Incomplete quantum process tomography and principle of maximal entropy

0.17 MB·English

by  Mario Ziman

Incomplete quantum process tomography and principle of maximal entropy
Ma´rio Ziman
Research Center for Quantum Information, Slovak Academy of Sciences, Du´bravsk´a cesta 9, 845 11 Bratislava, Slovakia
The main goal of this paper is to extend and apply the principle of maximum entropy (Max-
Ent) to incomplete quantum process estimation tasks. We will define a so-called process entropy
function being the von Neumann entropy of the state associated with the quantum process via
Choi-Jamiolkowski isomorphism. It will be shown that an arbitrary process estimation experiment
canbereformulatedinaunifiedframeworkandMaxEntprinciplecanbeconsistentlyexploited. We
willarguethatthesuggested choicefortheprocessentropysatisfiesnaturallistofpropertiesandit
reducesto thestate MaxEnt principle, if applied to preparator devices.
PACSnumbers: 03.65.Wj,03.67.-a,03.65.Ta
9
0
0
I. INTRODUCTION It seems that the complete knowledge about quantum
2
objects is not a very realistic dream and experimentally

n
we will not be able to perform all the desired tests [12–
a Physical objects and processes are described by pa-
14]. Fortunately, there are situations in which even the
J rametersthataredirectly,orindirectly,accessibleexper-
knowledge of only few parameters enables us to make
7 imentally and represent the maximal knowledge about
reasonableandnontrivialpredictionsaboutthe behavior
physical systems (according to physical theory used). In
] quantum theory (see for instance [1–3]) the complete in- andpropertiesofthe system. A typical(classical)exam-
h pleis theequilibriumthermodynamicsinwhichonlyfew
p formation (knowledge) is represented by the concepts of
parametersare used to describe the complex behaviorof
- quantum state (normalized positive operator), quantum
t a system of approximately 1023 degrees of freedom. Our
n observable (normalized positive operator valued mea-
aim is to describe the properties of quantum objects as
a sure) and quantum channel (completely positive linear
honestly as possible even in cases when the complete in-
u trace-preserving map). One of the main characteristics
q formationisnotavailable. Inparticular,inthispaperwe
of quantum system is its dimension d, i.e. the maximal
[ will focus on incomplete quantum process tomography.
number of mutually perfectly distinguishable states (in
InSectionIIwewilldefinetheconceptofprocessmea-
3 a single run of the experiment). These states form an
v orthogonalbasis ofthe associatedcomplex Hilbert space surement and shortly describe the idea of quantum pro-
2 cess tomography. The maximum entropy principle is de-
.
9 H scribed in Section III and also the idea is extended to
8 An arbitrary quantum state is described as a posi- process estimation problems by introducing the concept
3 tive trace-class linear operator with unit trace acting on
of process entropy. Finally, in Section IV the MaxEnt
. the Hilbert space, ̺ : : ̺ 0,Tr̺ = 1, i.e.
2 H → H ≥ procedureisappliedtoparticularexamplesofincomplete
a density operator, or a density matrix. The number
0 ancilla-free estimation of qubit channels.
8 of independent real parameters determining the quan-
0 tum states scales as = d2 1. The quantum
state
N −
: processes/operations correspond to completely positive
v
i trace-preserving linear maps defined on the set of all II. QUANTUM PROCESS MEASUREMENT
X linear operators including the set of all states ( ).
S H
r The number of independent real parameters determin- A general quantum process tomography experiment
a
ing the particular quantum operation equals process = consists of a test state ̺ that is transformed in some
N
d2(d2 1). Quantum measurements give us probabil- specific procedure involving the unknown channel
ity dis−tributions over the set of all possible outcomes into a state ̺′, anPd a measurement (POVM) perE-
x1,...,xL , where L is some positive integer. In the- formed on the state ̺′. This framework includesMall the
{ }
ory,the measuredprobabilities pj are determined by the possible strategies [5, 6, 14] via which the parameters
Born’s rule pj = Tr̺Fj, where Fj is a positive operator of quantum channels are accessible. We will define a
(quantumeffect)correspondingtooutcomexj. Theseop- process measurement Eas a particular choice of the test
erators form the so-called positive operator valued mea- state ̺, of the procedure and of the measurement .
sure(POVM),i.e. Fj arepositive(Fj 0)andtheysum Generally, the procedureP is composed of an appliMca-
up to identity operator (PjFj =I). T≥he number of pa- tion of some known quantPum channels on the test state
rametersspecifyingPOVMdepends onthe totalnumber and one usage of the unknown channel acting on d-
of outcomes L and equals measurement =(L 1)d2. dimensional quantum system (qudit). TEhat is itself
N − P
The goal of quantum tomography is to estimate and is a quantum channel that can be written as a product
fixalltheseparameters[4–6]. However,alreadyforsmall = ( ) ,where , canbeunder-
in anc out in out
P P ◦ F ⊗E ◦P P P
systems (in dimension) the number of parameters is in- stoodasbeingpartsofthepreparationoftheinitialstate
creasing rapidly, especially for quantum channels [7–11]. ̺, and of the final measurement , respectively. Con-
M
2
sequently, without loss of generality we can assume that form a POVM the numbers f represent the measured
j
= , where is a known quantum channel probabilities. The answer to incomplete state tomog-
anc anc
P F ⊗E F
actingonsomeancillarysystemandalsocanbeincluded raphy problem based on maximum entropy principle is
asbeingapartofeitherpreparationof̺,ormeasurement given by the following equation
performed. A process measurement is called ancilla-free
if either the initial state ̺ is factorized, or the ancillary ̺=argmax S(̺)̺ ( ),fj =Tr̺Fj,j =1,...,n .
