Complex Probability Measure and Aharonov's Weak Value

0.16 MB·English

by  Tatsuya Morita

Complex Probability Measure and Aharonov’s Weak Value
Tatsuya Morita∗
Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
Toshihiko Sasaki†
Photon Science Center, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
Izumi Tsutsui‡
Theory Center, Institute of Particle and Nuclear Studies,
High Energy Accelerator Research Organization (KEK), 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan
(Dated: October 5, 2012)
2 We present a complex probability measure relevant for double (pairs of) states in quantum me-
1 chanics, as an extension of thestandard probability measure for single states that underlies Born’s
0 statistical rule. When the double states are treated as the initial and final states of a quantum
2 process, we find that Aharonov’s weak value, which has acquired a renewed interest as a novel
t observablequantityinherentintheprocess,arisesasanexpectationvalueassociatedwiththeprob-
c ability measure. Despite being complex, our measure admits the physical interpretation as mixed
O
processes, i.e.,an ensembleof processes superposed with classical probabilities.

4
PACSnumbers: 03.65.-w,02.50.Cw,03.65.Aa,03.65.Ta,03.65.Ca

]
h
p Introduction. The weak value, which has long been the possible form of probability measure that can be as-
- advocatedbyAharonovetal.[1]asanovelandphysically sociatedwithpairsofstates,called‘doublestates’inthis
t
n measurable quantity, is currently attracting much atten- paper,inamannerconsistentwiththegivenstates. Em-
a tion. Theweakvaluearisesasanintrinsicphysicalentity ployinganargumentanalogoustothatusedinGleason’s
u inthetime-symmetricformulationofquantummechanics theorem [9] which determines the real probability mea-
q
[2], where one considers a quantum process specified by sure and lays a foundation of Born’s statistical rule for
[
both initial and final states (obtained by pre- and post- singlestates,weshowthatthe measurefordouble states

1 selections), instead of only an initial state (preselection) can also be determined properly. Although our measure
v in the usual formulation of quantum mechanics. turns out to be complex in general and contains a free
8
parameter, it reduces in the single state limit uniquely
9 There are basically two reasons for the renewed inter-
to the conventional real measure crucial to Born’s rule.
2 est in the weak value. One is that it offers an intrigu-
When appliedto quantumprocessesinwhich the double
1 ingpossibility ofextending the scopeof‘physicalreality’
0. in quantum mechanics, and thereby sheds a new light states are identified with the initial and final states, our
complex measure is found to admit the interpretation of
1 both on paradoxical phenomena such as Hardy’s para-
mixed processes, i.e., ensemble of processes superposed
2 dox [3, 4], and on fundamental issues such as the direct
1 detection of wave function [5] or tracing the trajectories with real probabilities, assuring that it is indeed phys-
: ically testable. The weak value is then obtained as the
v of Young’s double slit experiment [6]. The other, which
expectationvalueoftheobservablemeasuredinapartic-
i ismoreimportantforpracticalpurposes,isthatitserves
X ular ensemble corresponding to a pure process specified
to measure physical quantities by macroscopic appara-
r tuses with significantly amplified precision, as has been by the double states.
a
demonstrated in the strength of interaction in the spin In what follows, we begin by invoking the probabil-
Hall effect [7] or in the beam diffraction [8]. ity measure for single states and proceed to discuss the
probability measure for double states. We then apply
Despite these impressive developments and the en-
our measure to quantum processes to see how the weak
hancedrecognitionofitssignificance,theweakvaluestill
value arisesthere, and argue that the appropriatelygen-
remains obscure, not just because it is complex-valued
eralizedmeasure admits anoperationalinterpretationas
and defies our direct perception as physical reality, but
mixed processes.
also because it lacks a solid basis for the uniqueness of
the form, that is, the reason why it takes the form as it
Probability Measure for Single States. For our pur-
stands. The present work is an attempt to provide such
poses,wefirstrecallGleason’stheorem[9]whichdeduces
a basis from probability theory. Specifically, we consider
Born’sstatisticalrulebasedonaprobabilitymeasureful-
filling certain logical conditions. Given a Hilbert space
H of a finite dimension d ≥ 3, consider a real-valued
measure µ which is a map from the space of projection
∗Email:[email protected]
†Email:[email protected] operators P(H) of H to non-negatives, i.e., µ: P(H) →
‡Email:[email protected] [0,∞) ⊂ R. The theorem states that, if the map is
2
bounded |µ(P)| < ∞ and satisfies the partial additiv- Complex Measure for Double States. Now we extend
ity condition, theforgoingargumenttoameasurecharacterizedbydou-
blestates. Let{|ψi, |φi}betwostatesarbitrarilychosen
from H except that they are neither identical (up to a
µ P = µ(P ), (1)
i i
! phase) nor orthogonalto each other (i.e., hφ|ψi6=0).
