Optimal purification of a generic n-qudit state

0.43 MB·English

by  Giuliano Benenti

Optimal purification of a generic n-qudit state
Giuliano Benenti1,2,∗ and Giuliano Strini3,†
1CNISM, CNR-INFM & Center for Nonlinear and Complex Systems,
Universit`a degli Studi dell’Insubria, Via Valleggio 11, 22100 Como, Italy
2Istituto Nazionale di Fisica Nucleare, Sezione di Milano, via Celoria 16, 20133 Milano, Italy
3Dipartimento di Fisica, Universit`a degli Studi di Milano, Via Celoria 16, 20133 Milano, Italy
(Dated: January 13, 2009)
We propose a quantum algorithm for the purification of a generic mixed state ρ of a n-qudit
9 system by using an ancillary n-qudit system. The algorithm is optimal in that (i) the number of
0 ancillary quditscannot bereduced,(ii) thenumberof parameters which determinethepurification
0 state |Ψi exactly equals the number of degrees of freedom of ρ, and (iii) |Ψi is easily determined
2 from the density matrix ρ. Moreover, we introduce a quantum circuit in which the quantum gates
are unitary transformations acting on a 2n-qudit system. These transformations are determined
n
by parameters that can be tuned to generate, once the ancillary qudits are disregarded, any given
a
J mixed n-quditstate.

3
PACSnumbers: 03.67.-a,03.67.Ac
1

]
h I. INTRODUCTION pends on a number of parameters exactly equal to the
p number of degrees of freedom of a generic mixed state ρ
- andiseasilydeterminedasafunctionofρ. Furthermore,
t Purificationis one of the basic tools in quantuminfor-
n wedesignaquantumcircuitgivenbyasequenceofquan-
mation science [1, 2]: Given a mixed quantum system S
a tum gates acting on both the system and the ancillary
u described by a density matrix ρ it is possible to intro-
qudits. The parameters which determine such quantum
q duce anotherancillary systemA, suchthat the state Ψ
| i gatescanbe tunedto generate,once the ancillaryqubits
[ of the composite system S +A is a pure state and ρ is
are disregarded, any mixed state ρ of system S. Finally,
recovered after partial tracing over A: ρ=Tr (Ψ Ψ).
2 A | ih | our protocol works for any system of n qudits and re-
Theancillarysystemmaybeaphysicalenvironmentthat
v
quires n ancillary qudits. We show that the number of
7 must be taken into account when doing experiments on
ancillary qudits is optimal, that is, it cannot be reduced
1 the system S, but not necessarily so. It may be a fic-
if we wish to design a quantum circuit capable of gener-
4 titious environment that allows us to prove interesting
1 results about the system under investigation. ating any n-qudit mixed state.
. Our paper is organized as follows. To set the nota-
0 Purification is a tool of great value in quantum infor-
tions, we first briefly review the concept of purification
1 mation science, with countless applications, for instance
8 (Sec.II).Thenweproposeourpurificationscheme,mov-
in the study of the distance between quantum states [2],
0 ing from simple to more and more complex cases. We
of the geometry of quantum states [3] and of the quan-
: start with the purification of a single qubit mixed state
v tum capacity of noisy quantum channels [4]. Besides its
(Sec. III), then we proceedwith the qutrit (Sec. IV) and
Xi theoreticalrelevance,purificationisinterestingforexper- the two-qubit (Sec. V) cases and finally we illustrate the
imental implementations of quantum information proto-
r generic n-qudit case (Sec. VI). Appendix A provides a
a cols requiring mixed state. While the direct generation
shortaccountof the diagramsof states, namely of a tool
ofamixtureρofquantumstatesnecessarilyinvolvessta-
very useful for the purposes of the present paper.
tistical errors, this problem can be avoided if the purifi-
cationstate Ψ is generated. Ofcourse,the priceto pay
| i
is that one has to work with the enlargedsystem S+A,
II. PURIFICATION
including the ancillary system A.