{ | ∈S H }
system is trivial. Otherwise the process measurement is
ancilla-assisted and ̺′ = [̺]. The following state is the formal solutionof the MaxEnt
anc
I ⊗E problem [17, 21]
For example, consider is a qudit quantum chan-
E
nel and the test state ̺ = Ψ is a maximally entan-
+ 1
gled state of two qudits (Ψ+ = d1P|jihk| ⊗ |jihk|). ̺= Z exp(−XλjFj), (3.1)
The unknown channel is applied only on second of the j
qudits while the first one is transformed trivially, i.e.
̺al′ly=coωmEp=letIe⊗POEV[ΨM+](r[e6s]u.ltPinegrfoinrmcoinmgpltehteeisnpfeocrimficaattioionn- wtiphleireersZfi=xedTrb[eyxtph(e−sPysjteλmjFojf)]eaqnudatλiojnasreLagrangemul-
of ω ) the channel can be uniquely identified, because
E E ∂
the mapping E 7→ ωE = J[E] is the well known Choi- fj =Tr̺Fj =−∂λ lnZ(λ1,...,λn). (3.2)
Jamiolkowski isomorphism [15, 16] between the set of j
quantumquditchannelsandsetofquantumstatesoftwo
For example, consider a two-level quantum system
qudits. Hence, via general POVM measurements of the
(qubit) and observation levels
output state we can acquire either complete, or partial,
knowledge on the channel. Let us note that individual
= σ
1 z ̺
ancilla-freeprocessmeasurementscannotbeinformation- O {h i }
= σ , σ
allycomplete,buttheycanbecombinedtogethertogain ⊂O2 {h yi̺ h zi̺}
the complete information. In the following sections we 3 = σx ̺, σy ̺, σz ̺ .
⊂O {h i h i h i }
will concentrate onto situations in which the collection
Aqubitstatecanbeexpressedinaso-calledBlochsphere
of process measurements provides us with partial infor-
pictureas̺= 1(I+~r ~σ)withr =Tr̺σ . TheMaxEnt
mation, only. 2 · j j
principle applied for , sets mean values r of all
1 2 j
O O
the unobserved operators to zero. That is, for = z
1
we get ̺ = 1(I +zσ ) and for = y,z theOMaxE{n}t
III. PRINCIPLE OF MAXIMUM ENTROPY 2 z O2 { }
estimationgives̺= 1(I+yσ +zσ ). Observationlevel
2 y z
provides complete information about the quantum
3
Maximum entropy (MaxEnt) principle was originally O
state, hence the principle of maximum entropy is not
introducedinstatisticsinordertoestimateaprobability
needed in this case.
distribution providing that only partial information on
Clearly, there is a problem if we consider similar in-
that probabilityis available[17]. There aremany proba-
complete estimation task for processes, namely, which
bilitydistributionscompatiblewiththegivenconstraints
entropy should be maximized? Unlike quantum states
and our aim is to choose one of them that in some sense
the quantum channels are lacking some concept of en-
represents our knowledge the most honestly. This choice
tropy, or uncertainty. In fact, what does it mean that a
cannotbelogicallyderivedandsomeadditionalprinciple
quantum process is uncertain? Our goal is to introduce
mustbeintroduced. Usingtheresultsofinformationthe-
a suitable concept of a channel/processentropy S ( )
proc
ory [18] on the uniqueness of Shannon entropy, one can E
and investigate its properties. Before analyzing different
argue that [17, 19] the probability distribution maximiz-
choices let us discuss some (intuitive) properties of the
ing the Shannon entropyis the best choice we canmake.
process entropy.
Such probability maximizes the uncertainty (measured
by entropy) and, intuitively, also our predictions about 1. Uncertainty of unitary channels. Without any
the unspecified parameters are as uncertain as possible. doubts the unitary channels play a very specific
That is, a conclusion based on MaxEnt principle is in- roleamongallquantumprocesses. Forunitarypro-
troducingaslittle additionalinformationasitispossible cesses the interaction of the system with its envi-
[19]. ronment is trivial. The physical invertibility is the
This idea was generalized to the domain of quantum unique and characteristic property of the unitary
state tomography [20, 21] by using the concept of von channels. Insomesensethechannelentropyshould
Neumannentropy[1]S(̺)= Tr̺log̺,whichisconsid- reflecthowmuchnoisethechannelintroduces. Uni-
−
eredtobethequantumextensionofShannonentropy. A tary channels are noiseless and in what follows we
stateobservationlevel isdefinedasasetofn(n d2) will assume that the channel entropy is invariant
n
O ≤
mean values f ,...,f of linearly independent opera- under unitary preprocessing ( ) and unitary post-
1 n
{ } V
tors F ,...,F relatedtounknownstate̺viathetrace processing ( ), i.e., S ( ) = S ( ).
