Bell measurements and observables

17 July 2000

Physics Letters A 272 Ž2000. 32–38 www.elsevier.nlrlocaterpla

Bell measurements and observables G.M. D’Ariano, P. Lo Presti ) , M.F. Sacchi Theoretical Quantum Optics Group, UniÕersita` degli Studi di PaÕia and INFM Unita` di PaÕia, Õia A. Bassi 6, I-27100 PaÕia, Italy Received 27 April 2000; received in revised form 30 May 2000; accepted 12 June 2000 Communicated by P.R. Holland

Abstract A general matrix approach to study entangled states is presented, based on operator completeness relations. Bases of unitary operators are considered, with focus on irreducible representations of groups. Bell measurements for teleportation are considered, and robustness of teleportation to various kinds of non idealities is shown. q 2000 Elsevier Science B.V. All rights reserved.

1. Introduction Quantum mechanics builds up systems from subsystems in a fascinating way, through the tensor product, that allows one to set up the so called entangled states. These are states of the whole system that do not correspond to any state of the subsystems taken separately. This peculiar aspect of quantum world stands at the foundations of all the recent developments of quantum information theory, such as dense coding, teleportation, quantum computation, quantum cryptography, and so on w1x. These theoretical results have recently entered the realm of experimental physics w2–6x. Analogously to what happens for states, also quantum measurements on composite systems can be entangled when they are non local, namely they cannot be considered as a measurement jointly performed on the subsystems. In the general framework of positive operator valued measures ŽPOVM., en)

Corresponding author. E-mail address: [email protected] ŽP. Lo Presti..

tangled measurements correspond to non factorizable POVM’s. The so called ‘Bell measurements’ are the most relevant example w7x, corresponding to maximally entangled POVM’s. Entanglement and Bell measurements are the basic ingredients of quantum teleportation. In this Letter, we present a matrix approach to address bipartite-system pure states along with general operator-completeness relations. These allow us to write the most general Bell-like POVM in compact form. Bases of unitary operators are considered, with focus on irreducible representation of groups. The canonical role of the groups Z N = Z N and Weyl–Heisenberg is analyzed. We conclude with a study of robustness of teleportation to different kinds of non idealities.

2. Operator ‘basis’ Consider a set of linear operators  B Ž l., l g S , S Borel space4 on a finite dimensional Hilbert space H . This set is a ‘spanning set’ for the operator

0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 4 1 0 - 2

G.M. D’Ariano et al.r Physics Letters A 272 (2000) 32–38

space if it satisfies one of the following equivalent statements: 1. Completeness relation: Tr B † Ž l . B Ž lX . s D Ž l , lX . , X

X

HSd l D Ž l , l . B Ž l. s B Ž l . ,

where

and Tr B † Ž l . A s 0 ;l

m

A s 0 . Ž 1.

2. For any linear operator A on H ,

HSd l Tr

B † Ž l. A B Ž l. s A .

†

HSd l ² n < B Ž l. < m:² l < B Ž l. < k : s d

n k dm l

.

Ž 3.

4. For any linear operator A on H , †

HSd l B Ž l. A B Ž l. s Tr w A x | .

Ž 4.

3. General representation of bipartite-system pure states Chosen two orthonormal bases < i : 1 4 and < j : 2 4 for the Hilbert spaces H1 and H2 respectively, any vector < c :: g H1 m H2 can be written as < c :: s Ý c i j < i : 1 < j : 2 ( < C :: .

Ž 6.

ij

´

Proof of 1 2. To prove Ž . we define O s H Trw B † Ž l. A x B Ž l. d l y A. Then we evaluate the following trace Tr B † Ž lX . O s Tr B † Ž lX . B Ž l . Tr B † Ž l . A d l

H

Eq. Ž6. introduces a notation that exploits the correspondence between vectors in H1 m H2 and N = M matrices, where N and M are the dimensions of H1 and H2 , respectively Žcf. Ref. w8x.. The following relations are an immediate consequence of Eq. Ž6. A m B < C :: s < ACB T :: ,

y Tr B † Ž lX . A

²² A < B :: s Tr w A†B x ,

Tr 2 w < A::12 12 ²² B < x s Ž AB † .

