Structural Physical Approximation make possible to realize the optimal singlet fraction with two measurements
aa r X i v : . [ qu a n t - ph ] A p r Structural Physical Approximation make possible to realize the optimal singlet fraction with twomeasurements
Satyabrata Adhikari ∗ Delhi Technological University, Delhi-110042, Delhi, India
Structural physical approximation (SPA) has been exploited to approximate non-physical operation such aspartial transpose. It has already been studied in the context of detection of entanglement and found that ifthe minimum eigenvalue of SPA to partial transpose is less than then the two-qubit state is entangled. Wefind application of SPA to partial transpose in the estimation of optimal singlet fraction. We show that optimalsinglet fraction can be expressed in terms of minimum eigenvalue of SPA to partial transpose. We also show thatoptimal singlet fraction can be realized using Hong-Ou-Mandel interferometry with only two detectors. Furtherwe have shown that the generated hybrid entangled state between a qubit and a binary coherent state can be usedas a resource state in quantum teleportation. PACS numbers: 03.67.Hk, 03.67.-a
I. INTRODUCTION
Entanglement is a non-classical correlation [1] and is a nec-essary ingredient to build a quantum computer that can out-perform the classical computer. It has also been used as aquantum resource in various quantum communication taskssuch as teleportation [2], superdense coding [3], secret shar-ing [4] and quantum-key distribution (QKD) [5]. To performthe quantum communication task, we need to generate entan-glement but the issue in the generation of entanglement is thatit never free from imperfections and noise. Therefore, the gen-erated state may or may not be entangled. Thus it is importantto develop an efficient way to detect entanglement.Positive maps are stronger detectors of entanglement but theyare not physical and hence cannot be realized in the labora-tory. On the other hand, completely positive maps play animportant role in quantum information processing as they arecapable to describe an arbitrary quantum transmission channel[6]. Not only that but also it approximated non-physical op-erations such as quantum cloners or universal-NOT gate [7].A physical way by which positive maps can be approximatedby completely positive maps is called structural physical ap-proximation (SPA) [8]. SPA thus transform non-physical op-erations to physical operations.Here, we restrict our discussion only to the two-qubit sys-tem. Let P denote the positive map in 4-dimensional Hilbertspace and CP denote the completely positive map that trans-forms all quantum states ρ onto maximally mixed state I i.e. CP ( ρ ) = I [9, 10]. Since the positive map P is not physicallyrealizable map so we approximate it with CP map in such away that it would be a physical map. Therefore, SPA to themap P is given by e P = (1 − p ∗ ) P + p ∗ I (1)where p ∗ is the minimum value of p for which the approxi-mated map e P is a completely positive map [11]. ∗ Electronic address: [email protected]
Partial transposition (PT) is another strong entanglement de-tection criterion given by Peres [12]. Later, Horodecki [13]proved that the PT criterion is necessary and sufficient for × and × system. Although partial transposition cri-terion works well in qubit-qubit and qubit-qutrit system butit cannot be implemented in a laboratory for the detection ofentanglement as it is a non-physical operation. Therefore, tomake partial transposition map a physical operation we canapproximate it in such a way that it would be a completelypositive map. Let us consider that the partial transposition op-eration act on the second subsystem and is given by id ⊗ T ,where T denotes the positive transposition map and id rep-resent the identity operator. In general, partial transpositionoperation id ⊗ T can be approximated as [8] ^ id ⊗ T = (1 − q ∗ )( id ⊗ T ) + q ∗ I A ⊗ I B (2)where q ∗ = ν ν and ν = − min Q> T r [ Q ( id ⊗ T ) | ψ + ih ψ + | ] , | ψ + i = √ ( | i + | i ) . The map id ⊗ T is a non-physical map but its approximate map ^ id ⊗ T is acompletely positive map corresponds to a quantum channelthat can be experimentally implementable [8, 11].Let σ be a two qubit-state and the task is to determinewhether it is an entangled state or separable state. Since itis a two qubit system so we can apply partial transposition de-tection criterion. PT criterion states that if σ T B ( T B denotespartial transposition with respect to the second subsystem B)has a negative eigenvalue then the state σ is an entangledstate. But partial transposition is not a physical operation soapply SPA-PT operation (2) on σ and at the output, we have f σ . Since SPA-PT operation is completely positive so theoutput f σ also represents a state. Therefore, the practicalproblem of finding the eigenvalue of σ T B ( T B denotes partialtransposition with respect to the second subsystem B) reducesto determine the eigenvalue of f σ . Hence PT criterion mod-ified as SPA-PT criterion which states that if the minimumeigenvalue of f σ is less than then the state σ is entangledand vice-versa [8]. The minimum eigenvalue can be estimatedby the procedure given in [14, 15].Recently, H-T Lim