VOLUME 86, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 12 FEBRUARY 2001

Fidelity Balance in Quantum Operations

Konrad BanaszekCenter for Quantum Information and Rochester Theory Center for Optical Science and Engineering, University of Rochester,

Rochester, New York 14627and Instytut Fizyki Teoretycznej, Uniwersytet Warszawski, Hoza 69, PL-00-681 Warszawa, Poland

(Received 1 May 2000; revised manuscript received 30 June 2000)

I derive a tight bound between the quality of estimating the state of a single copy of a d-level system,and the degree the initial state has to be altered in the course of this procedure. This result provides acomplete analytical description of the quantum mechanical trade-off between the information gain andthe quantum state disturbance expressed in terms of mean fidelities. I also discuss consequences of thisbound for quantum teleportation using nonmaximally entangled states.

DOI: 10.1103/PhysRevLett.86.1366 PACS numbers: 03.67.–a, 03.65.Ta

As a general rule, the more information is obtained froman operation on a quantum system, the more its state has tobe altered. This heuristic statement was first exemplifiedby the Heisenberg microscope gedankenexperiment [1],where the spatial resolution of the apparatus was shownto scale inversely with the uncertainty of the momentumtransferred during the observation. Presently, the distur-bance caused by the information gain has become an is-sue of practical significance, as it underlies the security ofquantum key distribution [2].

The balance between the information gain and the statedisturbance attracts currently a lot of interest, particularlyin the context of quantum cryptography [3]. Informationtheory provides a selection of concepts to quantify boththe information gain and the state disturbance. The choiceof measures for these two effects is usually dictated by therelevance to a specific application. In most cases, however,derivation of the actual balance represents a highly nontriv-ial task, especially if one is tempted to resign from numeri-cal means. The purpose of this Letter is to present aformulation of the information gain versus state dis-turbance trade-off which is completely solvable usingelementary analytical techniques. This formulation ismotivated by recent works on quantum state estimation[4], where the information obtained from the operationis converted into an estimate for the initial state of thesystem.

The problem considered in this Letter can be formulatedas follows. Suppose we are given a single d-level particlein a completely unknown pure state jc�. We want to makea guess about the quantum state of this particle, but at thesame time we would like to alter the state as little as pos-sible. One can associate two fidelities with such a proce-dure. The first one, which we will denote by F, describeshow much the state after the operation resembles the origi-nal one. The second fidelity, denoted by G, characterizesthe average quality of our guess. It is natural to expecta trade-off between these two quantities: the more infor-mation is extracted from the system, i.e., the larger G, theless the final state should resemble the initial one, hence

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the smaller F should be. What is the actual quantitativebound between F and G?

Two extreme cases are well known: if nothing is doneto the particle we have F � 1, but then our guess aboutthe state of the particle has to be random, which yieldsG � 1�d. On the other hand, the optimal estimation strat-egy for a single copy [5] yields G � 2��d 1 1�, but thenthe particle after the operation cannot provide any moreinformation on the initial state; thus also F � 2��d 1 1�.I prove here that quantum mechanics imposes a generalconstraint between F and G in the form of the followinginequality:s

F 21

d 1 1#

sG 2

1d 1 1

1

s�d 2 1�

µ2

d 1 12 G

∂. (1)

I also show that this inequality cannot be further improved;i.e., there exist quantum operations saturating the equalitysign.

The most general strategy that can be applied to the par-ticle has the form of a trace-preserving operation describedby a set of operators Ar , where r � 1, . . . , N . These op-erators satisfy the completeness relation

NXr�1

Ayr Ar � ' . (2)

The classical information gained from this operation isgiven by the index r , which is subsequently used to es-timate the initial state of the particle. The outcome r ofthe operation performed on a state jc� is obtained with theprobability �cjAy

r Ar jc�. This corresponds to the follow-ing conditional transformation of the quantum state [6]:

jc� !Ar jc�q

�cjAyr Ar jc�

. (3)

We shall measure the resemblance of the transformed stateto the original one using the squared modulus of the scalar

© 2001 The American Physical Society

VOLUME 86, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 12 FEBRUARY 2001

product, equal to j�cjAr jc�j2��cjAyr Ar jc�. Summation

of this expression over r with the weights �cjAyr Ar jc�,

and integration over all possible input states jc�, yieldsthe complete expression for the mean operation fidelity F:

F �Z

dc

NXr�1

j�cjAr jc�j2. (4)

Here the integralR

dc over the space of pure states is per-formed using the canonical measure invariant with respectto the group of unitary transformations on the state vectorsof the particle.