s0 ,

Ž 5.

where integration has been carried out by means of the first line of Eq. Ž1.; the second line of Eq. Ž1. completes the proof. Converse implication: the first line of Eq. Ž1. follows immediately by replacing A with B Ž lX . in Eq. Ž2., whereas the second part is a direct consequence of Eq. Ž2.. Proof of 2 3. The proof of Ž . is immediate by substituting A with < m:² n < in Eq. Ž2. and taking the matrix element between ² l < and < k :. The converse is also straightforward: multiply both members of Eq. Ž3. by ² m < A < n:< l :² k < and take the sum over all indices k, l, m, n.

m

m

Proof of 3 4. The direct implication is derived multiplying both members of Eq. Ž3 . by ² m < A < l :< n:² k <, and summing the result over all the indices k, l, m, n. To prove the converse, let A s < m:² l < in Eq. Ž4. and take the matrix element between ² n < and < k :. Note that Eq. Ž2. is exactly the linear decomposition of the operator A on a set of operators  B Ž l.4 induced by the scalar product Ž B, A. s Trw B †A x. For infinite dimensional Hilbert spaces the previous relations have meaning for Hilbert–Schmidt operators. However, they still hold for all linear operators in a distribution sense.

Ž 2.

3. Chosen any orthonormal basis < i :4 for H ,

m

33

´

Ž1 .

Tr1 w < A::12 12 ²² B < x s Ž AT B ) .

,

Ž2 .

.

Ž 7.

Notice that the definition of the matrix C in Eq. Ž6. is base-dependent, hence the transposition and conjugation in Eqs. Ž7. are referred to the same fixed basis. These relations are very useful for derivations and to express the results in an index-free compact form. In the following we will focus our attention on bipartite systems whose Hilbert space is H m H , with N s dimŽ H .. As an application of the formalism just introduced, we give a direct proof of the existence of the Schmidt decomposition for a pure

G.M. D’Ariano et al.r Physics Letters A 272 (2000) 32–38

34

state of a bipartite system. Using a polar decomposition A s V'A†A , with V unitary, which holds for any matrix A w9x, we choose a unitary operator U so that UA†AU † is diagonal, then we can write < A:: s < V'A†A ::sVU † m U T < U'A†A U † :: sÝ

X XX l i < i :1 < i : 2

(

,

X XX where < i : 1 s VU † < i : 1 , < i : 2 s U T < i : 2 and l i is the † eigenvalue of A A with eigenvector < i :. Using Eq. Ž6., it is straightforward to characterize maximally entangled states. These are defined as the states < A:: whose partial trace on each of the two subsystems is proportional to identity; namely

and

|

N

Tr 2 w < A::²² A < x s AA† s

1 N

|,

Ž 9.

hence maximally entangled states are of the form < A:: s

1

'N

< U :: ,

Ž 10 .

with U unitary. Two maximally entangled states are always connected by means of a local unitary transformation. In fact < U :: s UV † m |< V :: .

Ž 11 .

Given a spanning set  B Ž l.4 , the set of vectors < B Ž l.::4 spans H m H in the sense that < A:: s d l Tr B † Ž l . A < B Ž l . :: .

H

Ž 12 .

Moreover one has

Ž l . s <|::²²|< ,

Ž 14 .

T

Ž 15 .

which are directly equivalent to Eq. Ž3. of statement 3.