Duetothepartialtracestructurethepurificationstate
Given a quantum system S described by the density
Ψ of a density matrix ρ cannot be uniquely defined, as
| i matrix ρ, it is possible to introduce another ancillary
any unitary transformationUA⊗11S acting non-trivially systemA,suchthatthestate Ψ ofthecompositesystem
on the ancillary system only maps the state Ψ into a | i
new state Ψ′ =(UA 11S)Ψ whichis again|a piurifica- is pure and
| i ⊗ | i
tion of ρ. In this paper, we propose a quantum protocol ρ=TrA(Ψ Ψ). (1)
| ih |
that selects a specific purification Ψ . Such purification
| i This procedure,knownas purification,allowsus to asso-
turns out to be very convenient since the state Ψ de-
| i ciatea pure state Ψ witha densitymatrix ρ. Ageneric
| i
pure state of the global system S+A is given by
M−1N−1
∗Electronicaddress: [email protected] Ψ = C α i , (2)
αi
†Electronicaddress: [email protected] | i αX=0 Xi=0 | i| i
2
with α and i basis sets forthe Hilbert spaces nonlinear. Finally, we will see in the next sections that
A
{| i} {| i} H
and , of dimensions M and N, associated with the ourschemesuggestaveryconvenientquantumcircuitfor
S
H
subsystems A and S. Given a generic density matrix for the preparation of a generic density matrix ρ. To illus-
system S, trate the working of our purification method, we discuss
cases of increasing complexity, from a single qubit state
N−1 to a generic n-qudit state.
ρ= ρ i j , (3)
ij
| ih |
iX,j=0
we say that the state Ψ defined by Eq. (2) is a purifi- III. QUBIT
| i
cation of ρ if
A. Mixed-state purification
M−1N−1
ρ=Tr (Ψ Ψ)= C C⋆ i j . (4)
A | ih | αi αj| ih | We consider M =N =2 and set
αX=0 iX,j=0
C R ,
The equality between (3) and (4) implies (cid:26)C0110 ∈R++, C11 =0. (8)
∈
M−1
ρ = C C⋆ . (5) We first determine C from (5):
ij αi αj 01
αX=0
ρ = C 2+ C 2 = C 2, (9)
11 01 11 01
It is clear that (5) always admits a solution, provided | | | | | |
the Hilbert space of system A is large enough. More
wherethelastequalityfollowsfromthesecondlineof(8).
precisely, it is sufficient to consider a system A whose
SincewehavealsochosenC toberealandnonnegative,
01
Hilbert spacedimension is the same as thatof systemS.
we simply obtain
Indeed, if we express the reduced density matrix using
its diagonal representation, C01 =√ρ11. (10)
ρ= pi i i, (6) Then we can determine C from the condition
| ih | 00
Xi
ρ =C C⋆ +C C⋆ =C C , (11)
a purification for the density matrix (6) is given by 01 00 01 10 11 00 01
since C is already known from (9). Finally, knowing
Ψ = √p i′ i , (7) 01
| i Xi i| i| i C00, we can derive C10 from the condition
with {|i′i} orthonormal basis for HA. This purification ρ00 =|C00|2+|C10|2. (12)
procedurerequiresthediagonalizationofthedensityma-
trixρandthereforeisingeneralonlynumericallyfeasible. Taking into account that C10 R+, we have
∈
In what follows, we propose a different purification
scheme, which is optimal in that, for the purification C10 = ρ00 C00 2. (13)
−| |
of a generic n-qudit state, the number m = n of qu- p
dits of the ancillary system A cannot be reduced. While Note that, in the special case in which ρ11 =0 we can
this is the case also for the well-known purification (7) removeanyambiguityinthedefinitionofthepurification
based on the spectral decomposition (6), we anticipate state Ψ by setting C00 = 0. In this case, ρ = 0 0 is
| i | ih |
thatourmethodreadilyprovidesapurificationstate Ψ already a pure state and its “purification” is Ψ = 10 .
