Entangling characterization of (SWAP) 1/m and Controlled unitary gates

Transcription

1 Entangling characterization of (SWAP) / and Controlled unitary gates S.Balakrishnan and R.Sankaranarayanan Departent of Physics, National Institute of Technology, Tiruchirappalli 65, India. We study the entangling power and perfect entangler nature of SWAP /, for, and controlled unitary (CU ) gates. It is shown that / SWAP is the only perfect entangler in the SWAP / faily. On the other hand, a subset of CU which is locally equivalent to CNOT is identified. It is shown that the subset, which is a perfect entangler, ust necessarily possess the axiu entangling power. PACS nubers: 3.67.Lx, 3.65.Ud, 3.67.Mn I. INTRODUCTION Entangleent [], a fascinating quantu echanical feature, has been recognized as a valuable resource for quantu inforation and coputation []. Much effort has been ade to know the production, quantification and anipulation of entangled state [3]. Required inforation processing can be achieved by the application of appropriate quantu operators (gates) on qubits prepared in a definite state. As two qubit gates have the ability to create entangleent, any research work focus on characterizing the entangling properties of the. The entangling capabilities of a quantu gate are quantified by the entangling power [4] which describes the average entangleent produced by the gate when it is acting on a given distribution of product states. In this description, linear entropy is used to easure the entangleent of a state. It is well known that the entangleent is a non-local property which is unaffected by local operations. Makhlin introduced local invariants to describe the non-local properties of quantu gates [5]. Two gates are said to be locally equivalent, possessing sae local invariants, if they differ only by local operations. Hence, local invariants are

2 convenient easures to identify the local equivalence class of quantu operators. In [6], Zhang et al. showed that the geoetric structure of non-local two qubit operations is a 3- Torus. To be precise, every non-local gate is associated with the coordinates of 3-Torus. In ters of the coordinates, it is easy to check whether a gate has the ability to produce axial entangleent when it acts on soe separable states. If the gate produces axial entangleent then it is known as a perfect entangler [5, 6]. As the axially entangled states are known to play a central role in the quantu inforation processing, it is of fundaental iportance to identify the perfect entanglers aong the non-local gates. It is known that one CNOT gate can be constructed using two / SWAP gates [7]. Since α SWAP faily of gates are recognized as the building blocks of universal two qubit gate [8], a detailed understanding of this faily is of fundaental iportance. In particular we focus on the entangling characterization, coplienting with geoetrical representation, of the above faily with α = / for. Further, we investigate entangling character of another iportant class of two-qubit gate naely, controlled unitary (CU ) gates. Using geoetrical representation, it is shown that / SWAP is the only perfect entangler in the SWAP / faily. On the other hand, we present a siplified expression for the entangling power and hence obtain conditions for iniu and axiu entangling power of an arbitrary two qubit gate. The siplification led to identify a subset of CU which is locally equivalent to CNOT. It is shown that the subset, which is a perfect entangler, ust necessarily possess the axiu entangling power as well. In the end, soe possible probles eerged fro this work are pointed out. II. PRELIMINARIES (A) Entangling power The entangling capability of a unitary quantu gate U can be quantified by entangling power (EP) which was introduced by Zanardi et al. [4]. For a unitary operator ( 4) U U the entangling power is defined as

3 EP U ) = average [ E( U ψ ψ )] () ( ψ ψ where the average is over all product states distributed uniforly in the state space. In the above forula E is the linear entropy of entangleent easure defined as E( ψ AB where ρ ( ψ ψ ) A B) tr B( A) ) = tr( ρ ) () AB A(B) ( = is the reduced density atrix of syste A(B). We ay note that EP ( U ) [8, 9]. It is to be noted that the linear entropy is related to the 9 well known easure of entangleent naely concurrence [, ] through the following expression E ( ψ ) = C ( ψ ) where the concurrence of a two qubit state ψ = α + β + γ + δ is defined as AB C ( ψ ) = αδ βγ. (3) While C = for product state, it takes the axiu value of for axially entangled state. (B) Perfect Entangler Two unitary transforations U, U SU (4) are called locally equivalent if they differ only by local operations: U = k U k where k, k SU () SU () [5]. The local equivalent class of U can be associated with local invariants which are calculated as follows. Any two qubit gates U SU (4) can be written in the following for [6,, 3] i U = k exp{ ( cσ xσ x + cσ yσ y + c3σ zσ z )} k. (4) Representing U in the Bell basis: as Φ Ψ U B = Q U Q with + + = = ( + ), Φ = ( ) i ( ), Ψ = ( ) i, 3