1 n proc proc
{ } U E U ◦ E ◦ V
rule f = Tr̺F = F . If the operators F ,...,F It follows that all unitary channels have the same
j j ̺ 1 n
h i { }
3
value of uncertainty that canbe set to zero. More- given by mean values x = TrX ω of n linearly inde-
j j E
over, we do require that S ( ) = 0 implies that pendent Hermitian operators X , the MaxEnt problem
proc j
E
is unitary. for processes can be formalized as follows
E
2. Uniqueness of maximum. For the purposes of in- =argmaxS(ω )
E
complete process estimation exploiting the Max- E ωE
Ent principle it is necessary that the maximum is
where the maximum of von Neumann entropy S(ω )
unique. Hence there must be a unique channel, for E
is taken over all states ω ( ) satisfying the
which the uncertainty is maximal. This channel constraints Tr ω = 1IE a∈ndSxH ⊗=HTrω X for all
should be the result of the incomplete estimation 2 E d j E j
X ,...,X . The resulting state ω determines the
if no data are available, i.e., when the observation 1 n ∈On E
quantum operation uniquely via the inverse relation
level is trivial, 0 = . Because of the unitary in- E
O ∅
variance the channel must be invariant under uni- [̺]=dTr [(̺T I)ω ]. (3.3)
anc E
tary preprocessing and postprocesing, i.e., = E ⊗
max
E
. Only the channel mapping the whole
max In what follows we will investigate the process entropy
Usta◦tEe spa◦ceVinto a total mixture (̺ 1I) is invari-
7→ d given as the von Neumann entropy of the state ωE asso-
antinthissense. Itisarguedin[22]thatthischan-
ciatedwith the channel via Choi-Jamiolkowskiformal-
nel is indeed the average channel over all possible E
ism.
qubit channels. To guarantee that in any process
It is straightforwardto see that only for unitary chan-
measurementthemaximumisuniqueitissufficient
nelsthestatesω arepureandhenceS ( )=S(ω )=
E proc E
thatthe processentropy is a concavefunction, i.e., E
0 only for unitary channels, = . Moreover, uni-
S (λ +[1 λ] ) λS ( )+[1 λ]S ( ). E U
proc 1 2 proc 1 proc 2 tary channels do not change the entropy of ω , i.e.,
E − E ≥ E − E E
S(ω )=S(ω ). The concavityof S follows from
3. Universality. This is not a condition on the con- E U◦E◦V proc
the concavity of von Neumann entropy and the maxi-
cept of process entropy itself, but rather on the
mum is achieved for ω = 1 I that is associated with
general possibility to employ such principle once E d2
thechannelmappingthewholestatespaceintothemax-
we agreeon a suitable measure of channel entropy.
imallymixedstate, :̺ 1I. Insummary,theprocess
It is important that the maximum entropy princi- E 7→ d
entropy
ple is applicable for all process measurements. We
shall discuss this issue later in more details.
S ( )= Trω logω (3.4)
proc E E
E −
In summary, a channel entropy is some concave func-
satisfies all the desired properties we have discussed pre-
tion (defined on the set of quantum channels) achieving
viously. Theonlyopenissueisitsapplicabilityingeneral
its maximum for the complete contraction to the total
process measurement.
mixture and vanishing only for unitary channels. A nat-
Choi-Jamiolkowski isomorphism is associated with a
ural choice of process entropy seems to be related to the
specific process measurement using as the test state a
conceptsofquantumchannelcapacity[23–26]. Quantum
maximally entangled state of two qudits. Second qudit
capacityquantifiesthedegreeofpreservationofquantum
is sent through the unknown channel while the first one
states during the transmission and this value is different
is evolving trivially to obtain the state [Ψ ]= ω ,
fordifferentunitarytransformations. Ontheotherhand, I⊗E + E
that is estimated in some state measurement described
the classicalcapacity is maximal also for noisy channels.
by POVM. In this case the process observation level can
For example, phase-damping channels ̺ diag[̺] max-
→ be defined as the following set of mean values
imize the transmission of classical information over the
quantum channels. Because of these properties, the ca-
proc = x ,...,x , (3.5)
pacities are notappropriatecandidates for the definition On { 1 n}
of process entropy usable in incomplete process tomog-
where
raphy tasks.
x =TrF ω = F F . (3.6)
j j E h jiI⊗E[Ψ+] ≡h jiE
A. Choi-Jamiolkowski process entropy Because of the identity Tr ω = 1I the process observa-
2 E d
tion level is equivalent to a state observation level
The Choi-Jamiolkowski isomorphism provides us nat-
urallywithanotionofchannelentropy. Ituniquelyasso- = x ,...,x ,0,...,0 .
