Quantum Logic and Partially Ordered Abelian Groups

69 Pages·2008·4.64 MB·English

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HANDBOOK OF QUANTUM LOGIC AND QUANTUM STRUCTURES: QUANTUM STRUCTURES 215
Edited by K. Engesser, D. M. Gabbay and D. Lehmann
O 2007 Elsevier B.V. All rights reserved
QUANTUM LOGIC AND PARTIALLY
ORDERED ABELIAN GROUPS
David J. Foulis and Richard J. Greechie
1 INTRODUCTION
Our purpose in this expository article is to give an account of the connection
between quantum logics1 and partially ordered abelian groups. In what follows,
we use the word "logic" in the sense of a partially ordered algebraic structure L
that could be interpreted as a semantic model for a formal symbolic logic L. This
usage is customary in the literature of algebraic logic [Halmos, 19621, and it is
consistent with the nomenclature of Birkhoff and von Neumann in their seminal
article on the logic of quantum mechanics [~irkhofafn d von Neumann, 19361. Our
emphasis is on the mathematical structure of L rather than on the interpretation
of its elements as propositions pertinent to a physical system. An account of L
and C from the point of view of logic as an instrument of reasoning, as well as
more comprehensive treatment of the general notion of a quantum logic, can be
found in [~allCah iara et al., 20041.
We begin, in Sections 2-5 below, with a brief sketch touching on some of the ba-
sic notions of orthodox quantum mechanics (QM). By orthodox QM, we mean non-
relativistic Hilbert-spaced-based QM without superselection rules and for which
the observables are represented by projection-valued (PV) measures. Readers al-
ready knowledgeable about quantum physics can skim these sections rapidly to
familiarize themselves with our notation. The relevant quantum logic (QL) for
orthodox QM is the complete atomic orthomodular lattice P(4j) of projection o p
erators on a Hilbert space Jj. The QL P(4) is a subset of the partially ordered
abelian group G(4) of Hermitian (i.e., bounded self-adjoint) operators on 4. A
comprehensive and authoritative account of the standard quantum logic P(4j) in
the context of orthodox QM can be found in [~eltramettain d Cassinelli, 19811.
In Section 6, we turn our attention to the much more general and flexible
positiveoperator-valued (POV) measures for which the corresponding QL is the
secalled effect algebra IE(4) of all positive semidefinite Hermitian operators on
4j that are dominated by the identity operator. Evidently P(4) C IE(4) C_ G(5).
As "quantum-logical propositions", effect operators in IE(4) can manifest "fuzzi-
ness" or "un~harpness'~w,h ile the projection operators in P(4) can be regarded as
l ~ foellow common usage in which the term "quantum logic" refers not only to a logic
associated with a quantum-mechanical system, but indeed to any physical system whatsoever.
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216 David J. Foulis and Richard J. Greechie
"sharp," even though their "truth values" might be subject to statistical fluctua-
tions. For an exposition of POV measures and the contemporary quantum theory
of measurements, see [~uschet al., 19911.
The mathematical structures of P(Ej), IE(fj), and G(4) are so rich and the
quantum-mechanical interpretations of these structures are so intriguing that one
cannot resist the temptation to formulate and study more general triples P C
c c
E G as abstractions or analogues of P(4) C IE(4j) G(4). Thus, in Sections
7 through 16 we conduct an abstraction process, motivated and guided by suit-
able examples, that will ultimately result in the notion of a so-called "CB-triple"
c c
P E G consisting of a regular orthomodular poset P, an effect algebra E,
and a CB-group G, i.e., a partially ordered abelian group enriched by a so-called
"compression base." The notion of a CB-group is both very general and mathe-
matically attractive, and may be considered as an appropriate basis for the general
study of quantum logics. In Sections 17 and 18, we introduce and study a class of
CB-groups, called archimedean RC-groups, for which a spectral theory has been
developed.
