Subjective and Objective Probabilities in Quantum Mechanics

0.15 MB·English

by  Mark Srednicki

Subjective and Objective Probabilities in Quantum Mechanics
∗
Mark Srednicki
Department of Physics, University of California, Santa Barbara, CA 93106 USA
We discuss how the apparently objective probabilities predicted by quantum mechanics can be
treated in the framework of Bayesian probability theory, in which all probabilities are subjective.
Our results are in accord with earlier work by Caves, Fuchs, and Schack, but our approach and
emphasisaredifferent. Wealsodiscusstheproblemofchoosinganoninformativepriorforadensity
matrix.
I. INTRODUCTION laws of nature) tell us that we must assign a probability
|ψ(x,t)|2dx to the statement “at time t, the particle is
5 between x and x+dx”. Different people do not appear
Probability plays a central role throughout human af-
0 to have a choice about this assignment. In this sense,
fairs, and so everyone has an intuitive idea of what it
0 quantum probability appears to be objective.
2 is. Moreover, because of the extreme generality and
Thegoalofthispaperistounderstandthehowtheap-
widespread use of the concept of probability, it cannot
n be easily defined in terms of anything more basic. For parentlyobjectiveprobababilitiesofquantummechanics
a
J example, the dictionary that I have in my office, Web- canbe fitintothe Bayesianframework,whichallowsdif-
ster’s Ninth New Collegiate, says that probability is “the ferent people to make different probability assignments.
4
1 state or quality of being probable”; that to be probable This issue has been addressed before by Caves, Fuchs,
is to be “supported by evidence strong enough to estab- and Schack [1], and our results are in broad agreement

2 lish presumption but not proof”; and that presumption with theirs. However, we emphasize a somewhat differ-
v is“theground,reason,orevidencelendingprobabilityto ent approachto certain issues that we will explain as we
9 a belief”. This is clearly unhelpful to anyone who does go along.
0
not already know what probability is. In section II, in order to fix the notation and key con-
0
cepts, we briefly review the axioms and basic theorems
1 Inmathematics andphysics,we are oftenfacedwith a
of probability theory. In section III, we introduce the
0 concept that is both simple enough to be clearly under-
5 stood, and fundamental enough to resist definition; for notion of a probability of a probability, and explain how
0 example, a straight line in euclidean geometry. To make it can be applied to experimental data to turn an origi-
/ nallysubjectiveprobabilityintoanincreasinglyobjective
h progress,wedonotattemptto deviseeverclearerdefini-
one, in the sense that all but strongly biased observers
p tions, but instead formulate axioms that our understood
- but undefined objects are postulated to obey. Then, us- agree with the final probability assignment. In section
t
n ing codified rules of logical inference, we prove theorems IV, we apply this formalismto the probabilitiesof quan-
a that follow from the axioms. tum mechanics. In section V, we discuss when and why
u it is preferable to assign probabilities to possible density
It is instructive to treat probability as one of these
q
matrices for a quantum system, rather than assigning
: primitive concepts. Dispensing, then, with any attempt
v a particular density matrix. In section VI, we discuss
at definition, we say that the probability that a state-
i theconstructionofnoninformativepriordistributionsfor
X ment is true is a real number between zero and one. A
density matrices. We summarizeandconcludein section
statement may be true or false; if we know it to be true,
r
VII.
a we assign it a probability of one, and if we know it to
be false, we assign it a probability of zero. If we do not
knowwhetheritistrueorfalse,weassignitaprobability
between zero and one. II. THE AXIOMS OF PROBABILITY
There is typically no definitive way to make this as-
signment. Different people could (and often do) assign
The statements to which we may assign probabilities
different numerical values to the probability that some
must obey a logical calculus. Some key definitions (in
particular statement (“the stock price of Microsoft will
which “iff” is short for “if and only if”):
be higher one year from now”) is true. In this sense,
S = a statement.
probability is subjective. This point of view is Bayesian.
Ω= a statement known to be true.