i i
X X Analogouslyto the single state case (3), giventhe two
forasetofprojectionoperators{P }whicharemutually
i states {|ψi, |φi} we wish to require
orthogonalP P =∅(nulloperator)fori6=j,thenithas
i j
the form, µ(Pψ)=1, µ(Pψ⊥)=0, (5)
µ(P)=tr(WP), (2) µ(Pφ)=1, µ(Pφ⊥)=0,
with a positive self-adjoint trace class operator W. where Pφ = |φihφ| and Pφ⊥ = |φ⊥ihφ⊥| with |φ⊥i ∈
Since we have µ(∅) = 0 from (2), if the range of the Hφ⊥. Obviously, in view of the uniqueness of W, this is
mapµisrestrictedto[0,1],andiftheconditionµ( )=1 impossible unless the two states are identical. However,
is further imposed, then obviouslythe map can be1inter- thecondition(5)canbemetifonepromotesthemeasure
pretedasaprobabilitymeasurewiththeattachedmean- to a complex one.
ing that ∅ and represent propositions which are iden- Toseethis,letusinvokethegeneralizedGleason’sthe-
tically false and1true, respectively. Note that µ( ) = 1 orem[10]whichextendstherangeofthemapfrom[0,∞)
implies tr(W)=1. 1 to the entire reals R. Demanding the condition (1), one
In quantum mechanics, the probability measure is in- finds that such a measure µR admits the same form,
deed realized by such a measure µ, where the operator
µ (P)=tr(W P), (6)
R R
W corresponds to the density operator ρ that character-
izes the state of the system. For instance, if the system but now WR is a self-adjoint trace class operator, not
is described by a pure state ρ = P := |ψihψ| for some necessarily positive. In order to extend the range of the
ψ
normalized |ψi∈H, our probability measure is required map to complex numbers C, we choose two such real
to yield maps µR, µ′R: P(H)→R and consider
µ(Pψ)=1, µ(Pψ⊥)=0. (3) µC(P)=µR(P)+iµ′R(P), (7)
Here,thefirstconditionstatesthattheprobabilityofbe- with the imaginary unit i. Clearly, the map µ still ful-
C
ing in the state |ψi is unity, whereas the second states fills (1) by linearity and is written as
thatthereisnoprobabilityassignedforanarbitrarystate
|ψ⊥i orthogonal to |ψi for which the projection is given µC(P)=tr(WCP), WC =WR+iWR′, (8)
byPψ⊥ :=|ψ⊥ihψ⊥|. Namely,themeasureµhasnosup- where WR and WR′ are the self-adjoint trace class oper-
port for the subspace P(H⊥) ⊂ P(H), where H⊥ is the ators associated with µ and µ′ , respectively. We then
ψ ψ R R
orthogonalcomplement to the one-dimensionalsubspace have:
Hψ = span{|ψi}. We note that, because of (1) the sec- Theorem If a map µC: P(H)→C satisfies the partial
ond condition in (3) actually follows from the first for a additivity condition (1) andtheconsistencycondition (5)
non-negative map µ, but this will no longer be the case for two non-identical states |ψi,|φi with hφ|ψi6=0, then
when the non-negativity is lifted. it has the form,
From (3) one finds that W is uniquely determined as
µ (P) = tr(W P),
W = ρ = P , and this shows that the probability of C C
ψ
the state |ψi being in the subspace H ⊂ H specified by |ψihφ| |φihψ|
i W = α +(1−α) , (9)
the projection Pi reads µ(Pi) = tr(ρPi), which is just C hφ|ψi hψ|φi
Born’sstatisticalrule. Itfollowsthat, if anobservableA
for some α∈C.
is measured in the pure state ρ, the expectation value is
Proof. Thecomplexmeasurefulfilling(1)isgivenby(8)
given by
with a trace class operator W . Let {|e i;i = 1,...,d}
C i
E(A):= aiµ(Pi)=hψ|A|ψi, (4) be a complete orthonormal basis in H with |e1i = |ψi.