Given the outcome r of the operation, we can make aguess jcr � what the state originally was. The quality ofthis guess, assuming that the initial state was jc�, can bequantified with the help of the overlap j�cr jc�j2. Themean estimation fidelity G is given by the average of thisexpression over all outcomes r with the probability distri-bution �cjAy

r Ar jc�, and by integration over states jc�:

G �Z

dc

NXr�1

�cjAyr Ar jc� j�cr jc�j2. (5)

We will start derivation of the trade-off between the fi-delities F and G by evaluating the integrals over jc�. Forthis purpose, let us introduce in Eq. (4) two decomposi-tions of unity in a certain orthonormal basis ji�:

F �NX

r�1

d21Xi,j�0

Zdc�c j i� �ijAy

r jc� �cjAr j j� � j jc�

�NX

r�1

d21Xi,j�0

�ijAyr MijAr j j� , (6)

where by Mij we have denoted the following integrals ofprojectors on the states jc� �cj:

Mij �Z

dc�c j i� � j jc� jc� �cj

�1

d�d 1 1��dij' 1 ji� � jj� . (7)

The second explicit form of the operators Mij has beenderived in Ref. [7]. This formula allows us to simplify theexpression for the mean operation fidelity F to the form

F �1

d�d 1 1�

√d21Xi�0

NXr�1

�ijAyr Ar ji�

1

NXr�1

Éd21Xi�0

�ijAr ji�

É2!

�1

d�d 1 1�

√d 1

NXr�1

jTrAr j2

!. (8)

Let us now consider the estimation fidelity G. The guessjcr � can be represented as a result of a certain unitarytransformation Ur acting on a reference state, which wewill take for concreteness to be j0�,

jcr � � Ur j0� . (9)

Using this representation, and changing the integrationmeasure in Eq. (5) according to jc� ! Ur jc�, we canevaluate the integral over jc�,

G �NX

r�1

Zdcj�0 jc�j2�cjUy

r Ayr ArUr jc�

�NX

r�1

Tr�Uyr Ay

r ArUrM00� . (10)

Inserting the explicit form of the operator M00 � �' 1

j0��0j���d�d 1 1�� yields

G �1

d�d 1 1�

√NX

r�1

Tr�Uyr Ay

r ArUr �

1

NXr�1

�0jUyr Ay

r ArUr j0�

!

�1

d�d 1 1�

√d 1

NXr�1

�cr jAyr Ar jcr�

!. (11)

This expression provides directly a recipe for optimal as-signment of guesses jcr � to outcomes of the operation:each of the components �cr jAy

r Ar jcr � in the sum over r ismaximized if jcr � is the eigenvector of Ay

r Ar correspond-ing to its maximum eigenvalue. Consequently, the maxi-mum value of the mean estimation fidelity G for a givenoperation �Ar can be written as

G �1

d�d 1 1�

√d 1

NXr�1

kArk2

!, (12)

where the operator norm is defined in the standard way,

kArk � sup�w jw��1

q�wjA

yr Ar jw� . (13)

In order to relate the fidelities F and G to each other,let us consider the singular-value decomposition [8] of theoperators Ar ,

Ar � VrDrWr , (14)

where Vr and Wr are unitary, and Dr is a semipositivedefinite diagonal matrix,

Dr �d21Xi�0

lri ji� �ij , (15)

with the diagonal elements put in a decreasing order:l

r0 $ · · · $ l

rd21 $ 0. We will first show that only the

diagonal matrices Dr are relevant to the trade-off. Indeed,the modulus of the trace of the matrix Ar appearing inEq. (8) is bounded by

jTrAr j �

Éd21Xi�0

�ijWrVrDr ji�

É

#

d21Xi�0

lri j�ijWrVr ji�j #

d21Xi�0

lri , (16)

and moreover any quantum operation can easily be

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VOLUME 86, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 12 FEBRUARY 2001

modified in such a way that the equality sign is reached.What needs to be done is to follow the operation �Ar withan extra unitary transformation Wy

r Vyr depending on the

outcome r . Let us note that this corresponds to the modi-fication of the operation according to Ar ! Wy

r Vyr Ar ,

which makes each element of the operation a semipositiveHermitian operator. As we are interested in the maximumvalue of F, we can further assume with no loss of gener-ality that