Because of the expository nature of this article, we give prooh only when we
regard them as being illuminating or when they could be difficult to locate in
the existing literature. Background material on Hilbert spaces can be found in
[Halmos, 19981, a source for basic facts in regard to a-fields, measurable functions,
and so on, is [Halmos, 19501, and [Halmos, 19631 is an authoritative exposition of
the theory of Boolean algebras-all three of these classic references were authored
ri ad is on
by P.R. Halmos. The monographs of R.V. Kadison and J.R. Ringrose
and Ringrose, 19831 provide comprehensive expositions of both the basic and ad-
vanced theory of operator algebras, and the treatise of G. Emch [~mch1,9 721
is an excellent reference for applications of operator algebras to physics. In our
treatment of partially ordered abelian groups, we follow the monograph [~oodearl,
19851 of K.R. Goodearl. For general lattice theory, the classic [~irkhoff1, 9791 of
G. Birkhoff is the standard reference.
2 ORTHODOX QUANTUM MECHANICS
Orthodox quantum mechanics (QM) is founded on the assumption that there is a
correspondence S -+ fi assigning a complex separable Hilbert space fi to a QM-
system S. Let (. , be the inner product2 on fj and denote the additive abelian
a)
group of all bounded Hermitian operators on 4j by G(fi). The tenets of orthodox
QM speclfy that bounded (real) observables for S are represented by operators
A E G(fj), (pure) states for S are represented by unit vectors E 4, and the
$J
expectation value of an observable A when the system S is in the state $J is given
by (A$J,$J).
DEFINITION 1. If $J is a state vector in 4, i.e., ll$Jll = 1, and W denotes the
system of real numbers, we define the expectation mapping w$: G(4j) -+ IR by
I
2~hysi~isutssu ally use the Dirac bra-ket notation (. .) for the inner product.
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Quantum Logic and Partially Ordered Abelian Groups 217
w+(A) := (A$, $) for all A E G(J~).~
The expectation value w+(A) is usually interpreted as the limit as n + oo of the
arithmetic mean of n independent measurements of the bounded observable A on
the QM-system S, or replicas thereof, in the fixed state $.
The expectation mappings w+ are used to organize the Hermitian group G(4)
into a partially ordered abelian group [Goodearl, 19851 as follows: if A, B E 6(4),
then by definition A 5 B iff4 w$(A) 5 w+(B) for every state vector $J E 4. In
fact, under the partial order 5, G(4) is an order-unit-nomed real Banach space
with the identity operator 1 E G(4) as the order unit [Alfsen, 1971, page 691.
F'urthermore, the order-unit norm coincides with the uniform operator norm 11 . 11
on G(fi), and therefore, for A E G(a),
(1) IlAll = inf{m/n I 0 5 m E Z, 0 < n E Z, and - ml 5 nA 5 ml),
where, as usual, Z is the ring of integers. If 0 is the zero operator in G(*), then
(by a slight abuse of language), observables A E G(4) with 0 5 A are called
positive. We note that the norm of a positive bounded observable is the maximum
expectation value of that observable for all state vectors, i.e.,
The positive operators A in G(4) are exactly the operators of the form A = B2
for B E G(4). Indeed, if B E G(4), then for every state vector $,
wahen ce 0 5 B2. Conversely, if 0 5 A E G(4), then there is a unique operator
E G(4) such that 0 5 and A = [~iesazn d Nagy, 1955, page 2651.
a,
If A E G(4), then 0 5 A', and one defines IAl := A+ := l2( IAl +A), and
A- := i(IAI - A). Then
Thus, every bounded observable is the difference of two bounded positive observ-
ables.
Suppose that A1 5 A2 5 . . . is an ascending sequence of bounded observables.
Then the sequence is bounded above in norm (i.e., there is a real number P with
llAnll 5 P for all n E M := {1,2,3, ...)) iff it is bounded above in G(4) (i.e.,
there is a bounded observable B E G(4) such that A, 5 B for all n E M).