Probabilityalsoentersquantummechanics,inaseem-
∅= a statement known to be false.
ingly more fundamental way. For example, given a
S = a statement that is true iff S is false.
wave function ψ(x,t) for a particle in one dimension,
S ∨S = a statement that is true iff either S or S
the rules of quantum mechanics (which are apparently 1 2 1 2
is true.
S ∧S = a statement that is true iff both S and S
1 2 1 2
are true.
∗Electronicaddress: [email protected] S1 and S2 are mutually exclusive iff S1∧S2 =∅.
2
S ,...,S are a complete set iff S ∨...∨S =Ω and is between 6.6 and 6.7×10−11m3/kgs2.” Some level of
1 n 1 n
S ∧S =∅ for i6=j. precision is typically insisted on, so that, for example,
i j
Elementarylogicalrelationshipsamongstatements in- “red is good” might be rejected as too vague.
clude S∨S =Ω, S∧S =∅, S∧Ω=S, S1∧(S2∨S3)= A major thesis of this paper is that the class of al-
(S ∧S )∨(S ∧S ), etc. Denoting the probability as- lowed statements should include statements about the
1 2 1 3
signed to a statement S as P(S), we can state the first probabilities of other statements. Some Bayesians (for
three axioms of probability. example, de Finetti [2]) reject this concept as meaning-
Axiom 1. P(S) is a nonnegative real number. less. However, it has found some acceptance and utility
Axiom 2. P(S) = 1 iff S is known to be true. Axiom in decision theory, where it is sometimes called a sec-
3. IfS andS aremutuallyexclusive,thenP(S ∨S )= ond order probability; see, e.g., [3]. In particular, it is
1 2 1 2
P(S )+P(S ). an experimental fact that people’s decisions depend not
1 2
From these axioms, and the logical calculus of state- only on the probabilities they assign to various alterna-
ments, we can derive some simple lemmas: tives,butalsoonthedegreeofconfidencethattheyhave
Lemma 1. P(S)=1−P(S). in their own probability assignments [3]. This degree of
Lemma 2. P(S)≤1. confidencecanbe quantifiedandtreatedasaprobability
Lemma 3. P(S)=0 iff S is known to be false. of a probability.
Lemma 4. P(S ∧S )=P(S )+P(S )−P(S ∨S ). To illustrate how we will use the concept,consider the
1 2 1 2 1 2
We omit the proofs, which are straightforward. followingproblem. Supposethatwehaveasituationwith
Wewillalsoneedthenotionofaconditional statement exactly two possible outcomes (for example, a coin flip).
S |S . S |S is a statementif and only if S is true; oth- Call the two outcomes A and B. In the terminology of
2 1 2 1 1
erwise, S |S is not a statement,and cannotbe assigned the logicalcalculus,A∨B =Ω andA∧B =∅,sothatA
2 1
aprobability. GiventhatS is true,the statementS |S and B are a complete set. The probability axioms then
1 2 1
is true if and only if S is true. The probability that require P(A)+P(B) = 1, but do not tell us anything
2
S |S is true is then specified by about either P(A) or P(B) alone.
2 1
Axiom 4. P(S |S )=P(S ∧S )/P(S ). In the absence of any other information, we invoke
2 1 1 2 1
Note that, if P(S ) = 0, then S = ∅ by Lemma 3, Laplace’s principle of insufficient reason (also called the
1 1
and so both sides of Axiom 4 are undefined: the right principle of indifference): whenwe haveno causeto pre-
side because we have divided by zero, and the left side fer one statement over another, we assign them equal
because S |∅ is not a statement. probabilities. Thus we are instructed to choose P(A) =
2
Another concept we will need is that of independence P(B) = 1. While this assignment is logically sound, we
2
between statements. Two statements are said to be in- clearly cannot have a great deal of confidence in it; typ-
dependent if the knowledge that one of them if true tells ically, we are prepared to abandon it as soon as we get
us nothing about whether or not the other one is true. some more information.
Thus, if S and S are independent, we should have Another (and, we argue, better) strategy is to retreat
1 2
P(S |S ) = P(S ) and P(S |S ) = P(S ). Using these from the responsibility of assigning a particular value to
1 2 1 2 1 2
relations and Axiom 4, we get a result that can be used P(A),andinsteadassignaprobabilityP(H)tothestate-
as the definition of independence, ment H =“the value of P(A) is between h and h+dh.”
