Table Of ContentQuantum Theory From Five Reasonable Axioms
Lucien Hardy
(cid:3)
1 Centre for Quantum Computation,
0
The Clarendon Laboratory,
0
2 Parks road, Oxford OX1 3PU, UK
p
e September 25, 2001
S
5
2 Abstract theory is simply a new type of probability theory.
Like classical probability theory it can be applied
4 Theusualformulationofquantumtheoryisbasedon to a wide range of phenomena. However, the rules
v rather obscure axioms (employing complex Hilbert of classical probability theory can be determined by
2 spaces, Hermitean operators, and the trace formula pure thought alone without any particular appeal to
1 for calculating probabilities). In this paper it is experiment (though, of course, to develop classical
0 shown that quantumtheory can be derived from(cid:12)ve probability theory, we do employ some basic intu-
1
very reasonable axioms. The (cid:12)rst four of these ax- itions about the nature of the world). Is the same
0
1 iomsareobviouslyconsistentwithbothquantumthe- true of quantum theory? Put another way, could a
0 ory and classical probabilitytheory. Axiom5 (which 19th century theorist have developed quantum the-
/ requires that there exist continuous reversible trans- ory without access to the empirical data that later
h
p formations between pure states) rules out classical became available to his 20th century descendants?
- probabilitytheory. IfAxiom5(orevenjustthe word In this paper it will be shown that quantum theory
t
n \continuous" from Axiom 5) is dropped then we ob- followsfrom(cid:12)veveryreasonableaxiomswhichmight
a tain classical probability theory instead. This work well have been posited without any particular access
u
provides some insight into the reasons why quantum to empirical data. We will not recover any speci(cid:12)c
q
theory is the way it is. For example, it explains the form of the Hamiltonian from the axioms since that
:
v need for complex numbers and where the trace for- belongs to particular applications of quantum the-
i
X mulacomes from. We also gaininsight into the rela- ory (for example - a set of interacting spins or the
r tionshipbetween quantumtheory andclassicalprob- motion of a particle in one dimension). Rather we
a abilitytheory. will recover the basic structure of quantum theory
along with the most general type of quantum evo-
lution possible. In addition we will only deal with
1 Introduction
the case where there are a (cid:12)niteor countablyin(cid:12)nite
number of distinguishable states corresponding to a
Quantumtheory,initsusualformulation,isvery ab- (cid:12)niteorcountablyin(cid:12)nitedimensionalHilbertspace.
stract. The basic elements are vectors in a complex WewillnotdealwithcontinuousdimensionalHilbert
Hilbert space. These determine measured probabil- spaces.
ities by means of the well known trace formula - a
The basic setting we will consider is one in which
formulawhich has no obvious origin. It is natural to
we have preparation devices, transformationdevices,
ask why quantum theory is the way it is. Quantum
and measurement devices. Associated with each
(cid:3)[email protected]. This is version 4 preparation will be a state de(cid:12)ned in the following
1
way: Axiom 2 Simplicity. K is determined byafunction
of N (i.e. K = K(N)) where N = 1;2;::: and
The state associated with a particular preparation
where, for each given N, K takes the minimum
is de(cid:12)ned to be (that thing represented by) any
value consistent with the axioms.
mathematical object that can be used to deter-
mine the probability associated with the out-
Axiom 3 Subspaces. A system whose state is con-
comes of any measurement that may be per-
strainedtobelongtoanM dimensionalsubspace
formedonasystempreparedbythegivenprepa-
(i.e. havesupportononlyM ofasetofN possi-
ration.
ble distinguishable states) behaves likea system
Hence, alistofallprobabilitiespertainingtoallpos- of dimension M.
sible measurements that could be made would cer-
Axiom 4 Composite systems. A composite system
tainlyrepresent the state. However, this would most
consisting of subsystems A and B satis(cid:12)es N =
likely over determine the state. Since most physical
N N and K =K K
theories have some structure, a smaller set of prob- A B A B
abilities pertaining to a set of carefully chosen mea-
Axiom 5 Continuity. There exists a continuous re-
surements may be su(cid:14)cient to determine the state.
versibletransformationonasystembetweenany
This is the case in classical probability theory and
two pure states of that system.