Furthermore, if the sequence is bounded above in either sense, then by Vigier's
theorem [Riesz and Nagy, 1955, page 2631, there is a uniquely determined bounded
observable A E G(4) such that, for every $ E 4, IIA,$ - A611 + 0 as n + oo,
i.e. the sequence A1, A2,. .. converges to A in the strong operator topology. But,
if A, -t A in the strong operator topology, then A, + A in the weak operator
3 ~ hneo tation := means "equal by definition."
4We use "iff' as an abbreviation for "if and only if!'
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218 David J. Foulis and Richard J. Greechie
topology, and it follows that w+(A,) converges monotonically up to w+(A) for
each state vector $ E 4j. The latter condition implies that A is the supremum
...
(least upper bound) of the sequence All A2, in G(4j); hence the Hermitian group
G(4j) is (Dedelcind) monotone a-complete, i.e., every bounded ascending sequence
in G(4) has a supremum.
Now suppose that B1, B2,. .. is a sequence of positive bounded observables, de-
fine the partial sum A, := Cy=LBi= flo r all n E N,a nd assume that the ascending
sequence A1 5 A2 5 . is bounded above. Then the sequence All A2, .. . has a
-0
supremum A to which A, converges in the strong operator topology. By defini-
tion, CEl Bi: = A, where the series converges in the strong (hence in the weak)
operator topology. In particular, for every state vector $, we have
and, in this sense, the expectation mappings w+ are countably additive on positive
observables.
According to orthodox QM, the spectrum spec(A) of an observable A E G(4j)
can be interpreted as the subset of the real number system R consisting of all pos-
sible numerical values of individual measurements of A in various states5. Thus,
if P E G(4j) and spec(P) G (0, I}, then we can regard P as a "two-valued propo-
sition" about the system S with possible truth values 1 (true) and 0 (false). As is
easily seen, if P E G(Ej), then spec(P) C {O,l) ifF P = P2. Thus, we define
and regard observables P E P(5) as experimentally testable truelfalse (or yes/no)
propositions about the QM-system S. The zero operator 0 and the identity o p
erator 1 on 4j belong to P(fi), and they can be regarded as propositions that
are always false and always true, respectively. The truth values of propositions
P E P(4j) other than 0 and 1 may vary from measurement to measurement, even
if the state $ remains constant. Operators P E P(5) are called projections (or, by
some authors, projectors).
If P E P(5) and $ is a state vector in 4, then by Equation (3), w+(P) =
11
w+(P2) = P$1I2. The expectation value w+(P) is the long-run relative frequency
with which the result 1 (true) will be obtained for independent measurements
of P in state $. In other words, w+(P) = llP$112 is the probability that the
proposition P is true (or more accurately, will be found to be true if tested) when
the QM-system S is in state $. Thus, we shall refer to the restriction of the
expectation mapping w+ to P(4) as the probability measure or as the probability
state determined by the state vector $.
Suppose that $ and 4 are state vectors in 4j and denote by C the system of
complex numbers. Then
5~osrh ort, a measurement of A when the system S is in state qlJ is often referred to as a
measurement of A in state qlJ.
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Quantum Logic and Partially Ordered Abelian Groups
(7) w$ = w+ on P(4) @ 30 E @ with 101 = 1 and I) = 04.
For a general quantum logic L, a state is often defined as a "probability mea-
sure" 7 ~ L: -+ [O, l] W (see Section 7 below). By (7), in the prototype quantum
logic P(fj), this entails disregarding phase relations among the state vectors. The
identification of physical states with probability states might be innocuous as long
as one is considering independent measurements of one and only one observable,
but the phase relations could be critical when non-independent or sequential mea-
surements are involved. However, we shall not address this issue in the present
article.