S and S are independent if and only if P(S ∧S )= Here dh is infinitesimal, and 0 ≤ h ≤ 1. Then P(H)
1 2 1 2
P(S )P(S ). takesthe formp(h)dh,where p(h)is anonnegativefunc-
1 2
Notethatindependenceisapropertyofprobabilityas- tion that we must choose, normalized by 1p(h)dh = 1.
0
signments,ratherthanthestatementsthemselves. Thus, We might choose p(h)=1, for example. R
peoplecandisagreeonwhetherornottwostatementsare Now suppose we get some more information about A
independent. and B. Suppose that the situation that produces either
A or B as anoutcome canbe recreatedrepeatedly (each
repetitionwillbecalledatrial),andthattheoutcomesof
III. PROBABILITIES OF PROBABILITIES thedifferenttrialsare(webelieve)independent. Suppose
thattheresultofthefirstN trialsisN A’sandN B’s,
A B
What limitations, if any, should be placed on the na- in a particular order. What can we say now?
tureofstatementstowhichweareallowedtoassignprob- The formula we need is
abilities? Bayes’ Theorem. P(H|D)=P(D|H)P(H)/P(D).
There are various schools of thought. Frequentists as- Bayes’theoremfollowsimmediately fromAxiom4;since
sign probabilities only to random variables, a highly re- H ∧D is the same as D∧H, we have P(H|D)P(D) =
stricted class of statements that we shall not attempt to P(H ∧D)=P(D|H)P(H). While H and D can be any
elucidate. Bayesians allow a wide range of statements, allowed statements, the letters are intended to denote
includingstatementsaboutthefuturesuchas“whenthis “hypothesis” and “data”. Bayes’ theorem tells us that,
coin is flipped it will come up heads,” statements about givenahypothesisH towhichwehavesomehowassigned
the past such as “it rained here yesterday,” and time- a prior probability P(H) (whether by the principle ofin-
less statements such as “the value of Newton’s constant difference, or by any other means), and we know (or can
3
compute) the likelihood P(D|H) of getting a particular probability; it is rather a limiting frequency or a propen-
set of data D given that the hypothesis H is true, then sity or a chance. Option two is to declare that p(h)dh is
we can compute the posterior probability P(H|D) that notactuallyaprobability;itisameasure oragenerating
the hypothesis H is true, given the data D that we have function.
obtained. Furthermore, if we have a complete set of hy- Let us explore optiontwo in more detail. Rather than
pothesesH ,thenwecanexpressP(D)intermsoftheas- assigning a second-order probability to H = “P(A) is
i
sociatedlikelihoodsandpriorprobabilities: startingwith between h and h+dh”, we assign a probability to ev-
D =D∧Ω=D∧(H ∨H ∨...)=(D∧H )∨(D∧H )∨..., ery finitesequence ofoutcomes;thatis,wechoosevalues
1 2 1 2
andnotingthatD∧H andD∧H aremutuallyexclusive for P(A), P(B), P(AB), P(BA), P(AAA), P(AAB),
i j
when i6=j, we have and so on, for strings of arbitrarily many outcomes. We
assume that all possible strings of N outcomes form
P(D)= P(D∧H )= P(D|H )P(H ), (1) a complete set. Our probability assignments must of
i i i
Xi Xi course satisfy the probability axioms, so that, for exam-
ple, P(A)+P(B) = 1. We also insist that the assign-
where the first equality follows from Axiom 3, and the
mentsbesymmetric;thatis,independentoftheordering
second from Axiom 4.