| i | i | i
that depends on a number of parameters exactly equal Alternatively, one can reshuffle the basis state according
to the number of degreesoffreedomof a genericρ. Note to 0 1 [8]. For a generic state ρ11 =0 and the state
| i↔| i 6
that,evenwhenthenumberofancillaryquditsisoptimal Ψ reads, in the αi = 00 , 01 , 10 , 11 basis, as
(m = n so that M = N), the number 2N2 2 of real f|olilows: {| i | i | i | i | i}
−
coefficients C determining the purification state (2) is
αi ρ
in general much larger than the number N2 1 of real 01
free parameters that must be set to determin−e a generic C00  √ρ11 
density matrix of size N. Different choices of the coef-  
ficients C are therefore possible. Our choice provides C01  √ρ11 
a purificaαtiion state Ψ depending on a number N2 1 Ψ = = . (14)
of real parameters e|xaictly equal to the number of r−eal | i C10   ρ00ρ11−ρ10ρ01 
freedoms of a generic mixed states ρ. Furthermore, the    r ρ11 
C   
coefficient Cαi in (2) can be easily determined from con-  11   
 0 
ditions (5), in spite of the fact that these equations are  
3
-Φ α θ
Φ
θ
00
LSB
01 Real
α
MSB 10 Real
11 0
FIG.1: Quantumcircuitforthepurificationofasinglequbit.
Thequbitsrunfrom top to bottom from theleast significant
(LSB) to the most significant (MSB). FIG.2: Diagramofstatesforthepurificationofasinglequbit.
Starting from the input state |00i, information flows on the
thick lines. We esplicitly indicate at theright hand side that
B. Mixed-state generation thecoefficientes C01 and C10 are real, while C11 =0.
In this subsection, we provide a quantum circuit gen-
eratingthestate Ψ of(14),namelythe purificationofa From the first equation we determine α, then from the
generic single-qub|itidensity matrix ρ. After disregarding second θ and finally from the third φ, as a function of
the ancillary qubit (this corresponds to performing the the coefficientsCαi,whichinturnaredeterminedbyour
partialtrace of the density matrix Ψ Ψ over the ancil- purification protocol from the density matrix elements
lary qubit), we obtain the mixed s|tatiehρ|. Therefore, we ρij. Note that, since by construction C01,C10 0, we
end up with an experimentally viable procedure for the can take α,θ ∈ 0,π2 . Finally, the phase φ∈[0,≥2π).
generationofagenericmixedsingle-qubitstatebymeans Forastraight(cid:2)forwa(cid:3)rdextensionoftheresultspresented
of a two-qubit state subjected??? to controlled unitary in this section from the purification of a single qubit
transformations. to more complex systems it is convenient to express
Thequantumcircuitgeneratingthestate Ψ isshown the quantum circuit in Fig. 1 in terms of diagrams of
in Fig. 1. A square box with a greek letter |insiide (here, states [5], of which a very brief account is given in Ap-
α or θ) stands for a rotation operator. Its matrix repre- pendix A. The diagram of states corresponding to the
sentation in the 0 , 1 reads as follows: purification circuit of Fig. 1 is shown in Fig. 2. Note
{| i | i} that, given the input state 00 , the output state (17) is
| i
cosα sinα immediatelywrittenfollowingtheinformationflowalong
R(α)= − . (15)
(cid:20) sinα cosα (cid:21) the thick lines of the diagram of states. We stress that
other optimal purifications, where the purification state
The full circle with φ above is the phase-shift gate, is determined by N2 1 =3 real parameters, are possi-
−
defined by the diagonal matrix ble. Our choice corres−ponds to setting C =0 and C ,
11 01
C real. We will see that the diagrams of states imme-
PHASE( φ)=diag(1,e−iφ). (16) 10
− diately lead to optimal purifications also for arbitrarily
As an overall phase factor is arbitrary, the action complex systems.
of this gate is equivalently represented by the matrix
diag(eiφ,1). In the controlled-gates, the empty circle on
the control qubit means that the gate acts non trivially C. “Invasion” of the Bloch ball
(differentlyfromidentity)onthe targetqubitifandonly
if the state of the control qubit is 0 . Using the quantum circuit in Fig. 1 we can write a
| i
Ontheotherhand,anysingle-qubitdensitymatrixcan generic single-qubit state as
be generated by means of the quantum circuit in Fig. 1.
