113d

A1q11, 113e

A21q 1. 113f

These operators Ei sum up to the identity, and for 0 q1

form a POVM. E0 and E1 are proportional to orthogonal

projectors; E2 , by contrast, leaves the state completely un-

changed. This POVM is like a measurement made with a

detector of efficiency q; there is a probability 1q that the

detector will fail to register anything at all, and hence give

no information about the state.Suppose the system q-bit is originally in a state of form

Sw 0w 111, 114

where an initial pure state would have w 01. Then a mea-

surement result of 0 has probability

p 0q w 0 22 cos2 w 1cos

2

q 1 1w 022 , 115

and leaves the system in a mixed state of the same form

0w 0 1w 122w 1 1w 0

22 1 1.116

The result 1 has probability

p 1q w 02 sin2 w 1sin

2 q 1w 02 2,

117

and leaves the system in the state 111; the result orrather, nonresult 2 has probability p 21q and leaves thesystem in the mixed state

2w 0 122 w 1w0

221 1.118

If the evolution began in a pure state, it will remain in a purestate unless outcome 2 occurs; but once this happens, the



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state will tend to remain mixed until either a result 1 occursin which case the system will be in state 1 henceforth or,in the limit of large n , the system approaches the state 0.The measurement process works much as before, but the null

results slow the convergence; if the efficiencyq is low, it cantake far longer for the state to settle down to either0or 1.

X. QUANTUM TRAJECTORIES, CONSISTENT

HISTORIES, AND COLLAPSE MODELS

As is clear from the previous sections, the formalism ofquantum trajectories calls on nothing more than standardquantum mechanics, and as framed above is in no way analternative theory or interpretation. Everything can be de-scribed solely in terms of measurements and unitary trans-formations, the building blocks of the usual Copenhagen in-terpretation.

However, many people have expressed dissatisfaction withthe standard interpretation over the years, usually due to therole of measurement as a fundamental concept in the theory.Measuring devices are large, complicated things, very farfrom elementary objects. What exactly constitutes a mea-surement is never defined, and the use of classical mechanicsto describe the states of measurement devices is not justified.

Presumably the individual atoms, electrons, photons, etc.,that make up a detector can themselves be described byquantum mechanics. If this reasoning is carried to its logicalconclusion, however, and a Schrodinger equation is con-structed for the measurement process, one obtains not clas-sical behavior, but rather macroscopic superpositions such asthe famous Schrodingers cat paradox.31

Three major approaches have been followed to tackle thisproblem. In the first, hidden variables are introduced thatdetermine the outcome of measurements, so that quantummechanics is a partial description of an underlying determin-istic theory. Unfortunately, as John Bell showed,32 such ahidden variables theory such as that of de Broglie andBohm33 must be nonlocal to give the same predictions as

quantum mechanics.The second approach is to modify quantum mechanics to

get rid of the unwanted macroscopic superpositions, whileretaining the usual quantum results on small scales. Theoriesof this nature have been proposed by Pearle,12 Ghirardi,Rimini, and Weber,13 Diosi,6 Gisin5 and Percival,14 andPenrose,15 among others; these theories are commonly calledcollapse models. These alternative theories usually replacethe Schrodinger equation with a new, stochastic equation,which includes a mechanism to produce wave function col-lapse onto some preferred basis. Such stochastic Schrodingerequations often strongly resemble those produced by a suit-able quantum trajectory description, in which the system in-teracts continuously with an environment which is repeatedly

measured. Although I do not know if all such alternativetheories are equivalent to a quantum trajectory description,certainly a large class of them is.

Because these theories are not equivalent to standardquantum mechanics, in principle they can be distinguishedexperimentally. Unfortunately, most such theories require ex-perimental sensitivities well beyond current technology, al-though Percival14,34 has made interesting proposals of pos-sible near-term tests.

Theoretically there are also problems in producing relativ-istic versions of such theories. Most such models producecollapse onto localized variables, such as position states.