quantumtheory. Central to the axiomsare two inte-
gers K and N which characterize the type of system The (cid:12)rst four axioms are consistent with classical
being considered. probability theory but the (cid:12)fth is not (unless the
word \continuous" is dropped). If the last axiom is
The number of degrees of freedom, K, is de(cid:12)ned
(cid:15) dropped then, because of the simplicity axiom, we
asthe minimumnumberofprobabilitymeasure-
obtain classical probabilitytheory (with K =N) in-
ments needed to determine the state, or, more
stead of quantum theory (with K = N2). It is very
roughly, as the number of real parameters re-
striking that we have here a set of axioms for quan-
quired to specify the state.
tum theory which have the property that if a single
The dimension, N, is de(cid:12)ned as the maximum word is removed { namely the word \continuous" in
(cid:15)
number of states that can be reliably distin- Axiom5{thenweobtainclassicalprobabilitytheory
guished from one another in a single shot mea- instead.
surement. Thebasicideaoftheproofissimple. Firstweshow
how the state can be described by a real vector, p,
We will only consider the case where the number
whoseentries areprobabilitiesandthattheprobabil-
of distinguishable states is (cid:12)nite or countably in(cid:12)-
ityassociatedwithanarbitrarymeasurementisgiven
nite. As will be shown below, classical probability
by alinearfunction, r p,ofthis vector (the vector r
theory has K = N and quantum probability theory (cid:1)
is associated with the measurement). Then we show
has K = N2 (note we do not assume that states are
that we must have K =Nr where r is a positive in-
normalized).
teger and that it follows from the simplicity axiom
The (cid:12)ve axioms for quantum theory (to be stated
that r=2 (the r=1 case being ruled out by Axiom
again,in context, later) are
5). We consider the N = 2, K = 4 case and recover
Axiom 1 Probabilities. Relative frequencies (mea- quantumtheory foratwodimensionalHilbertspace.
sured by taking the proportion of times a par- The subspace axiom is then used to construct quan-
ticular outcome is observed) tend to the same tum theory for general N. We also obtain the most
value(whichwecalltheprobability)foranycase general evolution ofthe state consistent with the ax-
where a given measurement is performed on a ioms and show that the state of a composite system
ensemble of n systems prepared by some given can be represented byapositiveoperatoronthe ten-
preparation in the limitas n becomes in(cid:12)nite. sor product of the Hilbert spaces of the subsystems.
2
Finally, we show obtain the rules for updating the because the button on the preparation device was
state after a measurement. not pressed) then it outputs a 0 (corresponding to a
This paper is organized in the following way. null outcome). If there is actually a physical system
First we will describe the type of situation we wish incident (i.e when the release button is pressed and
to consider (in which we have preparation devices, thetransformingdevicehasnotabsorbedthesystem)
state transforming devices, and measurement de- then the device outputs a number l where l =1 to L
vices). Then we will describe classical probability (we will call these non-null outcomes). The number
theory and quantum theory. In particular it will be ofpossible classicaloutputs, L, maydepend onwhat
shownhowquantumtheorycanbeputinaformsim- is being measured (the settings of the knobs).
ilarto classicalprobabilitytheory. After that wewill The fact that we allow null events means that we
forgetboth classicalandquantumprobabilitytheory will not impose the constraint that states are nor-
andshowhowtheycanbeobtainedfromtheaxioms. malized. This turns out to be a useful convention.
Various authors have set up axiomatic formula- It may appear that requiring the existence of null
tions of quantum theory, for example see references events is an additional assumption. However, it fol-
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10] (see also [11, 12, 13]). lowsfromthe subspace axiomthatwecanarrangeto
Much of this work is in the quantum logic tradition. have a null outcome. We can associate the non-null
The advantage of the present work is that there are outcomes with a certain subspace and the null out-
a small number of simple axioms, these axioms can come with the complement subspace. Then we can
easilybe motivatedwithoutany particularappeal to restrict ourselvestopreparingonlymixturesofstates
experiment, and the mathematicalmethods required which are inthe non-nullsubspace (when the button
toobtainquantumtheoryfromtheseaxiomsarevery ispressed) withstates whichare inthe nullsubspace
straightforward (essentially just linear algebra). (when the button is not pressed).