Given the input state Ψ = 00 , the output state is 2
i
| i | i ρ= p ψ ψ , (19)
k k k
cosαcosθeiφ kX=1 | ih |
cosαsinθ
Ψ = . (17)
| fi sinα with p1 = cos2α, p2 = sin2α, ψ1 generic single-qubit
| i
 0  pure state and ψ = 0 . The matrix representation of
  | 2i | i
ρ in the 0 , 1 basis is given by
Thisstateisequaltothepurificationstate(14),provided {| i | i}
we set cos2θ cosθsinθeiφ
ρ=cos2α
sinα=C , (cid:20) cosθsinθe−iφ sin2θ (cid:21)
10
 ccoossααscionsθθe=iφC=01C,00. (18) +sin2α(cid:20) 10 00 (cid:21)= 21(cid:20)X1++ZiY X1−ZiY (cid:21). (20)
−

4
1 1 IV. QUTRIT
YY00..55 YY00..55
00 00
--00..55 --00..55
--11 --11 We consider M =N =3 and set
1 1
0.5 0.5 C R ,
02 +
Z0 Z0  C11 ∈R+, C12 =0, (21)
-0.5 -0.5  C20 ∈R+, C21 =C22 =0.
∈
-1 -1 
--11 --11 We first obtain C from
--00..55 --00..55 02
00 00
XX 00..55 XX 00..55
11 11 ρ22 = C02 2+ C12 2+ C22 2 = C02 2. (22)
| | | | | | | |
1 1
YY00..55 YY00..55
00 00 Once C02 0 is determined as C02 = √ρ22, we obtain
--00..55 --00..55 ≥
C and C from
--11 --11 01 00
1 1
0.5 0.5 ρ12 =C01C0⋆2+C11C1⋆2+C21C2⋆2 =C01C02, (23)
Z0 Z0
-0.5 -0.5 ρ =C C⋆ +C C⋆ +C C⋆ =C C . (24)
-1 -1 02 00 02 10 12 20 22 00 02
--11 --11
--00..55 --00..55 Then we obtain C from
00 00 11
XX 00..55 XX 00..55
11 11
YY00..551 YY00..551 ρ11 =|C01|2+|C11|2+|C21|2 =|C01|2+C121, (25)
00 00
--00..55 --00..55 and, finally, C and C from
10 20
--11 --11
1 1
ρ =C C⋆ +C C⋆ +C C⋆ =C C⋆ +C C ,
0.5 0.5 01 00 01 10 11 20 21 00 01 10 11
(26)
Z0 Z0
-0.5 -0.5 ρ = C 2+ C 2+C2 . (27)
-1 -1 00 | 00| | 10| 20
--11 --11
--00..55 --00..55 We stress that conditions (21) lead to a purification
00 00
XX 00..55 XX 00..55 state determined by N2 1 = 8 free real parameters,
11 11 −
exactly corresponding to the number of real freedoms
needed to set a generic density matrix for a qutrit. Con-
FIG.3: “Invasion”oftheBlochball: cos2α=0(topleft), π
10 ditions (21) are readily derived if the purification of a
(topright),2π (middleleft),3π (middleright),4π (bottom
10 10 10 generic qutrit state is implemented by means of the di-
left), π (bottom right).
2 agram of states shown in Fig. 4. In this figure, a box
with two geek letters written on top of it (for instance,
α and α ) represents a unitary transformation whose
1 2
matrix representation has, in the 0 , 1 , 2 basis, the
{| i | i | i}
first column given by
cosα cosα
1 2
 cosα1sinα2 . (28)
sinα
1
 
ThelastequalitycorrespondstotheusualBloch-ballrep- Such transformation maps the input state 0 into
| i
resentation of the (generally mixed) single-qubit states.