The partial order on the Hermitian group G(4) induces (by restriction) a partial
order on P(4) and, for P, Q E IF(%), P 5 Q iff the probability w$(P) is less than
or equal to the probability w$(Q) for every state vector I). It is not difficult to
show that
If P, Q E P($) we say that P and Q are orthogonal, in symbols P I Q, iff
+
P Q E P(4j). Thus,
It is convenient to have a special symbol for the sum of two orthogonal projections,
and we use @ for this purpose. Thus:
We refer to PCBQ as the orthogonal sum, or for short the orthosum, of the orthog-
onal projections P and Q.
Suppose that P, Q E P(4). In regarding the system P(4) as a "logic" and P, Q
as "propositions," we regard 1 - P as a "negation" or "logical denial" of P, and
<
we interpret P Q to mean that P "implies" Q.6 Since
the condition P I Q means that, in some sense, P and Q "refute each other."
Thus, if P I Q, we may think of P@Q as a kind of "disjunction" of the mutually
refuting propositions P and Q. We shall refer to P(4), equipped with the "negation
mapping" P ct 1 - P, the partially defined binary operation @, and the partial
order relation 5, as the sharp quantum logic associated with the Hilbert space 4.
(In Section 6 below, an "unsharp" or "fuzzy" quantum logic will be introduced.)
Let PI,P 2,P 3,. . . be a sequence of pairwise orthogonal propositions in P(4).
Then, it can be shown that Cy=l Pi E P(4) for every positive integer n, whence
it is natural to define
n
'~oteth at the implication relation 5 is not a logical connective on the quantum logic P(4).
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220 David J. Foulis and Richard J. Greechie
ELl
or with alternative notation, Pi := C;=l pi. As the partial sums Pi
EEl
are bounded above by 1,i t follows that the infinite series Pi converges (in the
strong and weak senses) to an observable in G(4) which is, in fact, the supremum
of the partial sums. It can be shown that Pi E IF'($), whence it is natural to
define
00 C00
@pi := E P(4)
and to regard @El as a kind of "infinite orthogonal disjunction" of the propo-
sitions PI,P 2,P 3,. . . . By (5), the probability measures determined by the state
vectors are countably additive in the sense that
for each state vector I).
DEFINITION 2. Let B(W) be the a-field of Borel subsets of W. A projection-valued
(PV) measure is a mapping M H PMf rom Borel sets M E B(W) to projections
PM E P(9) that has the following properties: (i) fi = 1. (ii) If MI, M2, M3, ...
UZl
is a pairwise disjoint sequence in B(B) and M = @Mzil, then PMlP, M2..,. is
a pairwise orthogonal sequence in P(4) and PM = PMi. The PV-measure
M o PM is said to be bounded iff there are real numbers a < P such that
P[a,pl = 1.
As a consequence of the spectral theorem, a bounded observable A E G(4)
determines and is determined by a bounded PV-measure M H PM in such a way
that A = JR L dPM, where L is the identity function on B and the integral converges
in the uniform norm topology. Here the proposition PM is to be interpreted as
asserting that the measured value of A belongs to the Borel set M. Likewise,
unbounded observables also correspond to PV-measures, but without the bound-
edness condition. If M H PM is a PV-measure and $ is a state vector, then
M H w,+,,(PM)i s a countably additive probability measure on the a-field B(B);
thus by combining an observable (bounded or unbounded) and a state we obtain
a conventional Kolrnogorovian probability measure.
Operations that are only partially defined are always something of an annoyance,
and it is reassuring to keep in mind that the partial binary operation on P(4) can
+
be extended to a bona jide binary operation on the whole Hermitian group G(4);
furthermore, for each state vector I), the probability measure (or probability state)
P H w+(P) on P(4) can be extended to the expectation mapping A H w+(A) on
G(4).
3 PROJECTIONS AND COMPRESSIONS
Closed linear subspaces ?JJZ of 9 play important roles in the study of the QM-
system S. It is often necessary or desirable to drop down to a subspace ?JJZ 4
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Quantum Logic and Partially Ordered Abelian Groups 221
to eliminate degrees of freedom that are not of current interest. For instance, in
considering the spin components of a spin-112 particle S, it is appropriate to drop
down to a twedimensional subspace IM of 4 where the spin observables "live"
[Beltrametti and Cassinelli, 1981, Chapter 41.