The situation described here is quite generic. Al-
though we have described the set up as ifthe system
2 Setting the Scene
were movingalongone dimension,infact the system
could equally well be regarded as remaining station-
We will begin by describing the type of experimen- ary whilst being subjected to transformations and
tal situation we wish to consider (see Fig. 1). An measurements. Furthermore, the system need notbe
experimentalist has three types of device. One is a localizedbutcouldbeinseverallocations. Thetrans-
preparation device. We can think of it as preparing formationscouldbeduetocontrolling(cid:12)eldsorsimply
physical systems in some state. It has on it a num- duetothenaturalevolutionofthesystem. Anyphys-
ber ofknobs which can be varied to change the state ical experiment, quantum, classical or other, can be
prepared. The system is released by pressing a but- viewed as an experiment of the type described here.
ton. The system passes through the second device.
This device can transform the state of the system.
This device has knobs on it which can be adjusted 3 Probability measurements
toe(cid:11)ect di(cid:11)erent transformations(we mightthinkof
these as controlling (cid:12)elds which e(cid:11)ect the system). We will consider only measurements of probability
We can allow the system to pass through a number since all other measurements (such as expectation
of devices of this type. Unless otherwise stated, we values)canbecalculatedfrommeasurementsofprob-
will assume the transformation devices are set to al- ability. When, in this paper, we refer to a measure-
lowthe system through unchanged. Finally,we have ment or a probability measurement we mean, speci(cid:12)-
a measurement apparatus. This also has knobs on it cally,ameasurement of the probabilitythat the out-
which can be adjusted to determine what measure- comebelongstosomesubsetofthenon-nulloutcomes
ment is being made. This device outputs a classical with agivensetting ofthe knobon the measurement
number. If no system is incident on the device (i.e. apparatus. Forexample,we couldmeasure the prob-
3
Release button(cid:13)
Knob(cid:13)
System(cid:13)
Classical(cid:13)
information(cid:13)
out(cid:13)
Preparation(cid:13) Transformation(cid:13) Measurement(cid:13)
Figure 1: The situation considered consists of a preparation device with a knob for varyingthe state of the
system produced andarelease button forreleasingthe system, atransformationdevice fortransformingthe
state (and a knob to vary this transformation), and a measuring apparatus for measuring the state (with a
knob to vary what is measured) which outputs a classical number.
ability that the outcome is l = 1 or l = 2 with some a measurement. We can write
given setting.
p
1
Toperformameasurementweneedalargenumber
p
2
of identically prepared systems. 0 1
p
p= 3 (1)
A measurement returns a single real number (the B .. C
B . C
probability) between 0 and 1. It is possible to per- B C
Bp C
form many measurements at once. For example, we B NC
@ A
could simultaneously measure [the probability the
This vector can be regarded as describing the state
outcome isl =1]and [theprobabilitythe outcome is
of the system. It can be determined by measuring
l =1 or l =2] with a given knob setting.
N probabilities and so K = N. Note that we do
not assume that the state is normalized (otherwise
we would have K =N 1).
(cid:0)
ThestatepwillbelongtoaconvexsetS. Sincethe
set is convex it will have a subset of extremal states.
4 Classical Probability Theory
These are the states
1 0 0
A classicalsystem willhave availableto ita number, 0 1 0
0 1 0 1 0 1
N, of distinguishable states. For example, we could 0 0 1
p = p = p = etc:
1 2 3
consider a ball that can be in one of N boxes. We B..C B..C B..C
B.C B.C B.C
willcall these distinguishable states the basis states. B C B C B C
B0C B0C B0C
Associated witheach basisstate willbethe probabil- B C B C B C
@ A @ A @ A (2)
ity,p ,of(cid:12)ndingthe system inthat state ifwe make
n
4
and the state There must exist a measurement in which we sim-
ply check to see that the system is present (i.e. not
0 in the null state). We denote this by rI. Clearly
0
0 1
p =0= 0 (3) 1
null
B..C 1
B.C 0 1
BB0CC rI = rn = 1 (5)
B@ CA Xn BB...CC
The state 0 is the null state (when the system is not B C
B1C
present). Wede(cid:12)nethesetofpurestatestoconsistof B C
@ A
all extremal states except the null state. Hence, the Hence 0 rI:p 1 with normalized states saturat-
states in (2) are the pure states. They correspond to ing the u(cid:20)pper bo(cid:20)und.