It is clear that, once α is fixed, Eq. (20) represents a cosα1cosα2 0 +cosα1sinα2 1 +sinα1 2 . (29)
| i | i | i
surfaceinthe(X,Y,Z)-space,obtainedaftercontracting
the pure-states (unit radius) Bloch-sphere of the factor Finally, the box with the letter θ3 ontop ofit represents
cos2α and translating it in the positive Z-direction by the rotation R(θ3) acting on the two-dimensional sub-
sin2α. Plots of this surface for different values of α are space spanned by the state 0 and 1 , with R defined
| i | i
showninFig.3. ItisclearthatallthepointsoftheBloch by (15).
ball are recoveredwhen α goes from 0 to π. We can say Giventheinputstate 00 ,theoutputpurificationstate
2 | i
thatthereisan“invasion”oftheBlochballstartingfrom Ψ = C αi can be immediately written by fol-
| i α,i αi| i
the north pole. lowingPthe thick lines of the diagram of states in Fig. 4.
5
ball by means of suitably scaled and translated pure-
00 α1α2 θ1θ2 Φ1 qubit Bloch spheres may be generalized to the single-
qutrit case. The role of the Bloch sphere is here played
Φ
01 2 bythesurfaceofsingle-qutritpurestatesandthevolume
of all single-qutrit states is “invaded” when the parame-
02 Real ters p are varied, with the constraint p =1.
k k k
10 θ3 Φ3 P
V. TWO QUBITS
11 Real
We consider M =N =4 and set
12 0
C R ,
03 +
20 Real C ∈R , C =0,
21 0 CC1221 ∈∈RR++,, CC1232 ==CC23 ==C0, =0. (32)
30 + 31 32 33
∈
22 0 
We first obtain C from
03
ρ = C 2+ C 2+ C 2+ C 2 = C 2, (33)
33 03 13 23 33 03
FIG. 4: Diagram of states for the purification of a single | | | | | | | | | |
qutrit. To simplify the plot, only the thick lines correspond- then C , C , and C from
02 01 00
ing to the information flow are shown inside the boxes. The
angles αi,θj ∈ˆ0,π2˜, while thephases φk ∈[0,2π). ρ23 =C02C0⋆3+C12C1⋆3+C22C2⋆3+C32C3⋆3 =C02C03,
(34)
We obtain ρ13 =C01C0⋆3+C11C1⋆3+C21C2⋆3+C31C3⋆3 =C01C03,
(35)
C =cosα cosα cosθ cosθ eiφ1,
00 1 2 1 2
 C01 =cosα1cosα2cosθ1sinθ2eiφ2, ρ03 =C00C0⋆3+C10C1⋆3+C20C2⋆3+C30C3⋆3 =C00C03,
 CCCC01112012 ====ccc0ooo,sssααα111cssiionnsααα222cssioinnsθθθ313,,eiφ3, (30) thρe2n2 =C1|2Cf0r2o|2m+|C12|2+|C22|2+|C32|2 =|C02|2+C122, ((3367))
C =sinα ,
 CC222012 ==00,. 1 then ρC1121=anCd01CC100⋆2f+romC11C1⋆2=+CC21CC⋆2⋆2++CC31CC3⋆2, (38)
These relations can be easily inverted to obtain the pa- 01 02 11 12
rameters α ,θ ,φ as a function of the coefficients C
j k l αi
{ }
and, therefore, of the elements of the density matrix ρ. ρ =C C⋆ +C C⋆ +C C⋆ +C C⋆
02 00 02 10 12 20 22 30 32
Therefore, the quantum circuit represented by the dia-
=C C⋆ +C C . (39)
gramofstatesofFig.4canbeusedtogenerateanygiven 00 02 10 12
qutritstateρ,oncetheancillaryqutritisdisregarded. Fi-
then C from
21
nally, we point out that the number of real parameters
{αi,θk,φl} that determine the state |Ψi is equal to 8, ρ11 =|C01|2+|C11|2+|C21|2+|C31|2
that is, exactly to the number of parameters needed to = C 2+ C 2+C2 , (40)
determine a (generally mixed) single-qutrit state ρ. | 01| | 11| 21
It is clear from the purification drawn in Fig. 4 that then C from
20
we can write a generic single-qutrit state as
ρ =C C⋆ +C C⋆ +C C⋆ +C C⋆
01 00 01 10 11 20 21 30 31
3 =C C⋆ +C C⋆ +C C , (41)
ρ= p ψ ψ , (31) 00 01 10 11 20 21