of the outcomes, so that, for example,
To apply these results to the case at hand, recall that
our hypothesis is H =“P(A) is between h and h+dh”. P(AAB)=P(ABA)=P(BAA). (5)
We have assigned this hypothesis a prior probability
P(H) = p(h)dh. The data D is a string of N A’s and Furthermore, the assignments for strings of N outcomes
A
N B’s, in a particular order; each of the N =N +N must be consistent with those for N +1 outcomes; this
B A B
outcomes is assumed to be independent of all the oth- means that, for any particular string of N outcomes S,
ers. Using the definition of independence, we see that
the likelihood is P(S)=P(SA)+P(SB). (6)
P(D|H) = P(A)NAP(B)NB A set of probability assignments that satisfies these re-
quirements is said to be exchangeable. Then, the de
= hNA(1−h)NB. (2) Finetti representation theorem [4] states that, given an
exchangeablesetofprobabilityassignmentsforallpossi-
ApplyingBayes’Theorem,wegettheposteriorprobabil- ble strings of outcomes, the probability of getting a spe-
ity
cific string D of N outcomes that includes exactly N
A
A’s and N B’s can always be written in the form
P(H|D)=P(D)−1hNA(1−h)NBp(h)dh, (3) B
1
where P(D)= hNA(1−h)NBp(h)dh, (7)
Z
0
1
P(D)= hNA(1−h)NBp(h)dh. (4) where p(h) is a unique nonnegative function that obeys
Z
0 1
the normalization condition dhp(h) = 1, and is the
0
If the number oftrials N is large,andif the priorproba- same for every string D. NotRe that eq.(7) is exactly the
bility p(h) has been chosen to be a slowly varying func- same as eq.(4). Thus an exchangeable probability as-
tion, then the posterior probability P(H|D) has a sharp signment to sequences of outcomes can be characterized
peak at h = h ≡ N /N, the fraction of trials that by a function p(h) that can be (as we have seen) consis-
exp A
resultedinoutcomeA. Thewidthofthispeakispropor- tentlytreatedasaprobabilityofaprobability. Butthose
tionaltoN−1/2 ifbothN andN arelarge,andtoN−1 whofindthisnotionunpalatablearefreetothinkofp(h)
A B
if either N or N is small (or zero). Thus, after a large as specifying a measure, or a generating function, or a
A B
number of trials, we can be confident that the probabil- similar euphemism.
ity P(A) that the next outcome will be A is close to the To summarize, if we need to assigna prior probability
fraction of trials that have already resulted in A. The but have little information, it can be more constructive
only people who will not be convinced of this are those to abjure, and instead assign a probability to a range
whose choice of prior probability p(h) is strongly biased of possible values of the needed prior probability. This
against the value h=h . Thus, the value h for the probability of a probability can then be updated with
exp exp
probability h is becoming objective, in the sense that al- Bayes’ theorem as more information comes in.
most all observers agree on it. Furthermore, those who
do not agree can be identified a priori by noting that
their prior probabilities are strong functions of h. IV. PROBABILITY IN QUANTUM
Thosewhorejectthenotionofaprobabilityofaproba- MECHANICS
bility,butwhoacceptthepracticalutilityofthisanalysis
(which was originally carried out by Laplace), have two Suppose we aregivena qubit: a quantumsystemwith
options. Optionone is to declare thath is notactually a atwo-dimensionHilbertspace. (Wewillusethelanguage
4
appropriatetoaspin-one-halfparticletodescribeit.) We Now we use Bayes’ theorem. Our hypothesis is H =
are asked to make a guess for its quantum state. “the quantum state is within a volume dρ centered on
Without further information, the best we can do is ρ”. We have assigned this hypothesis a prior probability
invoketheprincipleofindifference. Inthecaseofafinite P(H) = p(ρ)dρ. The data D is a string of N +’s and
+
set of possible outcomes, this principle is based on the N− −’s, ina particularorder;eachof the N =N++N−
permutation symmetry of the outcomes; we choose the outcomesis assumedto be independent of allthe others.