4. Bell measurements A Bell measurement is a POVM whose elements are projectors on maximally entangled states. Referring to Eqs. Ž10. and Ž13. we argue that any POVM of this kind corresponds to a spanning set whose elements are proportional to unitary operators

P Ž d l . s < U˜ Ž l . ::²²U˜ Ž l . < d l ,

Ž 16 . ˜ ˜ where UŽ l. is a basis with UŽ l. s a Ž l.UŽ l., UŽ l. unitary, and a Ž l. c-number. As proved in Ref. w10x, Bell measurements are the only projector valued POVM’s capable of teleportation in the case of pure preparation of the shared resource, which turns out to be necessarily in a maximally entangled state. In the following we give a brief description of this kind of teleportation scheme. The Hilbert space H1 is prepared in an unknown state r Ž1., whereas H2 m H3 is in the maximally entangled state 1 <|:: 23

'N

Ž H1,2,3 have the same dimension.. Upon performing the measurement described by the POVM Ž16. on H1 m H2 , the Žunnormalized. state on H3 conditioned by the outcome l will be

D˜lŽ3. s Tr12 r Ž 1 . m <|:: 23 23 ²²|< < U˜l :: 12 12

Hd l < B Ž l. ::²² B Ž l. <

=²²U˜l < m |3

s d l B Ž l . m | <|::²²|< B † Ž l . m |

H

s Tr12 r Ž 1 . m <|:: 23 23 ²²|< U˜lŽ1. m |Ž23. <|::12 12

s Tr1 w <|::²²|< x s| ,

)

†

Ž 8.

1

Hd l B Ž l . m B

Hd l B Ž l. m B Ž l. < A:: s < A :: ,

i

Tr1 w < A::²² A < x s ATA) s

By explicit evaluation of the matrix elements, one can easily verify the following useful formulas

=²²|< m |Ž3. U˜l†Ž1. m |Ž23.

Ž 13 .

hence the projectors on < B Ž l.:: provide a resolution of the identity and a POVM.

s 12 ²²|< <|:: 23 U˜l†Ž1.r Ž1.U˜lŽ1. s U˜l†Ž3.r Ž3.U˜lŽ3. .

23

²²|< <|:: 12

Ž 17 .

G.M. D’Ariano et al.r Physics Letters A 272 (2000) 32–38

The normalized state writes

Dl s U † Ž l . r U Ž l . ,

Ž 18 .

and the teleportation can be completed upon applying the unitary transformation UŽ l. on the state Ž18.. If the shared entangled resource is prepared in another maximally entangled state, i.e. 1 < V :: 23 with

'N V unitary, it is enough to substitute UŽ l. with U Ž l. V ) . Notice that the product 12 ²²|< <|:: 23 that appears in Eq. Ž17. corresponds to the transfer operator t 31 of Ref. w11x, which for any vector < c : 1 of H1 satisfies the relation t 31 < c : 1 s < c : 3 .

Ž 19 .

If the set U˜Ž l.4 is an orthonormal operator basis ŽDirac-like orthonormality relations are allowed in the case of infinite dimensional spaces., it is possible to write the class of Bell observables, i.e. the self-adjoint operators that one has to measure in order to realize the Bell measurement. The Bell observables can be written as follows

35

describes a Bell measurement. For example, as noticed in Ref. w11x, the N-dimensional UIR of the group Z N = Z N whose elements are U Ž m,n . s Ý e 2 p i k m r N < k :² k [ n < ,

Ž 23 .

k

generates the Bell measurement corresponding to the teleportation scheme of Ref. w7x. As an example for the infinite dimensional case, consider the displacement operators of an electromagnetic field mode a Žw a,a† x s 1. D Ž z . s exp Ž za† y z ) a . ,

zgC .

Ž 24 .

Such operators are the elements of a projective UIR representation of the Weyl–Heisenberg group WH, and generate the Bell measurement corresponding to the Braunstein–Kimble teleportation scheme of Ref. w12x. For Z N = Z N the class of Bell observables defined by Eq. Ž20. is given by O s Ý f Ž g . < Ug ::²²Ug < g

O s f Ž l . < U˜ Ž l . ::²²U˜ Ž l . < d l ,

H

Ž 20 .

where f Ž l. must be an injective function Ži.e. O is non degenerate. in order to guarantee a univocal correspondence between the read eigenvalue f Ž l. and the unitary operator UŽ l. of Eq. Ž18. that completes the teleportation scheme.

s Ý f Ž g . Ug m |

Ý Ug m Ug) Ug† m |

g

X

g

2p i

s

Ý Ý X

f Ž m,n . e

X

X

N

Ž n mXym nX .