k k k
| ih |
Xk=1 and, finally, C from
30
wsinit2hαp,1 ψ= cogse2nαer1iccossi2nαgl2e,-qpu2tr=it pcuosr2eαs1tastine,2αψ2, pp3ur=e ρ00 =|C00|2+|C10|2+|C20|2+C320. (42)
1 1 2
| i | i
state residing in the two-dimensional subspace spanned As for the previous examples, we point out that the
by 0 and 1 , and ψ = 0 . The picture developed in number N2 1=15 of free parameters determining the
3
| i | i | i | i −
Sec. IIIC about the “invasion” of the single-qubit Bloch purification state Ψ cannot be reduced given a generic
| i
6
α1 α2 θ1 θ3 Φ can be written as
0000 1
Φ 4
0001 2
0010 θ4 Φ3 ρ= pk|ψkihψk|, (44)
kX=1
0011 Real
0100 θ2 θ5 Φ4 with p1 = cos2α1cos2α2, p2 = cos2α1sin2α2, p3 =
0101 Φ5 sin2α1cos2α3, and p4 = sin2α1sin2α3. The “invasion”
picture developed in Sec. IIIC can be extended also to
0110 Real
the present case, with the volume of all two-qubit states
0111 0 “invaded” when the parameters p are varied, under the
k
1000 α3 θ6 Φ6 constraint kpk = 1. Note that the number of real pa-
1001 Real rameters {αPi,θk,φl} used to determine the state |Ψi is
15,thatis,exactlythenumberofparametersrequiredto
1010 0
determine a generic two-qubit state.
1011 0
1100 Real
1101 0 VI. n QUDITS
1110 0
We consider M =N =dn and set
1111 0
C R ,
0,N−1 +
FIG. 5: Diagram of states for the purification of a two-qubit  C1,N−2 ∈R+, C1,N−1 =0
tαstoia,ttθheje.∈iTnoˆfo0sr,immπ2a˜p,tliiwofynhitfllheoewtphaleoretp,hsohanoswleysntφhinkes∈itdh[ei0ct,kh2eπlin)b.eosxecso.rrTeshpeoanndgilnegs  ...CC2,N−3 ∈∈RR+,, CC2,N−2 ==CC2,N−1 ==.0.,. (45)
N−1,0 + N−1,1 N−1,2
 =CN∈−1,N−1 =0.
two-qubit mixed state ρ. Moreover, conditions (32) are The general procedure for determining the coefficient
determined from the diagram of states for the purifica- C is clear from the previous examples.
αi
tion of a generic two-qubit state shown in Fig. 5. Fol-
lowing the information flow from the input states 0000 We first determine C0,N−1 from ρN−1,N−1,
| i •
we can immediately write down the output purification
then C from ρ , with j =N 2,...,0,
state |Ψi= α,iCαi|αii. We obtain • 0j j,N−1 −
P then C from ρ ,
1,N−2 N−2,N−2
•
C =cosα cosα cosθ cosθ eiφ1,
0000 1 2 1 3 then C from ρ , with j =N 3,...,0, ...,
1j j,N−2
 C0001 =cosα1cosα2cosθ1sinθ3eiφ2, • −
 CCCCCCC1000000011001101111000101010 =======s0ccccciooooo,nsssssαααααα111111csccssoiiioonnnsssαααααα322222csccssioooiinnnsssθθθθθθ261122,escsciiiooφnnss6θθ,θθ4545,eeeiiiφφφ534,,, (43) oodffroffeornTtrmehse•,hiaetaialtsynahnfpnemrdaeuc,naeinrlcfitilopifiranlrialcraxmayarlraltaoyysimlfo,yisznsCesyaittzNsteeietismr−oesdnmo1ctn,p0hactoinaoifsfmrtnnondodamm2tilntubiaρos−1es0tn0q1rtb.uehTed[dertus(ihtρencest)eud−imts=.o1bnTedot1rehet]tr.eosemufrnmffiTuqcmohuicnmiedbeerineetearsst-
C =sinα cosα sinθ , −
1001 1 3 6 to purify ρ, as a pure state Ψ in the Hilbert space of
 CCCCCC111111101011011110011001 ======s00000i,.,,,nα1sinα3, n−tFcTfooaih+2rgttii.tshao(e6nnnern,ym−fumatmdhc1iesteb)≥tdhte=nhuro2uade,2mitsntnibaosn−e≥tgoreh1ltxoe1oabqs.fncuuaftoOlffidrlreypinmctehsidaetar2hhnlesniaatee|zslafaotaip2sticdoa1ht22nr.eonadrrc−m2Fohni1rnnea−o−tdnm1|eΨidr2t−s,iitofahirin2nseseefa<oiodslrurlcobuhrd|misΨet2tpsrmnriuaa(aa−rrttyinthefii)dd1ec-.