4 Q = i i i, i the local invariants of the given two qubit gate can be calculated using the forula [6] where [, c, c3 ( M ( U )) ( U ) tr G = 6det (5a) G ( ) ( M ( U )) tr M ( U ) 4det( U ) tr = (5b) M ( U ) =. The relation connecting the local invariants G,G and a point T U B U B c ] on 3-Torus geoetric structure of non-local two qubit gates are [6] i G = cos c cos c cos c3 sin c sin c sin c3 + sin c sin c sin c3 (6a) 4 G = cos c cos c cos c3 4sin c sin c sin c3 cos c cos c cos 4 c 3. (6b) Therefore fro the values of G and G it is possible to find the point on the 3-Torus which corresponding to a local equivalence class of two qubit gates. Eploying Weyl group theory to reove the syetry on the 3-Torus, Zhang et al. [6] have obtained tetrahedron representation (Weyl chaber) of non-local two qubit gates (Fig.). FIG.: Tetrahedron OA A A 3, the geoetrical representation of non-local two qubit gates, is referred as Weyl chaber. Polyhedron LMNPQA (shown in dotted lines) corresponds to the perfect entanglers. Thick line OA 3, one edge of the Weyl chaber, corresponds to SWAP / gates. The points L = [ π /,,], A = [ π /, π /, / ] and = [ π / 4, π / 4, π / 4] 3 π P correspond to CNOT, SWAP and SWAP / respectively. The CNOT class of CU lies at the point L and they are all perfect entanglers. 4

5 A two qubit gate is called perfect entangler if it can produce a axially entangled state for soe initially separable input state. The theore for perfect entangler is the following: two qubit gate U is a perfect entangler if and only if the convex hull of the eigenvalues of M (U ) contains zero [5, 6]. Alternatively, if the coordinates satisfy the following condition π π 3π π c i + ck ci + c j + π or c i + ck ci + c j + π (7) where ( i, j, k) is a perutation of (,,3) then the corresponding two qubit gate is a perfect entangler. Thus the perfect entangler nature of a given two qubit gate U can be ascertained fro the corresponding geoetric representation. III. SWAP / FAMILY OF GATES It is well known that SWAP gate siply interchanges the input states i.e., SWAP ψ φ = φ ψ. Defining + exp( iπα) exp( iπα) α SWAP =, (8) exp( iπα) + exp( iπα) it is shown that three such gates with different values of α are the building blocks for the construction of an arbitrary two qubit operations [8]. In such a schee, α SWAP can be realized by Heisenberg exchange interaction where α is controlled by adjusting the strength and duration of the interaction. In this section, it is aied to introduce a faily of gates with α = / for and explore their entangling character. (A) Entangling power We use the following expression to calculate the entangling power of a two qubit gate U [4, 8, 9]: { U, T U T + ( SWAP. U ), T ( SWAP. U ) } 5 EP( U ) =,3,3,3 T,3 (9)

6 where A, B = tr( A B ), referred as Hilbert Schidt scalar product and T, 3 is the transposition operator defined as T,3 a, b, c, d = c, b, a, d on four qubit syste. The entangling power of α SWAP is given by [8] α ( SWAP ) = cos( πα ) EP. () For α = /, EP / π ( SWAP ) = cos EP SWAP It is worth entioning that ( / ) / 6 =. (), which is the axiu value in the above faily. By equating the above entangling power to that of CNOT, it is then possible to estiate the nuber of SWAP / gates (n) required to siulate CNOT for a given value of. Since EP ( CNOT ) = / 9, the axiu value, we have the following inequality / { EP( SWAP )} n. () 9 Alternatively, the nuber of gates n for a given is such that π 8 n cos. (3) 3 Following table shows soe integer values of and the corresponding n that satisfies the above inequality No. of gates n TABLE : Nuber of SWAP / gates (n) required for the construction of CNOT is shown for few integer values of. The nuber n is shown to increase with. Loss et al. [7] have shown that CNOT can be constructed using two / SWAP gates along with single qubit gates, which is understandable fro the point of entangling power. Moreover, since CNOT and / SWAP possess different local invariants, at least 6