Such localization, however, does not in general commutewith the Hamiltonian, and so such theories almost alwaysviolate energy conservationsometimes spectacularly so.35

Another class of models produces a collapse onto states ofdefinite energy,6,15,14 rather than localized variables. It is notclear, however, that such collapse models eliminate macro-scopic superpositions in either the short or long term. Cer-tainly the end result, a universe in an energy eigenstateastatic, unchanging state with no dynamicsdoes not greatlyresemble the world that we perceive, though in the context ofan expanding universe even this is unclear.36

The last approach is to retain the usual quantum theory,but to eliminate measurement as a fundamental concept,finding some other interpretation for the predicted probabili-ties. Although there are many interpretations that follow thisapproach, the one that is most closely tied to quantum tra-jectories is the decoherentorconsistenthistories formalismof Griffiths,37 Omnes,38 and Gell-Mann and Hartle.39 In thisformalism, probabilities are assigned to histories of eventsrather than measurement outcomes at a single time. Thesecan be grouped into sets of mutually exclusive, exhaustivehistories whose probabilities sum to 1. However, not all his-tories can be assigned probabilities under this interpretation;only histories which lie in sets that are consistent, that is,whose histories do not exhibit interference with each other,and hence obey the usual classical probability sum rules.

Each set is basically a choice of description for the quan-tum system. For the models considered in this paper, thequantum trajectories correspond to histories in such a consis-tent set. The probabilities of the histories in the set exactlyequal the probabilities of the measurement outcomes corre-sponding to a given trajectory. This equivalence has beenshown between quantum trajectories and consistent sets forcertain more realistic systems, as well.40,41

For a given quantum system, there can be multiple consis-tent descriptions which are incompatible with each other;that is, unlike in classical physics, these descriptions cannotbe combined into a single, more finely grained description.In quantum trajectories, different unravelings of the sameevolution correspond to such incompatible descriptions. Inboth cases, this is an example of the complementarity ofquantum mechanics.

We see, then, that although quantum trajectories can bestraightforwardly defined in terms of standard quantumtheory when the environment is subjected to repeated mea-surements, even in the absence of such measurements thereis an interpretation of the trajectories in terms of decoherenthistories. Because the consistency conditions guarantee thatthe probability sum rules are obeyed, one can therefore usequantum trajectories as a calculational tool even in caseswhere no actual measurements take place.

XI. CONCLUSIONS

In this paper I have presented a simple model of a systemand environment consisting solely of quantum bits, using nomore than single-bit measurements and one- and two-bit uni-tary transformations. The simplicity of this model makes itparticularly suitable for demonstrating the properties ofquantum trajectories. We see that these require no more thanthe usual quantum formalism, though they can also be ap-plied in cases that go beyond standard quantum mechanics,such as collapse models or the decoherent histories interpre-tation.



17/19

Quantum trajectories often can simplify the description ofan open quantum system in terms of a stochastically evolv-ing pure state rather than a density matrix. Although for theq-bit models of this paper, there is no great advantage indoing so, for more complicated systems this can often makea tremendous difference.8

In this model it is also possible to straightforwardly quan-tify the flow of information between system and environ-ment, in the form of entanglement and noise, as well as theinformation acquired in measurements and the decrease inentropy of the system. For realistic systems and environ-

ments this may be more difficult to do analytically, but onecan still form a qualitative picture of the information flow indecoherence and measurement.

I hope that this model will clear up much of the confusionthat currently surrounds the theory of quantum trajectories.Quantum trajectories are a very useful formal and numericaltool, and should find applications well outside their currentniches in quantum optics and measurement theory.

ACKNOWLEDGMENTS

The idea for this paper arose during a very productive visit

to the University of Oregon, and it has been encouraged bythe kindness of many people. I would particularly like tothank Steve Adler, Howard Carmichael, Oliver Cohen, LajosDiosi, Nicolas Gisin, Bob Griffiths, Jonathan Halliwell, JimHartle, Jens Jensen, Ian Percival, Martin Plenio, RudigerSchack, Artur Scherer, Andrei Soklakov, Tim Spiller, andDieter Zeh, for their comments, feedback, and criticisms ofthe ideas behind this paper. This work was supported in partby NSF Grant No. PHY-9900755, by DOE Grant No. DE-FG02-90ER40542, and by the Martin A. and Helen Chool-jian Membership in Natural Sciences, IAS.