the system de(cid:12)nitely being in one of the N distin- With a given setting of the knob on the measure-
guishable states. A general state can be written as mentdevicetherewillbeacertainnumberofdistinct
a convex sum of the pure states and the null state non-null outcomes labeled l = 1 to L. Associated
and this gives us the exact formof the set S. This is with each outcome will be a measurement vector r .
l
always a polytope (a shape having (cid:13)at surfaces and Since, for normalized states, one non-null outcome
a (cid:12)nite number of vertices). must happen we have
We will now consider measurements. Consider a
measurement ofthe probabilitythat the system is in L
r =rI (6)
the basis state n. Associated with this probability l
measurement is the vector rn having a 1 in position Xl=1
nand0’selsewhere. Atleastforthese cases themea-
This equation imposes a constraint on any measure-
sured probabilityis given by
ment vector. Let allowed measurement vectors r be-
longtothesetR. Thissetisclearlyconvex(byvirtue
p =r p (4)
meas (cid:1) of 1. above). To fully determine R (cid:12)rst consider the
set R+ consisting of allvectors which can be written
However,wecanconsidermoregeneraltypesofprob-
as a sum of the basis measurement vectors, r , each
ability measurement and this formula will still hold. n
multiplied by a positive number. For such vectors
There are two ways in which we can construct more
r p is necessarily greater than 0 but may also be
general types of measurement: (cid:1)
greater than 1. Thus, elements of R+ may be too
1. We can perform a measurement in which we long to belong to R. We need a way of picking out
decide with probability (cid:21) to measure r and those elements of R+ that also belong to R. If we
A
with probability 1 (cid:21) to measure r . Then can perform the probabilitymeasurement r then, by
B
(cid:0)
we will obtain a new measurement vector r = (6)wecanalsoperformthe probabilitymeasurement
(cid:21)r +(1 (cid:21))r . r rI r. Hence,
A (cid:0) B (cid:17) (cid:0)
2. We can add the results of two compatible prob- I(cid:11) r;r R+ and r+r=rI then r;r R
2 2
abilitymeasurements andtherefore add the cor- (7)
responding measurement vectors.
This works since it implies that r p 1 for allp so
(cid:1) (cid:20)
Anexampleofthesecond istheprobabilitymeasure- that r is not too long.
ment that the state is basis state 1 or basis state 2 is NotethattheAxioms1to4aresatis(cid:12)edbutAxiom
given by the measurement vector r +r . From lin- 5isnotsincethere area(cid:12)nitenumberofpurestates.
1 2
earity, it is clear that the formula (4) holds for such Itiseasytoshowthatreversibletransformationstake
more general measurements. pure states to pure states (see Section 7). Hence a
5
continuous reversible transformationwilltakea pure 2. The most general type ofmeasurement inquan-
statealongacontinuouspaththroughthepurestates tumtheoryisaPOVM(positiveoperatorvalued
which is impossible here since there are only a (cid:12)nite measure). The operator A^ is an element of such
number of pure states. a measure.
3. Two classes of superoperator are of particular
5 Quantum Theory interest. If $ is reversible (i.e. the inverse $ 1
(cid:0)
both exists and belongs to the allowed set of
Quantumtheorycanbesummarizedbythefollowing transformations) then it will take pure states
rules to pure states and corresponds to unitary evo-
lution. The von Neumann projection postulate
States The state is represented by a positive (and
takesthestate (cid:26)^tothe state P^(cid:26)^P^ whentheout-
therefore Hermitean) operator (cid:26)^ satisfying 0
(cid:20) come corresponds to the projection operator P^.
tr((cid:26)^) 1.
(cid:20) This is a special case of a superoperator evolu-
Measurements Probability measurements are rep- tion in which the trace of (cid:26)^decreases.
resented by a positive operator A^. If A^ corre-
l
4. It has been shown by Krauss [14] that one need
sponds to outcome l where l=1 to L then
only impose the three listed constraints on $ to
L fully constrain the possible types of quantum
A^l =I^ (8) evolution. This includes unitary evolution and
l=1 von Neumann projection as already stated, and
X
it also includes the evolution of an open system
Probability formula The probability obtained
(interacting with an environment). It is some-
when the measurement A^ is made on the state
times stated that the superoperator should pre-
(cid:26)^is
serve the trace. However, this is an unnecessary
p =tr(A^(cid:26)^) (9) constraint which makes it impossible to use the
meas
superoperator formalism to describe von Neu-
mann projection [15].