n+d2−1 1 n
O { }
ciatesaquantumstateω =( )[Ψ ]withaquantum
E +
channel ,hencewecanadoptIt⊗heEvonNeumannentropy The added zeros represent the mean values of d2 1 op-
E −
of ωE as being the channel entropy of E [27, 28]. Con- erators I ⊗Λj, where Λj are traceless Hermitian qudit
sider a quantum channel on d-dimensional system (qu- operators forming a basis of the set of traceless Hermi-
dit). Providing that for each process measurement we tian qudit operators, i.e., the general qudit state can be
are able to define uniquely a state observation level writtenas̺= 1(I+~r Λ~). Tobemorepreciseweassume
On d ·
4
thattheoperatorsF ,...,F arelinearlyindependentof ancilla process measurement with the maximally entan-
1 n
operators I Λ ,...,I Λ . gledstate Ψ being the test state anda measurementof
1 d2−1 +
⊗ ⊗
What if the test state is not the maximally entan- X =(A∗Ω⊗I)[Ianc⊗F], i.e.,
gled one? Is it possible to interpret the measured values
as linear constraints on the state ΩE defined by Choi- X = X(A†αl⊗I)(Ianc⊗F)(Aα,l⊗I)
Jamiolkowski isomorphism? Let us note that the lin-
α,l
earity is crucial, because we implicitly assume that the
= d λ µ ψ∗ ϕ ϕ ψ∗ F
constraints representing the incomplete information are X α l| lih α| αih l|⊗
linear,whichguaranteesthatthesetofpossiblesolutions α,l
isconvexand,hence,theentropyhasauniquemaximum. = d( µ ψ∗ ψ∗ ) F
X l| lih l| ⊗
l
= d̺T F ,
⊗
B. General quantum process experiment vs
Choi-Jamiolkowski isomorphism where we used that ̺ = µ ψ ψ , ξ =
λ ϕ ϕ and ̺T is the trPanlspol|seldihmla|trixa̺ncwith
Pα α| αih α|
Consider a general test state Ω of the qudit and an respect to basis k .
{| i}
arbitrary ancilla system. We will show that there exist We have shown that measuring the outcome associ-
a completely positive linear map : ( ) ( ) ated with F in the ancilla-free process measurement is
Ω d anc
such that [Ψ ] = Ω. AA gBeneHral →puBreHstate equivalent to measuring d̺T F in the process mea-
|Φi = Pα,jAΦΩαj⊗|αIianc+⊗|ji (α = 1,...,D; j = 1,...d) sisurtehmeeanntciwlliat-hfremeatxeismtasltlyatee.ntIa⊗tngmleedansstattheatΨt+h,ewahnecrilela̺-
can be written as Φ =A I Ψ , where the operator
Φ +
A : is| diefined a⊗s A| =i √d Φ α j . free process observation level consisting of mean val-
Φ H → Hanc Φ Pα,j αj| ih | ues F ..., F is equivalent to proc = d̺T
A general mixed state Ω can be written as convex com- F h,.1.i.̺,1d̺T h nFi̺n . On {h 1 ⊗
bination of pure states Ω = Pkλk|ΦkihΦk|, hence Ω = 1iE h n ⊗ niE}
λ (A I)Ψ (A† I)=( )[Ψ ]. Sincetheval-
Pk k j⊗ + j⊗ AΩ⊗I +
ues λ are positive the transformation is completely
k AΩ C. States as preparation channels
positive. Moreover,foreachstateΩthelinearmap is
Ω
A
unique. Hence, for a generaltest state Ω the mean value
Preparation devices play a completely different role
of an Hermitian operator F can be expressed as follows
than quantum channels. However, formally, they can
be understood as mappings that transform an arbitrary
F = F
h i(I⊗E)[Ω] h i(AΩ⊗E)[Ψ+] input state into a fixed output state ξ. In this sense
= ( ∗ )[F] , (3.7)
h AΩ⊗I i(I⊗E)[Ψ+] preparation channels ξ form a very specific convex sub-
E
setofquantumchannels. Letusapplytheproposedmax-
where ∗ isadualmappingto (Heisenbergpicture).
AΩ AΩ imum entropy based process tomography to preparation
As a result we get that an arbitrary ancilla-assistedpro-
channels, i.e., to preparators. The process measurement
cessmeasurementcanberewrittenwithintheframework
is ancilla-free consisting of all linearly independent test
of process measurement using the maximally entangled
states ̺ (j = 1,...,d2). and measurement of the mean
teststateΨ andmeasuringasuitableHermitianopera- j
+ value of Hermitian operator F. According to previous
tor ∗ [F],hence,themaximumentropyprinciplede-
AΩ⊗I paragraph the process observation level is described as
fined via Choi-Jamiolkowskientropy can be consistently proc = d̺T F ,..., d̺T F . The Choi-
employed in all incomplete process measurements. Od2 {h 1 ⊗ iEξ h d2 ⊗ iEξ}
Jamiolkowski entropy of the channel equals (up to a
ξ
In what follows we shall analyze the ancilla-free pro- E
constant)tothevonNeumannentropyofthestateξ,be-
cess measurement, hence only the qudit itself is used to cause [Ψ ]= 1I ξ impliesS(ω )=log d+S(ξ).