In terms of this we expand W and |ψ⊥i∈H⊥ as
Xi C ψ
where a is an eigenvalue of A, and P is the corre-
i i d d
sponding projection appearing in the spectral decom- W = β |e ihe |, |ψ⊥i= γ |e i, (10)
C ij i j i i
position A = a P . One notable consequence of
i i i i,j=1 i=2
this is that the expectation value satisfies the sum rule, X X
E(A+B)=E(AP)+E(B) foranyobservablesA, B which with βij,γi ∈C. From (5) we have
maynotcommutewitheachother. Thisimpliesthat,al-
thoughthesumoftheindividualmeasurementoutcomes 0=µC(Pψ⊥) = βijγkγl∗tr(|eiihej|ekihel|)
ofAandBmaynotbeaneigenvalueofA+B,onaverage Xi,j kX,l≥2
they coincide. = β γ γ , (11)
ij j i
i,j≥2
X
3
which implies β = 0 for i,j ≥2 since γ can be chosen We then notice that Aharonov’s weak value A =
ij i w
arbitrarily. The operator W is thus written, with some hφ|A|ψi/hφ|ψi arises at the choice α = 1 of the expec-
C
(unnormalized) states |ξ1i,|ξ2i∈Hψ⊥, as tation value λ(A). Although λ(A) is complex in gen-
eral,it becomes realat α=1/2 where W becomes self-
C
WC =|ψihψ|+|ψihξ1|+|ξ2ihψ|. (12) adjoint. This shows that one can find the measure µ
which meets the condition (5) by extending the range of
Defining
the map only to the entire R, but we shall soon realize
|ηi=|φi−hψ|φi|ψi∈H⊥, (13) that the particular measure obtained by α = 1/2 does
ψ
not account for all possible cases when applied to quan-
we further decompose
tumprocesses. Noticealsothat,asforE(A)thesumrule
|ξ i=z |ηi+|ξ′i, i=1,2, (14) holds, λ(A+B)=λ(A)+λ(B), for any α.
i i i
Interestingly, in the single state limit, that is, in the
with z ∈ C so that |ξ′i ∈ H⊥ in addition to |ξ′i ∈ H⊥.
i i η i ψ limit |φi → |ψi the ambiguity of α disappears and our
Similarly, if we define
complex measure µ reduces to the real measure µ in
C
|ζi=|ψi−hφ|ψi|φi∈H⊥, (15) (2) with the condition (3) enforced. Accordingly, the
φ
expectation value λ(A) also reduces to the conventional
we find that both |φi and |ζi belong to the linear space one E(A) in (4).
spanned by |ψi and |ηi. It follows that |ξ′i ∈ H⊥ and Another observation worth mentioning is that, since
i φ
|ξi′i∈Hζ⊥ fori=1,2aswell. Thisobservationmotivates under the single state |ψi the probability µ(Pφ) =
us to rewrite (12) in favor of |φi, |ζi and |ξ′i to find |hψ|φi|2 representsthe compatibility ofthe double states
i
|φi and |ψi, one obtains the overall expectation value of
WC = ωζζ|ζihζ|+ωφφ|φihφ|+ωφζ|φihζ|+ωζφ|ζihφ| A by the weighted product, |hψ|φi|2λ(A). The average
+ω |φihξ′|+ω |ξ′ihφ|+|ζihξ′|+|ξ′ihζ|, (16) value obtained after allowing the state |φi to vary freely
φξ1 1 ξ2φ 2 1 2
may then be evaluated by
where
ωζζ =1−z1∗hφ|ψi−z2hψ|φi, |hψ|φi|2λ(A)=hψ|A|ψi, (22)
ωφφ =1−ωζζ +ωζζ|hψ|φi|2, |φXi∈B
ωφζ =z2+ωζζhφ|ψi, (17)
ω =z∗+ω hψ|φi, where the summation is over the states of a complete
ζφ 1 ζζ
ω =hφ|ψi, ω =hψ|φi. basisB ofH. Thisoutcome(22)impliesthattheaverage
φξ1 ξ2φ
isactuallyindependentofthechoiceofthebasissetB as
On the other hand, an analogous argument for the well as the parameter α. Similarly, if one averages over
state |φi demanded by (5) shows that WC must also be |ψi instead, one obtains hφ|A|φi. In other words, the
of the form, information of one of the double states will disappear
once the average of the state is taken, and the outcome
WC =|φihφ|+|φihχ1|+|χ2ihφ|, (18)
becomes the standard expectation value with respect to
with some (unnormalized) states |χ1i,|χ2i ∈ Hφ⊥. Since the other state left over.