uniqueprobabilityassignmentthatisinvariantunderthis Using the definition of independence, we see that the
symmetry. The quantum analog of the permutation of likelihood is
outcomesisthe unitarysymmetryofrotationsinHilbert
space. The only quantum state that is invariant under P(D|H) = [P(σz=+1|ρ)]N+[P(σz=+1|ρ)]N−
this symmetry is the fully mixed density matrix
= [1(1+z)]N+[1(1−z)]N−. (12)
2 2
ρ= 1I. (8)
2
ApplyingBayes’Theorem,wegettheposteriorprobabil-
Thus we are instructed to choose eq.(8) as the quantum ity
state of the system. While this assignment is logically
sound, we clearly cannot have a great deal of confidence P(H|D)=P(D)−1[1(1+z)]N+[1(1−z)]N−p(ρ)dρ, (13)
2 2
in it; typically, we are preparedto abandon it as soon as
we get some more information. where
Another (and, we argue, better) strategy is to retreat
from the responsibility of assigning a particular state P(D)= [1(1+z)]N+[1(1−z)]N−p(ρ)dρ. (14)
Z 2 2
(pureormixed)tothesystem,andinsteadassignaprob-
ability P(H) to the statement H = “the quantum state
WhenthenumberoftrialsN islarge,andthepriorprob-
ofthesystemisadensitymatrixwithinavolumedρcen-
ability p(ρ) is a slowly varying function, the posterior
teredonρ”,whereρisaparticular2×2hermitianmatrix
probability P(H|D) has a sharp peak at z = z ≡
exp
with nonnegative eigenvalues that sum to one, and dρ is
(N+ −N−)/N. Thus, after a large number of trials in
asuitabledifferentialvolumeelementinthespaceofsuch
whichwemeasureσ ,wecanbeconfidentofthe valueof
z
matrices. Wecanparameterizeρwiththreerealnumbers
theparameterz inthedensitymatrixofthesystem. The
x, y, and z via
only people who will not be convinced of this are those
1 1+z x−iy whose choice of prior probability p(ρ) is strongly biased
ρ= , (9) against the value z = z . Furthermore, those who do
2(cid:18)x+iy 1−z (cid:19) exp
not agree can be identified a priori by noting that their
where where x2 + y2 + z2 ≡ r2 ≤ 1. We then take prior probabilities are strong functions of ρ.
dρ = dV, where dV = (3/4π)dxdydz is the normal- We can of course orient our Stern–Gerlach apparatus
ized volume element: dV = 1. P(H) takes the form alongdifferentaxes. Ifwechoosethexaxisorthey axis,
p(ρ)dV, where p(ρ) isRa nonnegative function that we the relevant predictions of quantum mechanics are
must choose, normalized by p(ρ)dV = 1. We might
choose p(ρ)=1, for example.R P(σx=+1|ρ) = Tr12(1+σx)ρ= 21(1+x), (15)
Nowsuppose we getsome moreinformationaboutthe
P(σ =−1|ρ) = Tr1(1−σ )ρ= 1(1−x), (16)
quantum state of the system. Suppose that the proce- x 2 x 2
dure that preparesthe quantumstate ofthe particle can
P(σ =+1|ρ) = Tr1(1+σ )ρ= 1(1+y), (17)
be recreated repeatedly (each repetition of this will be y 2 y 2
called a trial), and that the outcomes of measurements
P(σ =−1|ρ) = Tr1(1−σ )ρ= 1(1−y). (18)
performedoneachpreparedsystemare(webelieve)inde- y 2 y 2
pendent. SupposefurtherthatwehaveaccesstoaStern–
Foreachtrial,we canchoosewhether tomeasureσ , σ ,
x y
Gerlachapparatusthatallowsustomeasurewhetherthe
orσ . (We couldalsochooseto measurealonganyother
z
spin is + or − along an axis of our choice. We choose
axis.) Then, if the outcomes include N measurements
+z
the z axis. Suppose that the result of the first N trials
of σ with the result σ = +1, and so on, the posterior
z z
is N+ +’s and N− −’s. What can we say now? probability becomes
Givenadensitymatrixρ,parameterizedbyeq.(9),the
rules of quantum mechanics tell us that the probability P(H|D) = P(D)−1[1(1+x)]N+x[1(1−x)]N−x
2 2
thatameasurementofthespinalongthez axiswillyield
×[1(1+y)]N+y[1(1−y)]N−y
+1 is 2 2
×[1(1+z)]N+z[1(1−z)]N−zp(ρ)dρ, (19)
P(σ =+1|ρ)=Tr1(1+σ )ρ= 1(1+z), (10) 2 2
z 2 z 2
where P(D) is given by the obvious integral. Clearly
where σ is a Paulimatrix, andthe probability that this
z
the discussion in the preceding paragraphis simply trip-
measurement will yield −1 is
licated, and, when the number of trials is large, we have
P(σ =−1|ρ)=Tr1(1−σ )ρ= 1(1−z). (11) determined the entire density matrix to the satisfaction
z 2 z 2
5
ofallbutstronglybiasedobservers. Oursubjectiveprob- Thus, it is better to choose p(ρ)dρ when it is possible
abilities of probabilities have led us to an objective con- thatthere is something aboutthe preparationprocedure
clusion about quantum probabilities. thatconsistently prefers a particulardirectionin Hlibert
In [1], Caves et al arrived at an essentially identical space, but we do not know what that direction is. Since
result. The main difference in their analysis is that they this possibility can rarely be ruled out a priori, we are
regarded p(ρ)dρ as a measure rather than a probability. typically better served by choosing a prior probability
This approach required them to prove, first, a quantum p(ρ)dρ, rather than a particular vaue of ρ itself.