7 two / SWAP gates are needed to siulate CNOT. On the other hand, the above table is tepting to conjecture that CNOT can also be constructed using two / 3 SWAP gates. /3 SWAP are G i Note that the local invariants of = and G =. 5, which are different fro that of CNOT. Hence, the later conjecture is well supported by the Makhlin s concept of local equivalence [5]. We conclude this part by pointing that as increases, the entangling power reaches a axiu of / 6 at = and decreases for further increase of. Expressing (B) Perfect Entangler SWAP / in the Bell basis as S = Q SWAP / Q, one can B / find M ( SWAP ) = S T B S B. Using Eqs. (5a) and (5b) the local invariants are obtained as iπ i3π iπ G = 9exp + exp + 6exp 6 (4a) π G = 3cos. (4b) Subsequently the geoetrical points corresponding to SWAP / can be evaluated fro Eqs. (6a) and (6b) as π π π [ c, c, c3 ] =,,, (5) which lie along the line OA 3 in the Weyl chaber (see figure.). For these points, the inequality (7) can be rewritten as or It is easy to convince that the first inequality is satisfied only for = and the second inequality is not satisfied for any values of. Hence it is inferred that / SWAP is the only perfect entangler and the corresponding geoetrical point is = [ π / 4, π / 4, π / 4] P. This is also evident fro the fig. that the only point P of the line OA 3 belongs to the polyhedron of the perfect entanglers. In appendix A, this result is well justified with an explicit calculation of concurrence. 7

8 IV. CONTROLLED UNITARY GATES In this section we dwell upon the entangling character of another iportant class two qubit gate naely, the controlled unitary (CU ) operation: where α, β, θ and δ are real. CU = i( δ + α / + β / ) i( δ + α / β / ) e cos( θ / ) e sin( θ / ) (6) i( δ α / + β / ) i( δ α / β / ) e sin( θ / ) e cos( θ / ) (A) Entangling power Before calculating the entangling power of CU gates, we ake soe useful siplification in the expression (9). In what follows, we use the definitions: A =CU, S = SWAP, B = ( SWAP. CU ) and T = T, 3 tensor products [4]: ( A A ) ( B B ) = ( A B )( A ) B. Exploiting the property of, we can write B = S A. With + + this, we have B, TBT = tr( A S TSAT ) and the entangling power can be rewritten as 5 EP ( CU ) = [ tr( A TAT ) + tr( A S TSAT )] = [tr ( A TAT + A S TSAT )] 9 36 In the last step we use the fact that tr ( A) + tr( B) = tr( A + B). It is convenient to rewrite the above expression as 5 EP ( CU ) = [ tr( A RAT )] (7) 9 36 where R = T + S TS. Fro this expression we arrive the conditions for iniu and axiu entangling power of CU gates. The entangling power is iniu i.e., EP ( CU ) =, if and only if tr( A RAT ) =. Since tr ( RT ) =, EP ( CU ) = if RA = AR. It is easy to check that the later coutation relation is valid if A coutes with S andt. That is, if A coutes with S and T, the corresponding entangling power is zero. On the other hand, the entangling power is axiu i.e., EP ( CU ) = / 9, 8