F �1

d�d 1 1�

"d 1

NXr�1

√d21Xi�0

lri

!2#. (17)

The expression for the estimation fidelity written in termsof l

ri takes the form

G �1

d�d 1 1�

√d 1

NXr�1

�lr0�2

!. (18)

In addition, the trace of the completeness condition givenin Eq. (2) yields the following constraint on l

ri :

NXr�1

d21Xi�0

�lri �2 � d . (19)

To complete the proof of the inequality (1), it is convenientto introduce vector notation. Let us define d real vectorsvi � �l1

i , . . . , lNi �, where the index i runs from 0 to d 2 1.

Sums over r appearing in Eqs. (17) and (18) can be writ-ten as

f �NX

r�1

√d21Xi�0

lri

!2

�d21Xi,j�0

vi ? vj , (20)

g �NX

r�1

�lr0�2 � jv0j

2, (21)

where the dot denotes the scalar product, and j ? j is thestandard quadratic norm. The completeness condition (19)for the operation �Ar written in the vector notation takesthe form

d21Xi�0

jvij2 � d . (22)

Let us now suppose that the vector v0 is fixed. The estima-tion fidelity is then given by G � �d 1 jv0j

2���d�d 1 1��.What is the maximum operation fidelity F that can beachieved with this constraint? The answer to this questionis provided by an application of the Schwarz inequality toEq. (20):

f #

d21Xi,j�0

jvij jvjj �

√d21Xi�0

jvij

!2

�

√p

g 1

d21Xi�1

jvij

!2

.

(23)

We have excluded here from the sum over i the norm ofthe vector v0 which is fixed and equal to

pg. The sum

of the norms of the remaining vectors can be estimatedusing the inequality between the arithmetic and quadraticmeans,

1368

1d 2 1

d21Xi�1

jvi j #

vuut 1d 2 1

d21Xi�1

jvij2 �

sd 2 gd 2 1

, (24)

where we have evaluated the sumPd21

i�1 jvi j2 using

Eq. (22). Inserting this bound into Eq. (23) we finallyobtain the inequality

f # �p

g 1

q�d 2 1� �d 2 g� �2, (25)

which expressed in terms of the fidelities F and G takesthe form of Eq. (1).

The necessary and sufficient conditions for a quantumoperation to reach the equality sign can most easily beformulated in the vector notation. The Schwarz inequality(23) becomes equality if all the vectors v0, . . . , vd21 arecollinear. Furthermore, the equality sign in Eq. (24) holdsif and only if jv1j � · · · � jvd21j. It is straightforward tosee that an exemplary operation satisfying these conditionsfor a given estimation fidelity G � �1 1 g�d���d 1 1� isdefined by

Ar �

rgd

jr 2 1� �r 2 1j

1

sd 2 g

d�d 2 1��' 2 jr 2 1� �r 2 1j� , (26)

where the index r runs from 1 to d, and the projectors jr 2

1� �r 2 1j are constructed using any orthonormal basis.This confirms the inequality (1) is indeed a tight one andcannot be further improved.

A simple transformation of Eq. (1) shows that the quan-tum mechanically allowed region for the fidelities F andG is bounded by a quadratic curve, which turns out to bea fragment of an ellipse given by the equation

�F 2 F0�2 1 d2�G 2 G0�2 1

2�d 2 2� �F2F0� �G 2 G0� �d 2 1

�d 1 1�2 (27)

with F0 � �d 1 2���2d 1 2� and G0 � 3��2d 1 2�.The shape of the region for several values of d is depictedin Fig. 1.

The balance between the operation and estimation fi-delities derived in this Letter has interesting consequencesin quantum teleportation based on nonmaximally entan-gled states. If two parties share a pure bipartite stateof the Schmidt form jtele� �

Pd21k�0 mkjk� ≠ jk�, then the

maximum teleportation fidelity attainable using this stateis given by [7,9]

Ftele �1 1 �

Pd21k�0 mk�2

d 1 1. (28)

Furthermore, for a nonmaximally entangled state themeasurement performed during the teleportation protocolreveals some information on the teleported state. Thisinformation can be converted into an estimate for theinitial state, whose maximum average fidelity has beenshown to equal [7]

VOLUME 86, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 12 FEBRUARY 2001

1/d 2/(d+1)G2/(d+1)

1

F

FIG. 1. Rescaled bound for the operation fidelity F versusthe estimation fidelity G, plotted for d � 2 (solid line), d � 4(dashed line), and d � 8 (dotted line).