Let T be a bounded linear operator (not necessarily self-adjoint) on 4 and let
IM be a closed linear subspace of 4. Then IM is said to be invariant under T ifT
T(IM) C IM, in which case the restriction TIm of T to IM is a bounded linear oper-
ator on IM. If IML := {< E 4 1 (E, v) = 0, Vq E IM} is the orthogonal complement
of IM, and if both IM and DlL are invariant under T, then T decomposes into sort
of a "direct product" of TIzm and TIm*, and we say that Dl reduces T. If T is
Hermitian, then IM is invariant under T iff IM reduces T.
If P E P(4) and IM is a closed linear subspace of 4, then P is said to be the
projection onto IM ifT the range of P is IM, i.e., P(4) = IM. There is a one-teone
correspondence P * IM between projections P E P(4) and closed linear subspaces
IM of 4 given by P(4) = IM. If P is the projection onto Dl, then 1- P is the
projection onto IM1. If T is a bounded linear operator on 4 and P is the projection
onto IM, then IM is invariant under T iff PTP = TP. The mapping T H PTP,
called the Naimark compression determined by P, maps G(4) into itself, and we
shall refer to its restriction to G(4) as a compression. Compressions on G(4)
will play an important role in our subsequent development, and we shall use the
notation in the following definition for them.
DEFINITION 3. If P E P(4) we define the compression Jp: G(4) -+ G(4) by
Jp(A) := PAP for all A E G(4).
The compressions Jp provide a convenient way for dealing algebraically with
closed linear subspaces of 4. For instance, let P E P(4) and let IM = P(4). If
A E G(Jj), then the restriction JP(A)IW of Jp(A) = PAP t0.m is a Hermitian
operator on the Hilbert space IM. Moreover, under restriction to IM, the subgroup
I
Jp(G(8)) = {PAP A E G(fj)} of G(4) is isomorphic as a partially ordered
abelian group to the Hermitian group G(IM).
The compression mappings Jp on G(4) are characterized algebraically as follows
(see [Foulis, 2004al).
THEOREM 4. Let J : G(4) + G(4) and let P := J(1). Then P E P(4) and
J = Jp iff J is an order-preserving group homomorphism, P 5 1, and J(E) = E
for all E E G(4) with 0 5 E 5 P.
4 SYMMETRIES
In this section, we give a very superficial sketch to indicate how physical sym-
metries of the QM-system S interact with the Hermitian group G(4) and the
projection lattice P(4). According to the so-called "Heisenberg picture," a phys-
<
ical symmetry of S (e.g., a rotation of S in physical space) induces a mapping
A <A on the observables A for S, <A being the observable into which A is trans-
H
<.
formed when S is transformed by the symmetry If A is a bounded observable,
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222 David J. Foulis and Richard J. Greechie
then of course JA is again a bounded observable, and plausible arguments can be
made to show that the mapping A o JA is a "unital automorphism" as per the
next definition.
DEFINITION 5. A mapping E: G(9) G(4) is a unital automorphism iff E is
t
an automorphism of the abelian group G(9) such that J1= 1a nd both J and J-I
are order-preserving mappings on G(4).
In what follows, we shall assume that physical symmetries for S correspond to
unital automorphisms J of G(4).
THEOREM 6. Suppose that 5 is a unital automorphism on G(4). Then there
exists a unitary or antiunitary operator U on 4 such that [A = UP1AU for all
A E G(9). Conversely, each unitary or antiunitary operator U on 4 gives rise to
a unital automorphism J on G(4) defined by JA := U-lAU for all A E G(4).