Evolution The most general evolution is given by
the superoperator $
5. The constraint that $ is completely positive im-
poses not only that $ preserves the positivity of
(cid:26)^ $((cid:26)) (10)
! (cid:26)^butalsothat$A I^B actingonanyelementof
(cid:10)
where $ a tensor product space also preserves positivity
for any dimension of B.
Does not increase the trace.
(cid:15)
This is the usual formulation. However, quantum
Is linear.
(cid:15) theory can be recast ina formmore similarto classi-
Is completely positive. cal probability theory. To do this we note (cid:12)rst that
(cid:15)
the space of Hermitean operators which act on a N
Thiswayofpresentingquantumtheoryisrathercon-
dimensional complex Hilbert space can be spanned
densed. The following notes should provide some
by N2 linearly independent projection operators P^
clari(cid:12)cations k
for k = 1 to K = N2. This is clear since a general
1. It is, again,more convenient not to impose nor- Hermitean operator can be represented as a matrix.
malization. This, in any case, more accurately This matrix has N real numbers along the diagonal
models what happens in real experiments when and 1N(N 1) complex numbers above the diago-
2 (cid:0)
the quantum system is often missing for some nal making a total of N2 real numbers. An example
portion of the ensemble. of N2 such projection operators will be given later.
6
De(cid:12)ne or we can write D = tr(P^P^T). From (14,17) we
obtain
P^
1
P^ =0P^.21 (11) pS =DrS (19)
.
.
B C and
BP^ C
B KC
@ A p =DTr (20)
M M
Any Hermitean matrix can be written as a sum of
these projection operators timesrealnumbers,i.e. in
We also note that
theformaP^ whereaisarealvector(aisuniquesince
the operat(cid:1)ors P^k are linearly independent). Now de- D =DT (21)
(cid:12)ne
thoughthis wouldnot bethe case hadwechosen dif-
pS =tr(P^(cid:26)^) (12) ferent spanning sets of projection operators for the
state operators and measurement operators. The in-
HerethesubscriptS denotes‘state’. Thekthcompo-
verse D 1 must exist (since the projection operators
nentofthisvectorisequaltotheprobabilityobtained (cid:0)
are linearly independent). Hence, we can also write
when P^ is measured on (cid:26)^. The vector p contains
k S
thesameinformationasthestate(cid:26)^andcantherefore p =pT D 1p (22)
meas M (cid:0) S
be regarded asanalternativewayofrepresenting the
state. Note that K =N2 since it takes N2 probabil- Thestatecanberepresented byanr-typevectoror
ity measurements to determine pS or, equivalently, a p-type vector as can the measurement. Hence the
(cid:26)^. We de(cid:12)ne rM through subscripts M and S were introduced. We will some-
times drop these subscripts when it is clear from the
A^=r P^ (13)
M (cid:1) context whetherthe vectorisastateormeasurement
vector. Wewillsticktotheconventionofhavingmea-
The subscript M denotes ‘measurement’. The vector
surement vectors onthe left andstate vectors on the
r is another way of representing the measurement
M
A^. Ifwesubstitute (13)intothetrace formula(9)we right as in the above formulae.
We de(cid:12)ne rI by
obtain
I^=rI P^ (23)
p =r p (14)
meas M S (cid:1)
(cid:1)
We can also de(cid:12)ne Thismeasurement givesthe probabilityofanon-null
event. Clearlywe must have 0 rI p 1 with nor-
pM =tr(A^P^) (15) malized states saturating the u(cid:20)pper(cid:1)bo(cid:20)und. We can
also de(cid:12)ne the measurement which tells us whether
and rS by the state is in a given subspace. Let I^ be the pro-
W
(cid:26)^=P^ r (16) jector intoan M dimensionalsubspace W. Then the
(cid:1) S corresponding r vector is de(cid:12)ned by I^W = rIW P^.