|ψiand|φiarenotidentical,wehave|ζi=6 0. Comparing Sofar,wehaveconsideredcomplexmeasureswithWC
(16) and (18), we obtain of the type (9) which fulfils (5). As one can extend the
scope of single states from pure states to mixed states
ω =0, |ξ′i=|ξ′i=0, (19)
ζζ 1 2 by allowing W to be any positive self-adjoint operators
which implies with unit trace, one may similarly extend the scope of
double states by allowing W to be any operators with
C
WC = |ψihψ|+z1∗|ψihη|+z2|ηihψ| unittrace. If weletT(H) be the spaceofoperatorswith
= z1∗|ψihφ|+z2|φihψ|. (20) unit trace, we have
Since z1, z2 are free parameters but subject to ωζζ = 0, βWC +(1−β)WC′ ∈T(H), (23)
we arrive at (9) after putting α=z∗hφ|ψi.
1
Having found the complex measure µ for double for W ,W′ ∈ T(H) and β ∈ C. This shows that, if
C C C
states specified by (5), we may consider the expectation we regard T(H) as the space of general double states,
value of an observable A. Putting aside the question of the space is ‘convex’ in the complex sense. The measure
the meaning of complex probability for the moment, we µ (P) also provides the map Pr(P;W ) := µ (P) =
C C C
just follow the standard construction of the expectation tr(W P), which enjoys the affine property,
C
value as we had in (4) to find
Pr(P;βW +(1−β)W′ )
C C
λ(A) := aiµC(Pi) =βPr(P;W )+(1−β)Pr(P;W′ ), (24)
C C
i
X
hφ|A|ψi hψ|A|φi analogous to the conventional probability map. Along
= α +(1−α) . (21)
hφ|ψi hψ|φi with the property Pr(∅;WC) = 0 and Pr( ;WC) = 1,
1
4
this may be considered as a formal support for µ (P)
C
qualifying as a probability measure, albeit it is complex.
To explore the possible use of the complex measure, and
therebyexamine the physicalsignificance ofthe complex
parameterαin(9)or(21),wenowturntotheprobability
measure for a quantum process.
Probability Measurefor aProcess. Thecomplexmea-
sure for double states can be used to furnish the proba-
bilitymeasureforaquantumprocess|ψi→|φibytaking
thetimedependenceofthestatesproperlyintoconsider-
ation. Lett andt be the initialtime andthe finaltime FIG. 1: The double state WC(t) in (25) can equally be asso-
i f ciatedwitheitherofthetwopureprocesses,|ψi→|φishown
at whichthe states |ψi and |φi are realized,respectively,
bythesolid arrows or itsdualU(ti−tf)|φi→U(tf −ti)|ψi
and let t be the time of ‘measuring’ the obserbable A in
shown by the dashed arrows, representing the forward and
the period, t ≤ t ≤ t . To evaluate the outcome of the
i f backward time developments.
measurement results, we would like to have the complex
measure relevant at time t. Assuming that our system
is closed during the process, we have a unitary opera- the parameter of the superposition α. In particular, the
tor U to describe the time development in the period. choice α = 1 yields the pure process |ψi → |φi whereas
The forward time-developed state at t from the initial the choice α = 0 yields another pure process |φ˜i → |ψ˜i.
state is then given by U(t −ti)|ψi, and the backward Throughthisidentificationwecanestablishaone-to-one
time-developed state at t from the final state is given by correspondencebetweenthemeasureandthesuperposed
U(t−tf)|φi. Thissuggeststhat,insteadofthetwostates process,andthereby removethe ambiguityin the expec-
|ψi,|φi,weshouldusetheseforwardandbackwardtime- tationvalueλ(A). Asaresult,λ(A)agreespreciselywith
developedstatestocharacterizethemeasure(9). Weare Aharonov’s weak value A for W (t) when the process
w C
thus led to is |ψi→|φi.