APPENDIX: SOLUTIONS TO PROBLEMS

Problem 1. Any unitary operator can be written as U

exp(iH), where His an Hermitian operator. Any operator

on a single q-bit can be written in terms of the basis 1 , x ,

y , z :

Hc01c1xc 2yc3z . 119

For HH to be Hermitian, the c i must all be real. Choose

a convenient parametrization c 1,2,3nx,y ,z , where nx2ny

2

nz21. The expression for U is then

Uexp ic01in" exp ic01exp in" , 120

where the c0 term factors out because 1 commutes with ev-

erything else. This factor just contributes a global phasechange, and hence can be neglected. The second exponentialcan be expanded in a power series to yield

exp in" m0

in" m/m!. 121

Note now that

n" 2 nx2ny

2nz

21nxnyxyyx

nynzyzzy nznxzxxz

122a

1, 122b

where we have used the fact that x , y , z anticommute

and thatx2y

2z

2 1. If we substitute this result into the

power series, we obtain

U m0

i2m n" 2m/ 2m !

n" m0

i2m1 n" 2m/ 2m1 ! 123a

1 m0

i2m/ 2m ! n" m0

i2m1/ 2m1 !

1cosi n" sin, 123b

which was the result to be shown.Problem 2. The proof is the same as in Problem 1. Simply

expand the exponential in powers of its argument

UexpiU m0

iU m/m!. 124

Because U2U U 1, all the odd powers will be propor-

tional to Uand the even powers to 1. If we collect the even

and odd powers separately, we find the result

U 1cos iU sin , 125

just as in Problem 1.Problem 3. The average information gain from a measure-

ment is given by the Shannon entropy in Eq. 20,

Smeasp 0log2p 0p 1log2p 1 . 126

In this case, the relevant probabilities are p012 and

p 12. If we rewrite log2xlnx/ln 2, we can use the

expansion ln(1)2/23/3 to get

Smeas2 1/ln 2log2

2 . 127

Problem 4. Because is an Hermitian operator, wecan always find an orthonormal basis that diagonalizes it:

1 0 0 0

0 2 0 0

0 0 3 0

0 0 0 n, 128

where the i are all real. The positivity of requires that i0, and Tr 1 implies that i1. By putting theseresults together, we find that the von Neumann entropy be-comes

Trlog2 i1

n

ilog2 i. 129



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The right-hand side of Eq.129is exactly the expression fora Shannon entropy, as in Eq. 20. For an n-outcome mea-surement, the Shannon entropy has a single global maximum

at 12 n1/n . If we substitute these values intothe matrix expression for above, we find that 1/n1/Tr1.

Problem 5. We have

k

O k

O kj jk

kk jk j* A

j

Aj

j j

Mj jAj

Aj

j j

Aj

AjEBj

BjE

j j

TrenvAj

AjBj

Bj 1S EE

Trenv j

Aj

B

j

j

AjBj

1S EETrenvU

U 1S EE

Trenv 1S EE 1S.

Problem 6. A very direct way of showing this result is to

use the simple approximation 12exp(2) for 1.This approximation gives:

n0

p0 n n0

1n2

2

n0

2 122 expn2

22 expn2

n0

22 exp n1 2

22 expn2

limn

22 exp n1 2

22 2.

130

Problem 7. Using the definition Eq. 81, we see that

k

AkAk

1

2 00ei1 1 00ei1 1

00ei1 1

0 0ei11. 131

By rearranging and combining terms, we obtain

k

AkAk

1

2 00 cos isin 1 1

0 0 cos isin 11

0 0 cos isin 11

0 0 cos isin 11

00cos 1 1 0 0cos 1 1

sin2 1 11 1, 132

where the last equality is just A0A0A1A1

using the defi-

nition in Eq. 75.Problem 8. The procedure here is the same as in Problem

7. We write the new state in terms of the operators 90 andrearrange the terms to get the operators 75,

k

AkAk

1

2 0 0 cos sin 11

00 cos sin 11

0 0 cos sin 1 1

00 cos sin 11

00cos 1 1 0 0cos 1 1

sin 11 sin 1 1. 133

Of course, the evolutions given by Eqs. 75 and 81 havealready been shown to be equivalent, so the equivalence ofEqs. 90 and81 follows at once.

Problem 9. In this measurement, the system goes from the

maximally mixed state 1/2 to one of the two states 0,1given by Eq. 97. The von Neumann entropy of the maxi-mally mixed state is

Tr 12log2

12 12log2

12

12log2

121. 134

Because the states 0,1 are diagonal, it is straightforward toevaluate their von Neumann entropy as well; we find it to be

Tr0,1 log2 0,1p 0log2p 0p1log2p 1Smeas ,135

where p 0,1(1)/2. So we see that

Smeas112log2 1

12log2 1

12/ 2 ln 2 , 136

which implies that S2/ (2 ln 2) and S/Smeas0 as

0, as required. The entropy Smeas of the measurement is

nearly a full bit, but only a tiny part of it represents informa-

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