version of the de Finetti theorem [5], and, second, that
Bayes’ theorem can be applied to p(ρ)dρ [6]. Both steps
becomeunnecessaryifwetreatp(ρ)dρas,fundamentally, VI. NONINFORMATIVE PRIORS FOR
a probability. DENSITY MATRICES
Suppose we have decided to choosea priorprobability
V. PROBABILITIES FOR DENSITY MATRICES p(ρ)dρforthedensitymatrixρofsomequantumsystem.
VS. DENSITY MATRICES How should we choose this probability?
Inthecasewherewehavelittleornoinformationabout
If we assignan impure density matrix ρ to a quantum the quantum system, we would like to formulate the ap-
system,doesthisnotalreadytakeintoaccountourigno- propriateanalogoftheprincipleofindifference. Consider
ranceaboutit? Why is it preferableto assign,instead, a a qunit, a quantum system whose Hilbert space has di-
probabilityp(ρ)dρtothesetofpossibledensitymatrices? mensionnthatisknowntous. (We willnotconsiderthe
It depends on the nature of our ignorance. Suppose, even more general problem where n is unknown.) We
forexample,thesystemisthespinofanelectronplucked can always write the density matrix (whatever it is) in
from the air. Then we expect that eq.(8) will describe the form
it, in the sense that if we do repeated trials (plucking
ρ=U−1ρ˜U, (21)
a new electron each time, and measuring its spin along
an axis of our choice), we will find that x ≡ (N −
exp +x
where U is unitary with determinant one, and ρ˜ is di-
N−x)/(N+x+N−x), yexp ≡(N+y−N−y)/(N+y+N−y), agonal with nonnegative entries p ,...,p that sum to
and zexp ≡(N+z −N−z)/(N+z +N−z) all tend to zero. 1 n
one. There is a natural measure for special unitary ma-
Suppose instead that the spin is preparedby a techni-
trices, the Haar measure; it is invariant under U →CU,
cianwho(withtheaidofaStern–Gerlachdevice)putsit
where C is a constant special unitary matrix. In the
in either a pure state with σ =+1,or a pure state with
z simplest case of n = 2, we can parameterize U as
σ = +1, and each time decides which choice to make
byx flipping a coin that we believe is fair. In this case the U =eiα1σ3eiα2σ2eiα3σ3, with 0≤ α1 ≤ π, 0≤ α2 ≤ π/2,
0 ≤ α ≤ π; then the normalized Haar measure is
appropriate density matrix is 3
dU = π−2sin(2α )dα dα dα . This construction is ex-
2 1 2 3
ρ = 1[1(1+σ )]+ 1[1(1+σ )] tended to all n in [7].
2 2 z 2 2 x Supposeweknowthatthestateofthequantumsystem
is pure. Then we can set ρ˜ = δ δ , and parameterize
1 3 1 ij i1 j1
= . (20) ρ via U. Then it is natural to choose dρ = dU and
4(cid:18)1 1(cid:19)
p(ρ)=1,because this is the only choicethat is invariant
under unitary rotations in Hilbert space.