probe the action of the quantum channel. In fact, this I⊗Eξ + d ⊗ Eξ 2
Moreover,because of the identity
can be considered as an ancilla-assisted problem with
amefaascutorreimzeednttreesstuslttiantgeiΩnt=heξmanecan⊗v̺a,luaenodftahefaocptoerraizteodr hd̺Tj ⊗FiI⊗Eξ[Ψ+] =hd̺Tj ⊗Fid1I⊗ξ =hFiξ, (3.8)
of the form I F. Consider Ω is a pure factorized
anc it follows that finding a channel with the maximal Choi-
⊗
state Φ = ϕ ψ . Then the operator A takes
anc Φ Jamiolkowski entropy is equivalent to finding a state
| i | i ⊗| i
the following form A = ϕ ψ∗ , where ψ∗ is a com-
Φ maximizing the von Neumann entropy. As a result we
| ih | | i
plex conjugated state, i.e., k ψ∗ = k ψ for all basis get that the process MaxEnt procedure, if applied to
h | i h | i
vectors k , in which the maximally entangled state Ψ+ preparators, reduces to the state MaxEnt procedure.
| i
is defined. It follows that for general factorized state Thatis,theproposedChoi-Jamiolkowskiprocessentropy
Ω=Pα,lλαµl|ϕαiϕα|⊗|ψlihψl| the transformation AΩ isaconsistentextensionofthevonNeumannentropy. In
isexpressedviaKrausoperatorsA =√dλ µ ϕ ψ∗ . particular, the MaxEnt principle for states can be con-
αl α l| αih l|
Therefore, according to Eq. (3.7) the ancilla-free pro- sidered as being a special case of the MaxEnt principle
cess measurement of I F can be considered as an for channels.
anc
⊗
5
IV. EXAMPLES Asweseeinthiscasebothmethodstransformorthogonal
(in Hilbert-Schmidt sense) states to ψ into the total
| i
Inthissectionweshallpresentfewexamplesofincom- mixture, but for MaxEnt procedure also the orthogonal
plete quantumprocessestimationforancilla-freeprocess (in Hilbert space sense) state ψ⊥ is mapped into the
| i
measurements of a qubit channel. total mixture. In our opinion this feature (except the
universality) justifies the usage of MaxEnt procedure in
comparison with the scheme described in [22]. In fact,
A. O1proc={h2̺T ⊗σziE} the uncertainty introduced by the estimation procedure
onperfectly distinguishable (orthogonal)states fromthe
test states shouldbe as maximalas possible. And this is
In this case the collected data provides us about in-
not the case for the method used in [22], for which the
formation on the mean value of an observable σ , hence
z
estimated channel preserves the orthogonalstate.
theexperimentgivesussinglevaluem= σ . Unfor-
z E[̺]
h i
tunately, even in this simplest case we cannot give (see
Appendix A)ananalyticsolutioninits wholegenerality.
In particular, we found the solutions in following cases B. O3proc ={hI⊗σxiE,hI⊗σyiE,hI⊗σziE}
̺= 12I : E[ξ]= 12(I +mσz), (4.1) Consider a situation that the unknown qubit channel
̺=|ψihψ| : E[ξ]= 21(I + 21m(1+(~t·~r)σz), istestedbythetotalmixtureandthecompletetomogra-
phyofthe outputstate isperformed,i.e.,meanvaluesof
where ̺ = 21(I +~r·~σ) and ξ = 12(I +~t·~σ). It is inter- σx,σy,σz are known. The corresponding state observa-
esting that for pure test state the estimated channel is tionlevelis = ~σ I , I ~σ = ~0,m~ ,forwhich
6 ̺ ̺
notunital evenif there areunital channelssatisfying the O {h ⊗ i h ⊗ i } { }
thesolutionispresentedinAppendixB.Insuchcasethe
constraints.
proposed MaxEnt process tomography procedure leads
An alternative method for incomplete process estima-
us to the channel
tion was described in [22]. It is based on a different ad
hocruledemandingthatnoadditionalinformationabout 1
:̺ ̺ = (I +m~ ~σ), (4.3)
unobserved measurements (those completing the incom- Eest 7→ 0 2 ·
plete process observational level) is introduced. In par-
hence, the whole Blochsphere is contractedinto a single
ticular,forstatesηorthogonal(inHilbert-Schmidtsense,
point m~. As a result we get that if the total mixture
i.e., Trη̺ = 0) to given test states, the mean values are
is used to probe the channel action then according to
completely random, i.e., they are transformed into the
total mixture (η 1I). Hence the entropy of output MaxEnt procedure all the states are mapped into the
→ 2 output state ̺ = [1I]. In this case both the discussed
states for unmeasured inputs is maximal. It means that 0 E 2
procedures are giving the same estimation.
if possible (meaning there is no contradiction with the
data, or theory) the total mixture is preserved. Other-
wise an optimization procedure minimizing the average
distance from the total mixture is needed. This method C. O4proc={hI⊗σziE,h2(|xihx|)T ⊗σziE,h2(|yihy|)T ⊗
was analyzedonly for qubit channels and for ancilla-free σziE,h2(|zihz|)T ⊗σziE}
process measurements. The extension of the method to
In this case the process is probed with four test states
allprocessmeasurementswillrequireintroductionofad-
(total mixture and positive eigenvectors of σ ,σ ,σ
ditional rules. Let us compare the method proposed in x y z
forming a vector of pure states ~η), but only zth compo-
[22] and the one proposed in this paper.
nentofthe Blochvectorofthe output stateis measured.