−
As in the previous cases, we can invert these equations drawing in Fig. 5 we can also see that the n-qudit state
anddeterminetheparameters α ,θ ,φ intermsofthe can be written as
j k l
{ }
coefficientsC and,therefore,oftheelementsoftheden-
αi dn
sity matrix ρ.
ρ= p ψ ψ , (46)
k k k
We can see from Fig. 5 that a generic two-qubit state kX=1 | ih |
7
Number of freedoms is optimal, in that the number n of ancillary qudits used
{
forthepurificationcannotbefurtherreduced. Moreover,
ouralgorithmcanalsobeseenasaquantumprotocolfor
n
Cluster 1 2d -2
thegenerationofagenericn-quditstatebymeansofsuit-
Real ableunitaryoperationsappliedbothtothesystemandto
{ the ancillary qudits, with the ancillary qudits eventually
disregarded. While also the well-known purification (7)
n
Cluster 2 2d -4 based on the spectral decomposition (6) uses n ancillary
Real
qudits, our method is optimal in that it readily provides
0
a purification state that depends on a number of param-
{ eters exactly equal to the number of degrees of freedom
of the generic n-qubit state that we wish to purify.
n Real
Cluster d -1 0
2 It is well known that the purification Ψ of a generic
0 mixedstateρcannotbeuniquelydetermi|neid,asthepar-
0
{ tialtraceoverthe ancillaryqudits is invariantunder any
Real
0 unitary transformation UA 11S acting non trivially on
⊗
Cluster dn 0 0 the ancillary qudits only. Indeed, we have
0
0
ρ=Tr (Ψ Ψ)=Tr (Ψ′ Ψ′ ), (48)
A A
| ih | | ih |
FIG.6: Schetchofthediagramofstatesforthepurificationof with
a n-qudit state by means of 2n qudits. The diagram has d2n
lines, one for each state of the computational basis. We can
clustergroupsofdn lines. Theconstraintsandthenumberof |Ψ′i=(UA⊗11S)|Ψi. (49)
realfreedomsforeachclusterarehighlighted. Ifweaddthese
numbersplus the dn−1 independent weights in the mixture ItissufficienttoaddtheunitarytransformationUA 11S
(46),weobtain atotalnumberofd2n−1degreesoffreedom. at the end of our purification protocol to obtain any⊗2n-
quditpurification Ψ′ ofa n-quditstate ρ. Onthe other
| i
hand, we can say that our protocol selects a very conve-
swpiathcetshoefpduerceresatsaintegsdfirmomens|ψio1ni,tforo|mψddnni rteosi1d.ing in sub- nient purificationas the coefficients of the wave function
Ψ inthecomputationalbasisareeasilydeterminedfrom
We note that the tensor product structure of many- | i
the density matrix ρ andthe quantumcircuitgenerating
qudit quantum systems does not play any role in our
ρ can be immediately drawn.
purification protocol. That is to say, the same purifica-
tionschemeappliesto asystemofn quditsorto asingle
system of size N = dn. Of course, the practical im-
plementation of the protocol will depend on the specific
APPENDIX A: DIAGRAMS OF STATES
quantum hardware at disposal.