Comparing with eq.(9), we see that we now we expect
x , y , and z to approach +1, 0, and +1, respec- Now consider the more general case where we do not
exp exp exp 2 2 have information about the purity of the system’s quan-
tively.
tum state. Following [8], we define the volume element
Now suppose thatthe spinis preparedby a technician
via
whoputsitineitherapurestatewithσ =+1,orapure
z
state with σ = +1, and makes the same choice every
x dρ≡dUdF, (22)
time. We, however, are not aware of what her choice is.
If forced to assign a particular density matrix, we
where dU is the normalized Haar measure for U, and
would have to choose eq.(20). However, our situation
is clearly different from what it was in the previous ex- dF =(n−1)!δ(p +...+p −1)dp ...dp (23)
1 n 1 n
ample. In the present case, repeated experiments would
not verify eq.(20), but would instead converge on either is a normalized measure for the p ’s that we will call the
i
xexp = 0 and zexp = +1, or xexp = +1 and zexp = 0. Feynman measure (because it appears in the evaluation
Therefore,in this case,itis moreappropriateto assigna of one-loop Feynman diagrams). Eq.(23) assumes that
priorprobabilityofone-halfto ρ= 12(1+σz)anda prior each pi runs from zero to one; then eq.(21) is an over-
probability one-half to ρ = 1(1 + σ ). Then, as data completeconstruction,becauseU canrearrangethe p ’s.
2 x i
comes in, we can update these probability assignments This is easily fixed by imposing p ≥...≥p , and mul-
1 n
with Bayes’ theorem, as described in section IV. tiplyingdF byn!. However,eq.(23)asitstandsiseasier
6
to write and think about; the overcompleteness of this p(ρ)dρ overthe rangeofalloweddensity matrices,rather
construction of ρ causes no harm. thanby a specific choiceof density matrix. This method
In the case n = 2, we previously chose dρ = dV = isparticularlyappropriatewhen(1)thepreparationpro-
(3/4π)dxdydzfortheparameterizationofeq.(9). Inthis cedure may favora direction in Hilbert space, but we do
case,the eigenvaluesof ρ are 1(1+r) and 1(1−r), with not know what that direction is, and (2) we can recre-
2 2
0 ≤ r ≤ 1. After integrating over U, dV → 3r2dr; in ate the preparation procedure repeatedly, and perform
comparison, dF =dr for this case. measurementsofourchoice oneachpreparedsystem. In
The purity of a density matrix ρ can be paramertized thiscase,asdatacomesin,weuseBayes’theoremtoup-
by Trρ2, which for n=2 is 1(1+r2). Thus the volume datep(ρ)dρ. Eventually,allbutstronglybiasedobservers
2
measuredV ismorebiasedtowardspurestatesthanisthe (whocanbeidentifiedapriori byanexaminationoftheir
Feynman measure dF; we have dV =3(2Trρ2−1)dF. choiceofpriorprobability)willbeconvincedofthevalues
Ingeneral,wecanaccomodateanysuchbiasbytaking of the quantum probabilities. In this way, initially sub-
p(ρ)dρ to be of the form jective probability assignments become more and more
objective.
p(ρ)dρ=p(Trρ2)dUdF, (24) Inchoosingp(ρ)dρ,wecanusetheprincipleofindiffer-
ence, applied to the unitary symmetry of Hilbert space,
where p(x) is an increasing function if we are biased to-
to reduce the problem to one of choosing a probability
wards having a pure state. For n > 2, we can take p to
distribution for the eigenvalues of ρ. There is, however,
be a function ofTrρk for 2≤k ≤n. Arguments infavor
no compelling rationaleforany particularchoice;inpar-
of various choices of p have been put forth [9], but no
ticular, we must decide how biased we are towards pure
single choice seems particularly compelling. Of course,
states.
once we have done enough experiments, our original bi-
ases become largely irrelevant, as we saw in section IV.
Acknowledgments
VII. CONCLUSIONS
I am grateful to Jim Hartle for illuminating discus-
We have argued that, in a Bayesian framework, the sions, and prescient comments on earlier drafts of this
nature of our ignoranceabout a quantum systemcan of- paper. This work was supported in part by NSF Grant
ten be more faithfully representedby a prior probability No. PHY00-98395.
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