Onphysicalgrounds,onemayarguethatthisinterpre-
U(t−t )|ψihφ|U†(t−t )
W (t)=α i f tationis marredby the complexnatureofthe superposi-
C
hφ|U(t −t )|ψi
f i tion. This concerncanbe dismissed by invokingthe fact
U(t−t )|φihψ|U†(t−t ) that any operator W ∈T(H) admits the expansion,
f i C
+(1−α) , (25)
hψ|U†(t −t )|φi
f i d+1
|ψ ihφ |
i i
W = p , p ∈[0,1], (28)
from which we can obtain the time-dependent measure, C i hφ |ψ i i
µC(P;t) := tr(WC(t)P). The expectation value (21) Xi=1 i i
then acquiresthe correspondingtime-dependence by the in terms of pairs of states |ψ i, |φ i ∈ H for which
i i
use of µC(P;t), which is now characterized by the con- hφi|ψii6=0fori=1,...,d+1(seetheAppendix). Since
sistency condition at the initial and final times, tr(W )= p =1, one can interpret the double states
C i i
as a superposition of processes |ψ i→|φ i with the real
i i
µC(Pψ;ti)=1, µC(Pψ⊥;ti)=0, (26) probabilityPpi (hereafter we omit the time dependence
µC(Pφ;tf)=1, µC(Pφ⊥;tf)=0. for simplicity, pretending U = ). One therefore sees
1
that the complex superposition of processes can actu-
However,theidentificationoftheoperatorW by(25)
C ally be viewed as a classical probability mixture of dif-
with the process |ψi → |φi in the period [t ,t ] is not
i f ferent processes. Note that, since the expansion (28) is
quite correct, because our measure for double states is
far from unique, so is the set of processes superposed.
originally given at a single time t and does not involve
We may call those double states with W2 = W ‘pure’,
the time direction in any intrinsic manner. In fact, one C C
and otherwise call them ‘mixed’, knowing that the for-
can also associate the same W in (25) with the ‘dual’
C mer has only one term in the expansion (28) and reads
processU(t −t )|φi→U(t −t )|ψiinthesameperiod,
i f f i W =|ψihφ|/hφ|ψi for some pair ofstates, |ψi and|φi.
since the two states that determine the double states at C
The probabilistic interpretation admitted by the ex-
time t are equivalent in both cases (see Fig. 1). In fact,
pansion (28) explains how the expectation value of the
˜ ˜
with |φi := U(t −t )|φi, |ψi := U(t −t )|ψi, one can
i f f i observableAcanbe obtained. Indeed, ifwe considerthe
equally characterize our measure by
measurementofAinanensembleofprocesses|ψ i→|φ i
i i
each of which prepared with probability p , the expecta-
µC(Pφ˜;ti)=1, µC(Pφ˜⊥;ti)=0, (27) tion value reads i
µC(Pψ˜;tf)=1, µC(Pψ˜⊥;tf)=0,
hφ |A|ψ i
i i
p =λ(A), (29)
instead of (26). This indicates that the proper inter- i hφ |ψ i
i i
i
pretation of WC is that it is the measure correspond- X
ing to a linear superposition of the two processes, with in agreement with (21).
5
Our double states W in the context of quantum pro- the double states may have more applications than the
C
cesses are closely related to the ‘two-states’ proposed in one mentioned here.
[11] which are defined by W = c |e ihf | with In our probabilistic treatment, the weak value A is
T i,j ij i j w
c ∈ C. The states appearing in the expansion are obtained as the expectation value λ(A). However,it has
ij
P
from the bases B = {|e i}, B = {|f i} in H which are been argued that A should actually be regarded as an
e i f j w
mutually non-orthogonal he |f i 6= 0 for any |e i ∈ B , intrinsic value possessed by the system in the given pro-
i j i e
|f i∈B . Due to the non-orthogonalitycondition,these cess, which can be confirmed by measuring it without
j f
W , if normalized, form a subspace of T(H) and are (almost) disturbing the system, i.e., by weak measure-
T
argued to represent processes of an open system which ment. The gap between the two realizations of A , as
w
is realized as a subsystem of an extended closed system an expectation value and an intrinsic value, seems to in-
[11]. Incontrast,ourW isgivenadirectoperationalin- dicate that a deeper understanding of the value is still
C
terpretation of a probability ensemble of processes with in need. Along this line, the contextual value of the ob-
no need of extension, and it represents the most general servableAdeemedintheprocesshasbeenstudiedin[15]
situationsdeterminedbyapairofstatesinquantumme- requiringasetofconditionssimilarbutstricterthanours
chanics. and also asymmetric in time (in contrast to ours which
are symmetric). Despite this difference, both of them
Conclusion and Discussions. In this paper we have have pointed to the unique status of the weak value Aw
considered the complex probability measure for double as a physical value consistent in the quantum process.