(cid:1)
Using the trace formula(9) we obtain We willsay that a state p is in the subspace W if
p =p r =rT Dr (17) rIW p=rI p (24)
meas M (cid:1) S M S (cid:1) (cid:1)
whereT denotestransposeandDistheK K matrix so it only has support in W. A system in which
(cid:2)
with real elements given by the state is always constrained to an M-dimensional
subspace willbehave as an M dimensionalsystem in
D =tr(P^P^ ) (18)
ij i j accordance with Axiom 3.
7
The transformation (cid:26)^ $((cid:26)^) of (cid:26)^ corresponds to Each of these belong to one-dimensional subspaces
!
the followingtransformation for the state vector p: formed from the orthonormal basis set. De(cid:12)ne
p = tr(P^(cid:26)^) 1
mn = (m + n )
x
tr(P^$((cid:26)^)) j i p2 j i j i
!
= tr(P^$(P^TD 1p))
(cid:0) 1
mn = (m +in )
= Zp j iy p2 j i j i
where equations (16,19) were used in the third line
for m < n. Each of these vectors has support on a
and Z is a K K real matrix given by
two-dimensionalsubspace formedfromthe orthonor-
(cid:2)
Z =tr(P^$(P^)T)D(cid:0)1 (25) mal basis set. There are 12N(N (cid:0) 1) such two-
dimensionalsubspaces. Hencewecande(cid:12)neN(N 1)
(wehaveused thelinearityproperty of$). Hence, we further projection operators (cid:0)
see that a linear transformation in (cid:26)^corresponds to
alineartransformationinp. WewillsaythatZ (cid:0). mn x mn and mn y mn (27)
2 j i h j j i h j
Quantum theory can now be summarized by the
This makes a total of N2 projectors. It is clear that
followingrules
these projectors are linearly independent.
States Thestateisgivenbyarealvectorp S with Each projector corresponds to one degree of free-
2
N2 components. dom. There is one degree of freedom associated
with each one-dimensional subspace n, and a fur-
Measurements A measurement isrepresented bya
thertwodegreesoffreedomassociatedwitheachtwo-
real vector r R with N2 components.
2 dimensional subspace mn. It is possible, though not
Probability measurements The measured proba- actually the case in quantum theory, that there are
bility if measurement r is performed on state p furtherdegreesoffreedomassociatedwitheachthree-
is dimensional subspace and so on. Indeed, in general,
we can write
p =r p
meas
(cid:1)
K = Nx + 1N(N 1)x
Evolution The evolution of the state is given by 1 2! (cid:0) 2 (28)
+1N(N 1)(N 2)x +:::
p Zp where Z (cid:0) is a real matrix. 3! (cid:0) (cid:0) 3
! 2
Wewillcallthe vector x=(x ;x ;:::) the signature
The exact nature of the sets S, R and (cid:0) can be de- 1 2
of a particular probability theory. Classical proba-
duced from the equations relating these real vectors
bility theory has signature x = (1;0;0;:::)
andmatricestotheir counterparts inthe usualquan- Classical
and quantum theory has signature x =
tum formulation. We will show that these sets can Quantum
(1;2;0;0;:::). We will show that these signatures
also be deduced from the axioms. It has been no-
arerespectively pickedoutbyAxioms1to4andAx-
ticed by various other authors that the state can be
ioms1to5. Thesignaturesx =(1;1;0;0;:::)of
represented by the probabilities used to determine it Reals
realHilbertspacequantumtheoryandx =
[18, 19]. Quaternions
There are various ways of choosing a set of N2 (1;4;0;0;:::) of quaternionic quantum theory are
linearly independent projections operators P^ which ruled out.
k
Ifwehaveacompositesystemconsistingofsubsys-
span the space of Hermitean operators. Perhaps the
tem A spanned by P^A (i=1 to K ) and B spanned
simplest way is the following. Consider an N dimen- i A
sionalcomplexHilbertspacewithanorthonormalba- byP^jB (j =1toKB)thenP^iA(cid:10)P^jB arelinearlyinde-
sisset n forn=1toN. Wecande(cid:12)neN projectors pendent and span the composite system. Hence, for
j i the composite system we have K =K K . We also
A B
n n (26)
have N =N N . Therefore Axiom4 is satis(cid:12)ed.