If measuring σ and finding m = 1 the output state
z
must be pure and it corresponds to±an eigenvalue of σz. O7 ={hI⊗σzi̺,h2~ηT ⊗σzi̺,h~σ⊗Ii̺}={z,~ζ,~0} is the
In both mentioned scenarios we know the solution pro- corresponding state estimation problem and z,~ζ are the
vidingourknowledgeconsistsofcompleteinformationof experimentally identified mean values. Information en-
the actionofthe channelonthe pure teststate, thus, we coded in these parameters can be equivalently rewritten
knowthat : ψ z ,respectively. Asitwasargued into the form = I σ , ~σ σ , ~σ I =
7 z ̺ z ̺ ̺
E | i7→|± i O {h ⊗ i h ⊗ i h ⊗ i }
inthework[22]theestimatedtransformationshouldmap z,~ζ′,~0 , where ζ′ =ζ z.
thewholeBlochsphereintothelineconnectingnorthand { It is s}hownin [2j1] thja−t for suchstate observationlevel
south pole, i.e., ~t ~t′ = (0,0, tz). However, the pro- the estimated density matrix reads
→ ±
posed MaxEnt estimation procedure gives different re-
tsrualtn.sfoInrmpaatriotincuilsarn,o~tt→uni~tt′al=, b(0u,t0,th±e(1to+tatlz)m/2ix).turTehiiss ω = 41(cid:16)I⊗I +zI⊗σz +(ζ~′·~σ)⊗σz(cid:17) . (4.4)
mapped to the state ~t′ = (0,0,1/2). A state ~t = ~r
− Hence, the process is described by the following state
orthogonalto the test state~r is transformed as follows
transformation (̺ [̺])
est
→E
Scheme in [22]: m=1 : ~r ~r
MaxEnt: m=1 Eest :−~r7→~−0 (4.2) ~t ~t′ =(0,0,z+ζ~′ ~t), (4.5)
Eest − 7→ → ·
6
i.e.,̺′ = 1[I+(z+ζ~′ ~t)σ ]. Asinallpreviouscases,also Section III.B. This idea goes beyond the applications in
2 · z
inthiscasethewholestatespaceismappedontoasubset incomplete process estimation and is further developed
of the line connecting states +z and z . However, in [30].
| i |− i
in this case the final state depends also on parameters Recently, Olivares et al. [29] proposed and analyzed
t ,t . a state estimation problem combining incomplete in-
x y
formation with some nontrivial apriori knowledge. In
their approach the maximization of entropy is replaced
V. CONCLUSION AND DISCUSSION by minimization of Kullback relative entropy S(̺̺ ) =
0
|
Tr[̺(log̺ log̺ )] with a bias ̺ representing the prior
0 0
−
We have addressed the problem of incomplete process knowledge. This approach can be directly extended
estimation based on maximum entropy principle [19]. In to the case of channels by introducing the quantity
general the maximum entropy principle is an intuition- S(ωE ω0)= Tr[ωE(logωE logω0)]with 0 playingthe
|| − − E
based ad hoc principle related to quantification of igno- roleofpriorinformation. Ifweset 0tobethestatespace
rancecontainedinprobabilitydistributionsthatseemsto contraction into the total mixtureE(i.e. ω0 = d12I), then
agree with our experience. This ignorance measured in Tr[ω logω ] = logd2Trω = 2logd. Consequently,
E 0 E
− −
entropycanbeextendedtodomainofquantumstatesby S(ω ω ) = 2logd S(ω ) and the biased estimation
E 0 E
|| −
introducingthevonNeumannentropy. Ourattempthere problem reduces to the unbiased maximum process en-
wastodevelopsimilarapproachforprocesses. Weargued tropy estimation.
that capacities are not good candidates for quantifying Let us give a simple example based on the observa-
the uncertainty of quantum channels and we exploited tion level discussed in Section IV.A. Suppose that out
the Choi-Jamiolkowskiprocess entropy defined as of the performed measurement we acquire the informa-
tion 0 0 . We shall consider three different pri-
| i 7→ | i
Sproc(E)=−Tr[ωElogωE], ωE =(I⊗E)[Ψ+], (5.1) ors: i) identity channel E0 =I; ii) diagonalisation chan-
nel = diag transforming each state into its diagonal
0
wlahr,erweeΨsh+owisedthtehmatatxhiemsaullgygeesntteadncgolendcesptattcea.nIbnepuanrtiviceur-- for[mξE] i=n σtheξσba.siLse|t0iu,s|1ni;otieiit)hEa0t S=(̺21̺(I)+isUfixn)i,tewohnerlye
x x x 0
sally applied in all possible process measurements. The Uif the support of ̺ is included in the s|u|pport of ̺ . For
0
procedure is demonstrated on three incomplete ancilla- the case of identity channel ω is a pure state, hence
0
free estimation problems of a qubit channel: i) pure test S(ω ω ) < only if ω = ω , i.e. = . For-
E 0 E 0 est
state and projective measurement, ii) the total mixture tunat|e|ly, the i∞dentity channel is in accorEdance wIith the
asthe teststate andcomplete tomographyofthe output constraint 0 0 ,hencetheestimationgivestheiden-
state, and iii) four test states and the same projective tity channe|l.i7→In| tihe second case it is straightforward
measurement. to verify that the channel diag fullfils the constraints.