Proof. If the dimension of 4 is three or more, then the desired conclusion follows
from G. Ludwig's version of Wigner's theorem on symmetry transformations (see
[Casinelli et al., 19971). That the conclusion also holds when the dimension of 4 is
2 follows from [~olnharn d Phles, 20011. If 4 has dimension 1, then W is isomorphic
under X A 1 to G(4) as an ordered group, and the desired conclusion follows
H
from the fact that the only order-preserving group automorphisms of B are of the
form X H KX where K E W is a positive constant. The converse is obvious.
COROLLARY 7. If J is a unital automorphism on G(B), then J maps P(4) onto
itself, and the restriction of J to P(4) is a order automorphism that preserves
orthogonality and orthogonal sums.
In the sense of Corollary 7, a physical symmetry induces a "logical symmetry" on
the sharp quantum logic P(5).
z
As a consequence of Theorem 6, if is a multiplicatively-written group (e.g.,
a group of physical symmetries of S), then a unitarylantiunitary representation
of z on 4 is mathematically equivalent to a unital action of E on G(4), i.e., a
mapping (J, A) JA from E x G(4) to G(4) satisfying the following conditions
H
for all J, J' E E and all A, B E G(4): (i) (<<')A = <(<'A). (ii) If 1 is the identity
in=, then lA=A. (iii) Jl = 1. (iv) J(A+B) =JA+JB. (v) A 5 B+JA 5EB.
In the theory of unitarylantiunitary group representations, the notion of re-
duction of a representation by a closed linear subspace is of special significance
[~chroeck1, 996, Chapter 111, Section 11. If a unitary/antiunitary representation of
a group E on Ij is expressed as a unital action of E on G(4), then a closed linear
subspace 2132 of 4 reduces the representation iff the compression Jp corresponding
to the projection P onto commutes with the unital action in the sense that
J(Jp (A)) = Jp( <A) for all J E Z and all A E G(4).
Sometimes it is convenient to adopt an alternative point of view, namely the
secalled "Schrodinger picture." According to the Schrodinger picture, a physical
symmetry J induces a mapping Ijl o JIjl on the state vectors Ijl to be interpreted
as follows: if the system S is in state Ijl before the application of the physical
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Quantum Logic and Partially Ordered Abelian Groups 223
<,
symmetry it will be in state <$ after the application of J. The Heisenberg and
Schrodinger pictures are physically equivalent, and the relation between them is
simply that w+(JA) = w<*(A) for all A E G(4) and all state vectors $. From
this relation, it follows that if U is the unitary or antiunitary transformation on
4 representing 5 as in Theorem 6, then <$ = U($).
Temporal transformations of states (Schrodinger picture) or observable5 (Heisen-
berg picture) according to a (reversible) dynamical law are represented by a one-
ct#
parameter family (<t)tEwo f symmetries satisfying <t+t~= & o for all t, t' € W.
For instance, in the Schrodinger picture, starting from an initial state $0, the
state of the system S evolves in time t' to the state 1Jlt1 = <t~lJloI. n an ad-
ditional t units of time, the system will evolve from the state $ t ~to the state
<t$tl = Et(<tl$o) = Jt+t40.
By Theorem 6, a (reversible) dynamical law corresponds to a one-parameter
family (Ut)tEwo f unitary or antiunitary transformations on 4j such that UtSt1 =
Ut o Utl for all t, t' E W. However, since the square of a unitary or antiunitary
operator is always unitary, and since Ut = (UtI2)' for all t E W,a ll operators in
the family are necessarily unitary. If the one-parameter unitary group (Ut)tEw
is continuous in the strong operator topology, then by Stone's theorem [Kadison
and Ringrose, 1983, Theorem 5.6.361, there is a (possibly unbounded) self-adjoint
operator H on 4j such that Ut = e-iHt for all t E B. The operator H is the
Hamiltonian for the physical system S, and it corresponds to the energy observable
for S.