j ih j A B
8
The set S is convex. It contains the null state 0 We can write this as
(if the system is never present) which is an extremal
1 0 1 a 1 b
state. Purestatesarede(cid:12)nedasextremalstatesother (cid:0) (cid:0)
0 1 a b
thanthenullstate(since theyareextremaltheycan- D =0 1 (34)
1 a a 1 c
not be written as a convex sum of other states as we (cid:0)
B 1 b b c 1 C
expect of pure states). We know that a pure state B (cid:0) C
@ A
can be represented by a normalized vector . This where a and b are real with (cid:12) = paexp(i(cid:30)3), (cid:14) =
is speci(cid:12)ed by 2N 2 real parameters (Njcoimplex pbexp((cid:30)4), and c = (cid:11)(cid:13)(cid:3)+(cid:12)(cid:14)(cid:3) 2. We can choose (cid:11)
(cid:0) j j
numbers minus overall phase and minus normaliza- and (cid:13) to be real (since the phase is included in the
tion). On the other hand, the full set of normalized de(cid:12)nition of (cid:12) and (cid:14)). It then followsthat
states is speci(cid:12)ed by N2 1 real numbers. The sur-
(cid:0) c=1 a b+2ab
face ofthe set ofnormalizedstates musttherefore be (cid:0) (cid:0) (35)
+2cos((cid:30) (cid:30) ) ab(1 a)(1 b)
N2 2dimensional. Thismeansthat,ingeneral,the 4(cid:0) 3 (cid:0) (cid:0)
(cid:0)
pure states are of lower dimension than the the sur- Hence, byvaryingthecompplexphaseassociatedwith
face of the convex set of normalizedstates. The only (cid:11), (cid:12), (cid:13) and (cid:14) we (cid:12)nd that
exception to this is the case N =2 when the surface
c <c<c (36)
oftheconvexsetis2-dimensionalandthepurestates +
(cid:0)
are speci(cid:12)ed by two real parameters. This case is il- where
lustrated by the Bloch sphere. Points on the surface
of the Bloch sphere correspond to pure states. c 1 a b+2ab 2 ab(1 a)(1 b) (37)
(cid:6) (cid:17) (cid:0) (cid:0) (cid:6) (cid:0) (cid:0)
In fact the N = 2 case will play a particularly
This constraint is equivaplent to the condition
important role later so we will now develop it a lit-
Det(D) > 0. Now, if we are given a particular D
tle further. There will be four projection operators
matrixofthe form(34)then we can gobackwards to
spanning the space ofHermitean operators whichwe
the usual quantum formalismthough we must make
can choose to be
some arbitrary choices for the phases. First we use
P^1= 1 1 (29) (35) to calculate cos((cid:30)4(cid:0)(cid:30)3). We can assume that
j ih j 0 (cid:30) (cid:30) (cid:25) (this corresponds to assigning i to
4 3
(cid:20) (cid:0) (cid:20)
one of the roots p 1). Then we can assume that
P^2=j2ih2j (30) (cid:30)3 = 0. This (cid:12)xes(cid:0)(cid:30)4. An example of this second
choiceiswhenweassignthestate 1 (+ + )(this
p2 j i j(cid:0)i
P^ =((cid:11)1 +(cid:12) 2 )((cid:11) 1 +(cid:12) 2) (31) has real coe(cid:14)cients) to spin alongthe xdirection for
3 (cid:3) (cid:3)
j i j i h j h j a spin half particle. This is arbitrary since we have
rotationalsymmetry aboutthe z axis. Havingcalcu-
P^4 =((cid:13)j1i+(cid:14)j2i)((cid:13)(cid:3)h1j+(cid:14)(cid:3)h2j) (32) lated (cid:30)3 and (cid:30)4 from the elements of D we can now
calculate (cid:11), (cid:12), (cid:13), and (cid:14) and hence we can obtain P^.
where (cid:11)2+ (cid:12) 2 = 1 and (cid:13) 2+ (cid:14) 2 = 1. We have
j j j j j j j j We can then calculate (cid:26)^, A^ and $ from p, r, and Z
chosen thesecond pairofprojections tobe moregen-
and use the trace formula. The arbitrary choices for
eral than those de(cid:12)ned in (27) above since we will
phases do not change any empirical predictions.