9 if and only if tr( A RAT ) =. In ters of the paraeters α, β, θ and δ the conditions for the iniu and axiu entangling power for CU gate can be expressed as follows. Since tr( A RAT) 4cos θ = [ + cos( α + β )] +, (8) EP ( CU ) = if cos ( θ / )[ + cos( α + β )] =. That is, if the paraeters are such that θ = ( π ) and α + β = ( π ) the entangling power is zero. Siilarly, for the axiu value of entangling power the angles ust satisfy the expression cos θ [ + cos( α + β )] =. (9) The above expression gives two distinct cases naely, (i) θ = π for any values of α and β (ii) θ π andα + β = π, for which CU possesses the entangling power / 9. (B) Perfect Entangler In order to calculate the local invariants, it is convenient to transfor CU in the Bell basis as CU B = Q T CU Q and hence we calculate M ( CU ) = ( CU B ) ( CU B ). Then using Eqs. (5a) and (5b) the invariants are found to be θ α + β G = cos cos (a) θ α + β G = cos cos +. (b) It ay be noted that the local invariants of CNOT are G = and G =, which correspond to the geoetrical point = [ π /,,] L. Since this point satisfies Eq. (7), CNOT is a perfect entangler [6]. By equating the above expressions to the local invariants of CNOT, we have cos θ [ + cos( α + β )] =. () That is, CU gates satisfying the above condition are locally equivalent to CNOT, and they correspond to the sae point L. In other words, the CU satisfying Eq. () are locally equivalent class of CNOT and hence they are also perfect entanglers, as shown in appendix B. 9

10 It is interesting to note that Eqs. (9) and () are identical, iplying that CU gates which are locally equivalent to CNOT ust necessarily possess the axiu entangling power. V. DISCUSSION In this paper we have studied the entangling character of two qubit gates naely, SWAP / and controlled unitary, using entangling power and perfect entanglers as tools. The first part of the investigation shows that SWAP / faily lies along one edge (OA 3 ) of the geoetrical representation of non-local two qubit gates. It is also observed that, / SWAP possesses the axial entangling power as well as the only perfect entangler in the faily. Further, fro the entangling power point of view, it is conjectured that CNOT can also be constructed using two / 3 SWAP gates. The possibility of such a construction is left for future investigation. In later part of the paper, we have addressed the entangling properties of controlled unitary (CU ) which is an iportant class of two qubit gates. In particular, without loss generality, a siplified expression for the entangling power of two qubit gate is presented. This siplification facilitates to obtain condition on CU to possess the iniu and axiu entangling power. Further, a subset of CU which is locally equivalent to CNOT is explicitly identified. We refer the subset as a CNOT class. Interestingly, the CNOT class is shown to possess the entangling power of CNOT, which is the axiu value. This result provokes to conjecture that, locally equivalent gates will have the sae entangling power, which warrants a detailed study. We also note that, since CNOT is a perfect entangler, all its locally equivalent gates are also perfect entanglers. It is well known that an arbitrary CU gate can be constructed using two CNOT and single qubit gates [5]. Since the subset CU and CNOT are locally equivalent, in principle it is possible to construct an eleent of the subset with single CNOT. Such a construction would be of fundaental iportance in the circuit coplexity.

11 Appendix A Here we present a siple technique to explicitly show that / SWAP is the only perfect entangler. Consider two single qubit states as ψ = a + b and / ψ =e + f and denoting η = SWAP ψ ψ, the concurrence can be calculated using Eq. (3) as C 4 π i ( η ) = exp ( af be) Fro the above expression, we observe that ( η) = choices of input states like (A) a =, b =, e =, f =,(B) C only for = with appropriate a = e =, b = f = ±. Appendix B As seen fro Eq. (9) CU is a perfect entangler for (i) θ = π for any values of α and β (ii) θ π andα + β = π. Following the earlier technique, here we show that the CU can generate axially entangled state for soe input states. (i) θ = π for any values of α and β : Adopting the notations used in appendix A, we denote η =CU ψ ψ and the concurrence is α β α β C ( η) = ab e exp i f expi. It is easy to verify that ( η) = C for the following choices of input states: (A) a =, b = ±, e =, f = (B) a =, b = ±, e =, f =. (ii) θ π, α + β = π. In this case the concurrence can be obtained as C θ θ ( η) i abef cos i absin ( e exp( iα) f exp( iα) ) =.

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