5 MIXED STATES AND DENSITY OPERATORS
If the QM-system S is coupled with another QM-system system S' to form a
composite system S +St,t hen the tenets of QM stipulate that the Hilbert space
+
for S Sti s the Hilbert-space tensor product 4j @fito f the Hilbert spaces Sj and
3' for S and St,r espectively [Aerts et al., 19781. Even if the composite system
+
S S' is in a pure state represented by a unit vector in 4 @ fit, the subsystem
S need not be in a pure state. In general, the subsystem S will be in a secalled
mixed state in the sense that the expectation values for an observable A measured
on S will be a convex linear combination of expectation values w+(A), w+(A), .. .
for suitable state vectors $, 4, ... E 4. Such a mixed state is represented by a
secalled density operator.
Let T(3) be the (real) linear subspace of G(4j) consisting of the Hermitian
trace-class operators, and let D(4) be the convex subset of T(4j) consisting of
the positive trace-class operators D with tr(D) = 1. Operators D E D(4j) are
called (von Neumann) density operators on 4, and the (possibly) mixed states for
the QM-system S are represented by these density operators7. By definition, the
expectation value wD(A) of the bounded observable A E G(4) when the system S
is in the state D E D(4j) is given by the Born formula
'physicists usually denote density operators by the Greek letter p.
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224 David J. Foulis and Richard J. Greechie
As in Equation (5), the expectation mapping WD is countably additive.
If $ E Sj is a state vector, then the projections P$ E P(Sj) onto the one-
dimensional subspace of Sj spanned by $ is a density operator on Sj, and the
ll$ll
density operators of the form P$ for $ E Sj with = 1 are the extreme points
of the convex set D(9). Furthermore, wp, as given by the Born formula (1)
coincides with the expectation mapping wq as in Definition 1. Thus (neglecting
phase factors), the pure states of the QM-system S are represented by the extreme
points of the convex set ID@).
Mixed states also arise in connection with ensembles of replicas of the QM-
system S. If $i, i = 1,2, ..., n, are state vectors, and if for each i a fraction pi of
the systems in the ensemble are in the pure state Qi, then the state of the ensemble
can be considered to be the mixed state represented by the convex linear sum
If A E G(Sj), then wD(A) can be interpreted as the expectation value for a mea-
surement of the bounded observable A on a system chosen at random from the
ensemble.
If P E P(8) and D E D(4) is a density operator, then the expectation value
wD(P) is the long-run relative frequency with which the result 1 (true) will be
obtained for independent measurements of P in the mixed state Dl i.e., wD(P) =
tr(DP) is the probability that the proposition P is true (or, more accurately, will
be found to be true if tested) when the QM-system S is in the mixed state D.
Thus, we shall refer to the restriction of the expectation mapping WD to P(3)
as the probability measure or as the probability state determined by the density
operator D.
An alternative interpretation for the mixed state D in Equation (2) is afforded
by the notion that an agent might be in possession of information that the QM-
system S is in one of the pure states $i, and that pi is the probability that S is
in state $if or i = 1,2, ..., n. Then wD(P) = tr(DP) is the probability that the
agent should assign to the proposition P E P(5). According to this interpretation,
although the pure states P$, may be ontic, i.e., "real physical states," the mixed
state D is epistemic, i.e., it is a state of incomplete knowledge about the ontic state
of S. For an exposition of the epistemic interpretation of states, see [F'uchs, 2002;
Spekkens, 20041.
With the partial order inherited from G(Sj), T(Sj) is a base-nonned Banach
space with the density operators D(Sj) as the cone base [Alfsen, 1971, page 771,
and the base norm coincides with the trace norm. The base-normed Banach space
T(B) is in separating order and norm duality with the order-unit Banach space
G(Sj) under the bilinear form (., .) defined by (C, A) := tr(CA) for C E T(Sj) and
A E G(Sj); in fact, under this duality, G(Sj) may be identified with the Banach
s~hysicistsu se the Dirac notation !$)(+I for the projection PJ,.