need to consider this more general case later. We
can calculate D using (18)
6 Basic Ideas and the Axioms
1 0 1 (cid:12) 2 1 (cid:14) 2
(cid:0)j j (cid:0)j j
0 1 (cid:12) 2 (cid:14) 2
D =0 j j j j 1 We will now forget quantum theory and classical
1 (cid:12) 2 (cid:12) 2 1 (cid:11)(cid:13) +(cid:12)(cid:14) 2
(cid:0)j j j j j (cid:3) (cid:3)j probability theory and rederive them from the ax-
BB1(cid:0)j(cid:14)j2 j(cid:14)j2 j(cid:11)(cid:13)(cid:3)+(cid:12)(cid:14)(cid:3)j2 1 CC ioms. Inthissectionwewillintroducethebasicideas
@ (3A3)
and the axioms in context.
9
6.1 Probabilities 6.2 The state
As mentioned earlier, we willconsider only measure- We can introduce the notion that the system is de-
mentsofprobabilitysinceallothermeasurementscan scribed byastate. Eachpreparationwillhaveastate
be reduced to probability measurements. We (cid:12)rst associated with it. We de(cid:12)ne the state to be (that
need to ensure that it makes sense to talk of prob- thingrepresented by)anymathematicalobjectwhich
abilities. To have a probability we need two things. canbeusedtodeterminetheprobabilityforanymea-
First we need a way of preparing systems (in Fig. 1 surement that could possibly be performed on the
this is accomplished by the (cid:12)rst two boxes) and sec- systemwhenprepared bytheassociatedpreparation.
ond, we need a way of measuring the systems (the It is possible to associate a state with a preparation
third box in Fig. 1). Then, we measure the number because Axiom 1 states that these probabilities de-
of cases, n , a particular outcome is observed when pend on the preparation and not on the particular
+
agiven measurement isperformed onan ensemble of ensemble being used. It follows from this de(cid:12)nition
n systems each prepared by agiven preparation. We of a state that one way of representing the state is
de(cid:12)ne byalistofallprobabilitiesforallmeasurements that
could possibly be performed. However, this would
n
+ almost certainly be an over complete speci(cid:12)cation
prob = lim (38)
+ n n of the state since most physical theories have some
!1
structure whichrelatesdi(cid:11)erentmeasured quantities.
Inorder foranytheoryofprobabilitiestomakesense
We expect that we will be able to consider a subset
prob must take the same value foranysuch in(cid:12)nite
+ of all possible measurements to determine the state.
ensembleofsystemspreparedbyagivenpreparation.
Hence,todeterminethestateweneedtomakeanum-
Hence, we assume
ber of di(cid:11)erent measurements on di(cid:11)erent ensembles
of identically prepared systems. A certain minimum
Axiom 1 Probabilities. Relative frequencies (mea- numberofappropriatelychosenmeasurementswillbe
sured by taking the proportion of times a particular both necessary and su(cid:14)cient to determine the state.
outcome is observed) tend to the same value (which Let this number be K. Thus, for each setting, k=1
we call the probability) for any case where a given to K, we will measure a probability p with an ap-
k
measurement is performed on an ensemble of n sys- propriate setting of the knob on the measurement
tems prepared by some given preparation in the limit apparatus. These K probabilitiescan be represented
as n becomes in(cid:12)nite. by a column vector p where
p
1
Withthisaxiomwecanbegintobuildaprobability p
2
0 1
theory. p
p= 3 (39)
Some additional comments are appropriate here. B .. C
B . C
There are various di(cid:11)erent interpretations of proba- B C
Bp C
bility: as frequencies, as propensities, the Bayesian B KC
@ A
approach, etc. As stated, Axiom 1 favours the fre- Now, this vector contains just su(cid:14)cient information
quency approach. However, it it equally possible to todeterminethestateandthestatemustcontainjust
cast this axiominkeeping withthe other approaches su(cid:14)cientinformationtodeterminethisvector(other-
[16]. Inthispaperweareprincipallyinterested inde- wise it could not be used to predict probabilities for
riving the structure of quantum theory rather than measurements). In other words, the state and this
solving the interpretational problems with probabil- vector are interchangeable and hence we can use p
ity theory and so we will not try to be sophisticated as a way of representing the state of the system. We
with regard to this matter. Nevertheless, these are will call K the number of degrees of freedom associ-
importantquestions which deserve further attention. ated with the physical system. We will not assume
10