states as an extension of the real probability measure
for single states. Our intention is to provide an appro-
Acknowledgments
priate framework for discussing probabilities and phys-
ical quantities in a given circumstance specified by two
conditions, which are basically determined by two states We appreciate Professors Y. Aharonov, H. F. Hof-
suchastheinitialandfinalstatesprescribingaquantum mann, A. Hosoya, M. Iinuma, N. Imoto, Y. Shikano and
process. In order to find out the form of the measure, A. Wipf for useful discussions.
we haveemployedananalogousargumentusedfor Glea-
son’s theoremwhich establishes Born’s rule of statistical
interpretation for single states, implementing the com- Appendix
plex extension and the additional condition required by
the double states. The complex measure µ turns out In this Appendix we show that any double state W
C C
to be rather simple (9) and contains a free parameter canbe expandedas(28) interms ofstates|ψ i,|φ i∈H
i i
α. Withthis measurewe haveevaluatedthe expectation for which hφ |ψ i 6= 0 under the probabilities p ∈ [0,1]
i i i
valueλ(A)ofanobservableAandfoundthatitcoincides with Σd+1p = 1. We do this by furnishing the states
i=1 i
with Aharonov’s weak value at α=1. explicitly as follows.
In the application of the measure for describing quan- Let {|wii;i = 1,...,d} be a complete orthogonal basis
tum processes, we are alluded to the interpretation that B of H. A double state WC is then expanded as
the measure represents a superposition of two processes
d
with the weight specified by α. The case α = 1 is then
W = w |w ihw |, (30)
found to be one of the two exceptional situations of re- C ij i j
i,j=1
alizing a pure process (the other being α= 0), in accor- X
dance with the original interpretation of the weak value with w ∈ C and w = 1. Consider then the states
ij i ii
[1]. We alsohaveextendedourmeasuretoaccommodate |ψ i,|φ i for i=1,...,d given by
i i
the most general case of superpositions, in parallel with P
thecaseofsinglestates. Ourdoublestatesadmitadirect |ψ i = |w i,
i i
operational interpretation in terms of classical probabil- d
ity ensembles of processes, which is important to assure |φ i = β∗ |w i+|w i, (31)
i ij j i
thatthecomplexmeasureismeaningfulindescribingac-
j=1,j6=i
X
tual physical situations.
with β ∈C and also for i=d+1 by
We mention that the notion of complex probability in ij
quantum mechanics appeared a long time ago in the ex-
d−1
pansion of density states in terms of two complete basis
|ψd+1i = αi|wii+|wdi,
states [12],where the coefficients areinterpretedascom-
i=1
plex joint probability (see also [13]). In fact, this may X
d−1 d−1
be thought of just as our complex measure with positive |φd+1i = |wii+ 1− α∗j |wdi, (32)
W in such decomposition. The double states W , at
α=C 1/2,alsoappearinthe weakmeasurementtomCogra- Xi=1 (cid:16) Xj=1 (cid:17)
phyasthetransientdensitymatrix[14]. Theseinstances with α ∈ C. Note that these pairs of states fulfill
i
suggest that our complex probability measure based on hφ |ψ i = 1 for all i = 1,...,d+1. Choose p ∈ [0,1]
i i i
6
d+1
such that pd+1 6= 0, i=1 pi = 1, and pi = 0 iff wik = 0 WC in (30) from (28) as required.
for all k. We then put
P This shows that WC can generically be decomposed
1 into rank(WC)+1 processes, with properly chosen {pi},
α = (w −p ), i=1,...,d−1, (33)
i ii i and as such the processes are highly non-unique, as
pd+1
are the density operators which generically admit non-
and thereby set unique decompositions. Incidentally, we remark that, if
weemploythesingularvaluedecomposition,wehavethe
βij = 1(wij −pd+1αi), i6=j, unique expansion WC = di ri|uiihvi| with two bases
pi Bu = {|uii} and Bv = {|vii} and ri ≥ 0 in terms of d
P
1 d−1 pairs of states at most. However, the two bases Bu and
βid = p wid−pd+1αi 1− αi , Bv may have hui|vii = 0 for some i, in which case the
i(cid:18) (cid:16) Xj=1 (cid:17)(cid:19) twostates cannotappearasthe initialandfinalstatesof
1 a process (i.e., the latter cannot be observed by postse-
βdj = (wdj −pd+1), (34) lection when the former is preselected), and accordingly
p
d
they cannot specify a physical process. Such unphysical
fori,j =1,...,d−1. Thesecoefficientsallowustoobtain cases are excluded in our expansion (28).
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