X

m, n m , n

=U Ž mX ,nX . m U ) Ž mX ,nX . 4.1. The role of group representations

s Ý f˜Ž g . UgŽ1. m UgŽ2.) ,

Unitary irreducible representations ŽUIR. of groups provide a method to generate a spanning set of unitary operators in the sense of statements Ž1–4.. In fact, if Ug , g g G4 are the elements of a projective UIR of the group G, from the first Schur’s lemma it follows that

HG

dg Ug A Ug† s Tr

w A x| ,

Ž 21 .

where dg is a Žsuitably normalized. group invariant measure on G . Recalling Eq. Ž16., it follows that the POVM

P Ž dg . s < Ug ::²²Ug < dg

Ž 22 .

Ž 25 .

g

where we used Eq. Ž14. along with the relation U Ž m,n . U Ž mX ,nX . U † Ž m,n . 2p i

se

N

Ž n mXym nX .

U Ž mX ,nX . ,

Ž 26 .

and we introduced the Fourier transform f˜ over the group 2p i

f˜Ž m,n . s

Ýe X

X

m ,n

N

Ž n mXym nX .

f Ž mX ,nX . .

Ž 27 .

G.M. D’Ariano et al.r Physics Letters A 272 (2000) 32–38

36

By applying Eq. Ž14., the analogous relation for WH reads as follows: Os

HCd

2

z f Ž z . < D Ž z . ::²² D Ž z . <

s

HCd

2

z f Ž z . DŽ z . m | d 2a

=

HC

s

s

p

d 2a

†

DŽ a . m DŽ a ) . DŽ z . m |

d2 z

HC HC

f Ž z . ea z

p

)

ya ) z

HCd a f˜Ž a . D Ž a . m D Ž a 2

DŽ a . m DŽ a ) .

)

..

Ž 28 .

However, in this case, one can derive a more explicit expression for the Bell observables. In fact, from Eq. Ž14. one has d 2b

<|::12 12 ²²|s <

HC

d 2b s

HC

p d 2b

s

HC

p

exp

p

D1Ž b . m D 2 Ž b ) .

Ž b a†1 y b ) a1 . q Ž b ) a†2 y b a2 .

† exp b Z12 y b ) Z12 ( pd Ž2. Ž Z12 . ,

5. Robustness of ‘pure’ teleportation ‘Pure’ teleportation schemes rely on projector valued POVM’s and pure preparations for the shared resource. As proved in Ref. w10x, this kind of teleportation works properly if and only if the elements of the POVM are proportional to projectors on maximally entangled states and the resource itself is maximally entangled. However, for practical purposes, one is interested in the evaluation of the robustness of this kind of schemes to non ideality. Looking at Eq. Ž17., it is evident that the state on H3 conditioned by the measurement is a continuous function of the shared resource preparation and of the element of the POVM related to the outcome. Since the teleported state is again a continuous function of this conditioned state and of the ‘adjusting’ unitary transformation, we conclude that teleportation is robust to non ideal entanglement preparation, non ideal measurement, and non ideal adjusting transformation. Let’s suppose that before the measurement the maximally entangled resource evolves according to a trace preserving CP map E owing to some kind of noise. In the following, we will simply evaluate the state on H3 after the measurement in presence of such a noise. By means of the Kraus’s decomposition w15x of E , the noisy state of H2 m H3 can be written as

Ž 29 . with Z12 s a1 y a†2 . Using the relation DaŽ z . a Da† Ž z . s a y z, one obtains 1

p

< D Ž z . ::12 12 ²² D Ž z . < s d Ž2. Ž Z12 y z . ,

Ž 30 .

r 23 s E

ž

1 N

<|:: 23 23 ²²|<

s Ý AmŽ23. m

1 N

/

<|:: 23 23 ²²|< AmŽ23.† ,

Ž 32 .

and finally Os

HC

d 2 zf Ž z .

1

p

< D Ž z . ::12 12 ²² D Ž z . < s f Ž Z12 . .