From the mathematical viewpoint, our purification
Diagramsofstates[5]graphicallyrepresenthowquan-
protocol can be seen as the Cholesky decomposition [6]
of the density matrix ρ. If we consider the coefficients tum information is elaborated during the execution of a
C aselementsofadn dn matrixC,thenitstranspose quantum circuit. In the usual way of drawing a quan-
αi
D CT is a upper tria×ngular matrix and Eq. (5) reads tum circuit [1, 2] each horizontalline represents a qubit.
≡ In contrast, in diagrams of states we draw a horizontal
ρ=CTC⋆ =DDT∗ =DD†, (47)
lineforeachstateofthecomputationalbasis. Therefore,
namely it is the Cholesky decomposition of the density diagrams of states are less synthetic but may help us to
matrix ρ. Such decomposition is unique when ρ is pos- clearlyvisualizequantuminformationflowinaquantum
itive, while the ambiguities arising when ρ is singular circuit.
may be removed by suitable prescriptions or reshuffling For the purposes of the present paper, it will be suffi-
as previously discussed for the single-qubit case. cienttoshowthediagramsofstatesforelementarysingle-
ItisinterestingtoremarkthattheCholeskydecompo- qubit quantum gates. The phase-shift gate PHASE(φ),
sition has already been used in the context of quantum defined by Eq. (16), and the rotation gate R(θ), defined
information science in parametrizing the density opera- by Eq. (15), are shown in Fig. 7 and Fig. 8, respec-
tor in order to guarantee positivity [7]. The purpose of tively. Finally, the generation of a generic single-qubit
Ref.[7]wastopresentauniversaltechniqueforquantum- state cosθ 0 +sinθeiφ 1 starting from the input state
| i | i
state estimation. 0 isshowninFig.9. Theinformationflowsonthethick
| i
VII. FINAL REMARKS lines,fromlefttoright,whilethinnerlinescorrespondto
absence of information. Note that, following the thick
In summary, we have proposed an algorithm for the lines, the final state can here be immediately written.
purification of a generic n-qudit state. This algorithm
8
0
Φ
Φ
1
FIG. 7: Quantum circuit (left) and diagram of states (right)
for the phase-shift gate.
θ
0
θ
1
FIG. 8: Quantum circuit (left) and diagram of states (right)
for the R(θ) gate.
θ
0
Φ
θ
Φ
1
FIG. 9: Quantum circuit (left) and diagram of states (right)
for thegeneration ofageneric single-qubit state. Tosimplify
theplot,onlythethicklinescorrespondingtotheinformation
flow are shown inside theboxes.
[1] G. Benenti, G. Casati and G. Strini, Principles of Quan- nal of UnconventionalComputing, in press.
tum Computation and Information,Vol.I:Basic concepts [6] G. H. Golub and C. F. Van Loan, Matrix Computations
(World Scientific, Singapore, 2004); Vol. II: Basic tools (thirdedition)(TheJohnsHopkinsUniversityPress,Bal-
and special topics (World Scientific, Singapore, 2007). timore, 1996).
[2] M. A. Nielsen and I. L. Chuang, Quantum computation [7] K.Banaszek, G. M. D’Ariano, M. G. A. Paris, and M. F.
and quantum information (Cambridge University Press, Sacchi, Phys. Rev.A 61, 010304(R) (1999).
Cambridge, 2000). [8] Similar procedures can be applied to higher dimensional
[3] I. Bengtsson and K. Z˙yczkowski, Geometry of quantum cases whenever diagonal elements of thedensity matrix ρ
states (Cambridge University Press, Cambridge, 2006). are equal to zero. For the sake of simplicity, we will not
[4] H.Barnum,M.A.Nielsen,andB.Schumacher,Phys.Rev. discuss any longer in ourpaper such special cases.
A, 57, 4153 (1998).
[5] S. Felloni, A. Leporati, and G. Strini, International Jour-