Gtele �1 1 m

20

d 1 1, (29)

where m0 denotes the largest Schmidt coefficient for thestate jtele�. As the procedure of teleportation can beviewed as a special case of a quantum operation [10], thebound (1) applies as well to the pair of fidelities Ftele andGtele. Consequently, for a given teleportation fidelity Ftele,the maximum value of the estimation fidelity is achievedfor the state jtele� satisfying the condition m1 � · · · �

md21 �q

�1 2 m20���d 2 1�. This condition defines a

class of pure bipartite states which are optimal from thepoint of view of the trade-off between the teleportation fi-delity and the estimation fidelity.

In conclusion, I have obtained a tight bound for thefidelities describing the quality of estimating the state of asingle copy of a d-level particle, and the degree the initialstate has to be changed during this operation. This resultseems to be one of very few cases, when the trade-offbetween the information gain and the state disturbance canbe derived in a closed analytical form.

I acknowledge useful discussions with J. H. Eberly,C. A. Fuchs, N. Lütkenhaus, V. Vedral, and I. A. Walm-sley. This research was partially supported by ARO-

administered MURI Grant No. DAAG-19-99-1-0125,NSF Grant No. PHY-9415583, and KBN Grant No. 2P03B 089 16.

[1] W. Heisenberg, The Physical Principles of the QuantumTheory (Dover, New York, 1930). Let us note that theHeisenberg uncertainty relation can be assigned two mean-ings. The first one refers to variances of the position andthe momentum when these quantities are determined fromdisjoint subsets of an ensemble of identically prepared par-ticles. The second meaning, illustrated by the microscopeexperiment and relevant in the context of information gainversus disturbance, describes a bound on the resolution ofan apparatus measuring simultaneously both the conjugatevariables. For a detailed discussion, see K. Wódkiewicz,Phys. Lett. A 124, 207 (1987); S. Stenholm, Ann. Phys.(N.Y.) 218, 233 (1992) and references therein.

[2] C. H. Bennett and G. Brassard, in Proceedings of the IEEEInternational Conference on Computers, Systems, and Sig-nal Processing, Bangalore, India (IEEE, New York, 1984),pp. 175–179.

[3] A. K. Ekert, B. Huttner, G. M. Palma, and A. Peres, Phys.Rev. A 50, 1047 (1994); C. A. Fuchs and A. Peres, ibid.53, 2038 (1996); E. Biham and T. Mor, Phys. Rev. Lett.78, 2256 (1997); N. Lütkenhaus, Phys. Rev. A 59, 3301(1999); H. Bechmann-Pasquinucci and N. Gisin, ibid. 59,4238 (1999).

[4] S. Massar and S. Popescu, Phys. Rev. Lett. 74, 1259 (1995);R. Derka, V. Bužek, and A. K. Ekert, ibid. 80, 1571 (1998);G. Vidal, J. I. Latorre, P. Pascual, and R. Tarrach, Phys. Rev.A 60, 126 (1999).

[5] D. Bruss and C. Macchiavello, Phys. Lett. A 253, 149(1999); A. Acín, J. I. Latorre, and P. Pascual, Phys. Rev.A 61, 022113 (2000).

[6] We restrict our attention to the class of ideal operations,which preserve the purity of the state for each conditionaltransformation. This introduces no loss of generality, as forthe purpose of further considerations all nonideal opera-tions can be represented as ideal ones with an additionalconstraint on the assignment of the guesses to the classicaloutcomes.

[7] K. Banaszek, Phys. Rev. A 62, 024301 (2000).[8] K. E. Atkinson, An Introduction to Numerical Analysis

(Wiley, New York, 1978), Sec. 7.2.[9] G. Vidal, D. Jonathan, and M. A. Nielsen, Phys. Rev. A 62,

012304 (2000).[10] M. A Nielsen and C. M. Caves, Phys. Rev. A 55, 2547

(1997).

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