We haveshownthat unlike the concepts ofcapacityof Since S(diag ) = 0 is the minimal possible value we
0
quantum channels the process entropy defined above is get = di|a|Eg. In the third case the support of ω is
est 0
compatible with the following properties: a linEear span of vectors ψ = (00 + 11 )/√2 and
+
| i | i | i
φ = (01 + 10 )/√2. Therefore, only for chan-
1. Uniqueness of maximum: unique maximumforthe | +i | i | i
nels with ω = aφ φ +bψ ψ the relative en-
channelcontractingwholestatespaceintothetotal E | +ih +| | +ih +|
tropy is finite. However, the constraint requires that
mixture.
0 0 = (a +b )[0 0] = a0 0 +b1 1, i.e. nec-
x
| ih | I U | ih | | ih | | ih |
2. Minimum: minimum is achieved only for unitary essarily b = 0. In such case a = 1, because otherwise
processes. a is not a valid quantum channel. That is, only the
I
identitychannelsatisfiesthemeasuredconstraint,thusit
3. Unitary invariance: invariant under unitary trans- minimizes the Kullback relativeentropy. The estimation
ufonrimtaartyiotnrsa,nis.feo.,rmSpartoioc(nUs ◦,E ◦.V) = Sproc(E) for all mgivaetisonEesstas=inI.thIenuanlblitahseedsemcaasxeismwume fienndtrodpiffyeraepnptroeastcih-
U V
(see Eq.(4.2)). The role of prior information in incom-
4. Concavity: function S ( ) is concave, i.e.,
proc E plete process estimation deserves much deeper analysis
S (p +q ) qS ( )+pS ( ).
proc E F ≥ proc E proc F than it is presented in these simple examples. However,
such task is beyond the scope of this manuscript.
The proposed Choi-Jamiolkowski process entropy serves
asaveryvaluabletoolinincompleteprocesstomography
deserving future testing and investigation. Moreover, as
itisshowninSectionIII.B,theproposedprocessentropy Acknowledgments
principle,ifappliedtostatepreparatordevices,isequiva-
lenttothestateentropyprinciplebasedonvonNeumann This work was supported by in part by the European
entropy. The key feature discussed in this manuscript is UnionprojectsQAP,by the SlovakAcademyofSciences
theuniversalityoftheproposedprocedurefollowingfrom via the project CE-PI and by the Slovak grant agency
the unification of all process measurements described in APVV and VEGA.
7
APPENDIX A: MAXENT SOLUTION FOR σ =m=0theMaxEntresultsinthestateω = 1I I
O1proc={L 2̺T ⊗σziE} FOR QUBIT CHANNELS hanxdi, consequently, the estimated channel acts as fo4llo⊗ws
1
According to Section III this process observation level ξ ξ′ =2Tr [(ξT I)ω]= I, (A7)
1
→ ⊗ 2
is equivalent to the following state observation level
i.e., the whole Blochsphere is transformedinto the total
= 2̺T σ , σ I , σ I , σ I
4 z ̺ x ̺ y ̺ z ̺ mixture.
O {h ⊗ i h ⊗ i h ⊗ i h ⊗ i }
= m,0,0,0 . Forthecase~λ=~0thefirstequationimpliesthateither
{ }
d=0(leadstosamesolutionasbefore),or~r =~0. Ifthe
T
Maximumentropyestimationgivesusthefollowingstate the test state is chosen to be in total mixture (~r =~0),
T
the second equation leads to d = arctanh(m), hence
ω = 1 exp[ ~λ (~σ I) 2d̺T σ ] (A1) the estimated state reads −
z
Z − · ⊗ − ⊗
1
where Z =Tr[exp[ ~λ (~σ I) 2d̺T σ ]] and~λ,d are ω = e−dI⊗σz
z 4coshd
− · ⊗ − ⊗
Lagrange multipliers that can be determined by solving 1
the system of algebraic equations = I (coshd sinhdσz)
4coshd ⊗ −
1 1
∂ ∂ = I (I +mσ ). (A8)
~0= lnZ m= lnZ. (A2) 2 ⊗ 2 z
−∂~λ −∂d
The corresponding process is given by the identity
est
U(rsi,ngrt,hre))extphreessstiaotne c̺aTn b=e w21ri(tIte+n i~rnTth·e~σf)or(m~rTω == E[ξ]=dTr1[(ξT ⊗I)ω], i.e.,E
x y z
1e−−R with 1
Z [ξ] = Tr [(ξT I)(I (I +mσ ))] (A9)
est 1 z
E 2 ⊗ ⊗
R = A 0 0 +B 1 1 (A3) 1
⊗| ih | ⊗| ih | = Tr [̺T (I +mσ )] (A10)
A = [(~λ+d~r ) ~σ+dI] 2 1 ⊗ z
T
· 1
B = [(~λ d~r ) ~σ dI]. = (I +mσ ). (A11)
T z
− · − 2
Since the operators A 0 0 and B 1 1 commute This transformationmaps the whole state space into the
⊗| ih | ⊗| ih |
we can write single point ξ = 1(I +mσ ).