Ž 31 . Hence, in order to realize the Bell measurement generated by WH, we have to measure an injective function of the operator Z12 , or simply Z12 itself. This measurement can be easily performed by unconventional heterodyne detection Žcf. Ref. w13,14x..

where Am are operators on H2 m H3 satisfying Ým A†m Am s |. For any Žgenerally non local. operator A acting on H m H one has A <|:: s < AˆT :: s | m Aˆ<|:: , where ² i < Aˆ< j : s Ý l ² j <² i < A < l :< l :.

Ž 33 .

G.M. D’Ariano et al.r Physics Letters A 272 (2000) 32–38

Therefore it is possible to write E

ž

1 N

<|:: 23 23 ²²|<

s Ý AmŽ23. m

1 N

ing that Fmin is independent of the normalization of S, we can choose S to be

/

Se s Ž 1 q e . <0:²0 < q Ž 1 y e . <1:²1 < .

< c Ž x . : s cos x <0: q sin x <1: ,

s Ý |2 m AˆmŽ3. m

1 N 1

ž

N

<|:: 23 23 ²²|< |2 m AˆmŽ3.†

/

<|:: 23 23 ²²|< ,

˜

ˆ

Ž U Ž l.

†

r U Ž l. . .

Ž 34 .

Ž 35 .

Now, we will restrict our attention to qubit teleportation with non ideal resource preparation < S :: 23 , it is possible to give an explicit expression for the minimum fidelity achieved by teleportation on pure states. If < c : is the original state, apart from a normalization factor, the teleported state will be < cl : s Ul S T Ul† < c : ,

Ž 36 .

where Ul is the unitary operator related to the outcome l. The minimum fidelity achieved by teleportation can be written as follows Fmin s min

< c :g H

s min < c :g H

x g w 0,2 p . ,

Ž 39 .

because any phase would be irrelevant. Substituting Eqs. Ž38. and Ž39. in Eq. Ž37. and minimizing respect to x, one obtains

where Eˆ is the map whose Kraus’s decomposition is given by the operators Aˆm . This last equation shows how the action of any CP map on a maximally entangled state of a bipartite system can be written as the result of the application of a local CP map. Recalling Eq. Ž17., it results that the local CP map Eˆ Ž3., which describes noise, commutes with all other maps and with the partial trace, so that the unnormalized conditioned state after the measurement, in presence of such a noise, can be simply written as

DlŽ3. s E Ž3.

Ž 38 .

The minimization can performed only on states < c Ž x .: of the form

<|:: 23 23 ²²|< AmŽ23.†

s I Ž2. m Eˆ Ž3.

37

<² c < Ul S T Ul† < c :< 2

Fmin s 1 y e 2 .

Ž 40 .

With some little algebra, Eq. Ž40. can be cast in a compact form independent of basis and normalization as follows Fmin s 4

det Ž S˜. Tr 2 w S˜x

,

where

S˜s 'S †S .

Ž 41 .

6. Conclusions We studied the problem of characterizing Bell measurements, as maximally entangled POVM’s for measurements on composite systems. We introduced operator-completeness relations and a simple matrix approach to deal with bipartite systems. These allow us to write the most general Bell-like POVM in a compact form. The role of spanning sets of unitary operators has been emphasized, with attention to unitary irreducible representations of groups. Bell observables related to Bell POVM’s have been explicitly derived. As direct application of the matrix formalism, we evaluated the robustness of teleportation to non maximality of the shared entangled resource and to non ideality of the measurement.

Acknowledgements

² c < Ul S ) Ul†Ul S T Ul† < c : <² c < S T < c :< 2 ²c < S )ST < c :

.

Ž 37 .

Using a basis of H2 m H3 for which < S :: 23 is in the Schmidt-form, i.e. diagonal and positive, and notic-

This work has been supported by European Union under the project EQUIP Žcontract IST-1999-11053. and by Ministero dell’Universita` e della Ricerca Scientifica e Tecnologica under the project Quantum Information Transmission and Processing: Quantum Teleportation and Error Correction.

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G.M. D’Ariano et al.r Physics Letters A 272 (2000) 32–38

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