2 z
The last family of solutions of MaxEnt conditions is
e−R = e−A⊗|0ih0|e−B⊗|1ih1| (A4) that the vectors ~λ and ~r are collinear. In this case we
T
= e−A 0 0 +e−B 1 1 . reduced the number of unknown parameters to λ = ~λ
⊗| ih | ⊗| ih | | |
and d. The first condition out of Eqs. (A6) then reads
Having in mind the operator identity
sinh (λ+d)r sinh (λ d)r
sinh ~y 0 = e−d | |(λ+d)+ed | − |(λ d)
exI+y~·~σ =ex(cosh ~y + | |~y ~σ) (A5) λ+d λ d −
| | ~y · | | | − |
| |
where we used r = ~r = ~r . Analyzing all possible
T
we obtain | | | |
valuesforλ ditfollowsthatthe absolutevaluescanbe
±
omitted and the equations simplify to
Z =Tre−R =2(e−dcosh ~λ+d~r +edcosh ~λ d~r ).
T T
| | | − |
0 = e−ds +eds (A12)
Inserting this expression into Eqs.(A2) we get + −
e−dc edc 2re−ds
+ − +
m = − − , (A13)
~0 = S (~λ+d~rT) +S (~λ−d~rT) e−dc++edc−
+ ~λ+d~r − ~λ d~r
T T where s = sinh[(λ d)r], c = cosh[(λ d)r]. After a
| | | − | ± ±
(~λ+d~r ) ~r short algebra these ±equations can be rew±ritten into the
T T
m(C +C ) = C C 2S · (A6)
+ − +− −− + ~λ+d~r form
T
| |
eλrcosh[d(1 r)]=e−λrcosh[d(1+r)] (A14)
where S = e∓dsinh ~λ d~r and C = e∓dcosh ~λ −
± T ±
| ± | | ±
d~rT . Fromthefirstoftheseequationsitfollowsthat~λ+ and
|
d~r and~λ d~r arecollinear,i.e.,~λ+d~r =k(~λ d~r ).
T T T T
Thisispos−sibleonlyifeither~λ=0,ord=0,or~λ−=λ~rT. m(eλrcosh[d(1−r)]+e−λrcosh[d(1+r)])=
The case d = 0 requires that ~λ =~0,m = 0 whatever = eλrsinh[d(1 r)] e−λrsinh[d(1+r)])
− − −
test state ~r is used. Thus, measuring the mean value 2re−dsinh[(λ+d)r]. (A15)
T
−
8
Unfortunately, we cannot give a general solution in a InthelanguageofBlochvectorsthetransformationreads
closed form. Consider therefore a special case and let us
assume that the test state is pure, i.e., r = 1. In such 1
~t ~t′ =(0,0, m[1+~t ~r]). (A20)
case the solution reads → 2 ·
1 1 1 m
d= arctanh( m)= ln − (A16)
2 − 4 1+m
APPENDIX B: MAXENT SOLUTION FOR
O6 ={L I⊗~σi̺,L ~σ⊗Ii̺}={m~,~0}
1
λ= lncosh(2d). (A17)
2 According to maximum entropy principle the
state maximizing the entropy has the form
As a result we get ω = 1e−(~λ·~σ)⊗I−I⊗(µ~·~σ) with
Z
1
ω = e−d[cosh(λ+d)I sinh(λ+d)~r ~σ] 0 0
Z − T · ⊗| ih | Z = Tre−(~λ·~σ)⊗I−I⊗(µ~·~σ) (B1)
1
+ ed[cosh(λ d)I sinh(λ d)~rT ~σ] 1 1 = (Tre−~λ·~σ)(Tre−µ~·~σ)=4cosh ~λ cosh ~µ .
Z − − − · ⊗| ih | | | | |
with Z = 2(eλ +e−λcosh(2d)). Let us denote by ~t the The values of ~λ,~µ are given by the following system of
Bloch vector corresponding to a general input state ξ, equations
then the estimated operation is given by the following
prescription 1 ∂Z ~λ
~0= ~0= tanh ~λ (B2)
−Z ∂~λ ⇒ − | | ~λ
ξ ξ′ = 2Tr1[(ξT I)ω] | |
→ ⊗ 1 ∂Z ~µ
1 1 m~ = m~ = tanh ~µ , (B3)
= (1+ m(1+~tT ~rT))0 0 −Z ∂~µ ⇒ − | | ~µ
2 2 · | ih | | |
1 1
+ (1 m(1+~t ~r ))1 1 (A18) and for the estimated state we get
T T
2 − 2 · | ih |
where wehaveusedξ = 1(1+~t ~σ). Taking intoaccount ω = 1e−(~λ·~σ)⊗I−I⊗(µ~·~σ) = 1I (I +m~ ~σ). (B4)
2 · Z 4 ⊗ ·
that~t ~r =~t ~r we can write
T T
· ·
Asaresultwefoundthattheestimatedchannel maps
ξ ξ′ = 1(I + 1m(1+~t ~r)σ ). (A19) the whole Bloch sphere into the point 1(I +m~Ee~σst).
→ 2 2 · z 2 ·
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