et.al. [16] have demonstrated the experi-mental realization of SPA-PT for photonic two qubit photonicsystem using single-photon polarization qubits and linear op-tical devices. They provided the decomposition of SPA-PT fora two-qubit state σ as f σ = ^ I ⊗ T ( σ ) = [ 13 ( I ⊗ ˜ T ) + 23 ( ˜Θ ⊗ D )] σ (3)where ˜ T denote SPA for transpose operation, Θ denotes theinversion map and works as Θ( σ ) = − σ , ˜Θ denote its SPAand can be constructed by the prescription given in (1) and D ( σ ) = I denote the polarization. Since I ⊗ ˜ T and ˜Θ ⊗ D are local operations and are completely positive operators so ^ I ⊗ T is a physically realizable operators.The motivation of this work is two fold: Firstly, the method offinding the eigenvalues in [14, 15] require more than one copyof the given state and the method described in [14] for esti-mating the eigenvalues works well asymtotically. Therefore,to circumvent these problems we take the approach of witnessoperator to determine the minimum eigenvalue, which requirea single copy of SPA-PT of the given state. Also, we show thatthe minimum eigenvalue determined by our method require aset up that need only two measurements. Secondly, Verstraeteand Vershelde [17] have established a relationship between theoptimal singlet fraction and partial transpose of a given stateand using the derived relation they have shown that the two-qubit state is useful as a resource state for teleportation if andonly if the optimal singlet fraction is greater than . But thepartial tranposition is an non-physical operation and cannot beimplemented in a laboratory so it would be not easy to realizethe optimal singlet fraction. Also the filtering operation usedin [17] to achieve the optimal singlet fraction depends on thequantum state under investigation. Thus information about thestate under investigation is needed. To overcome these prob-lems, we apply SPA-PT method and show that optimal singletfraction does not depend on the state under investigation andalso can be realized in experiment.This paper is organized as follows: In section-II, we haveconstructed the witness operator to determine the minimumeigenvalue of SPA-PT of a given state ρ . In section-III, wehave shown that the number of measurements needed to de-termine the minimum eigenvalue is two. In section-IV, weshow that the teleportation fidelity can be determined exper-imentally using Hong-Ou-Mandel interferometry with onlytwo detectors. In section-V, we have studied the hybrid en-tangled state between a qubit and binary coherent state andhave shown that the mixed hybrid entangled state can be usedas a resource state for teleportation and lastly, we conclude insection-VI. II. WITNESS OPERATOR THAT DETERMINE THEMINIMUM EIGENVALUE OF SPA-PT OF TWO QUBITSTATE
Any arbitrary two qubit density operator in the computa-tional basis is given by ρ = t t t t t ∗ t t t t ∗ t ∗ t t t ∗ t ∗ t ∗ t , X i =1 t ii = 1 (4)where ( ∗ ) denotes the complex conjugate.SPA-PT of ρ is given by f ρ = [ 13 ( I ⊗ e T ) + 23 ( e Θ ⊗ e D )] ρ = E E E E E ∗ E E E E ∗ E ∗ E E E ∗ E ∗ E ∗ E (5)where E = 19 (2 + t ) , E = 19 ( − it + t ∗ ) ,E = 19 ( t − i ( t ∗ + t ∗ )) , E = 19 ( − it + t ) ,E = 19 (2 + t ) , E = 19 ( t + it ) ,E = − i t ∗ + t ∗ ) , E = 19 (2 + t ) ,E = 19 ( − it + t ∗ ) , E = 19 (2 + t ) (6)We note that the matrix f ρ is not only a Hermitian matrix butalso has non-negative eigenvalues. The trace of the matrix isequal to unity. So, it possesses all the properties of a state andthus the matrix can be regarded as a density matrix f ρ . Theminimum eigenvalue of f ρ detect whether the state ρ is en-tangled or not? Therefore, our task is to construct the witnessoperator that detect whether the minimum eigenvalue of f ρ is less than ?To start, we consider the operator O = f ρ − ρ T , T de-note the partial transpose with respect to the second subsys-tem. The expectation value of the operator O in the state | φ i = α | i + β | i ( α + β = 1) is given by h φ | O | φ i = T r [( f ρ − ρ T ) | φ ih φ | ] = 29 ⇒ T r [ | φ ih φ | f ρ ] − T r [ | φ ih φ | T ρ ] = 29 ⇒ T r [ W opt ρ ] = T r [ f ρ | φ ih φ | ] − (7)where W opt = | φ ih φ | T is the optimal witness operator thatdetect whether the state ρ is entangled or not.Let λ min be the minimum eigenvalue of f ρ and | φ i be theeigenvector corresponding to the minimum eigenvalue, thenthe eigenvalue equation is given by f ρ | φ i = λ min | φ i (8)Using (8) in (7), we have T r ( W opt ρ ) = λ min − (9)For all separable state ρ s , we have [8] T r ( W opt ρ s ) ≥ ⇒ λ min ≥ (10)If ρ is an entangled state and W opt detect that entangledstate then T r ( W opt ρ ) < ⇒ λ min < (11)The inequality (11) gives us the condition that when ρ is anentangled state.Since the above condition is a purely mathematical conditionso naturally one can ask a question that can we achieve thisinequality experimentally? To investigate this, let us againrecall (7) and write it in a different form as T r ( W opt ρ ) = T r [( | φ ih φ | − I ) f ρ ] (12)When W opt detect an entangled state ρ then T r ( W opt ρ ) < and hence we arrive at a conditiongiven by T r [( | φ ih φ | − I ) f ρ ] < (13)The above condition (13) is equivalent form of the condition(11) and hence the inequality implies that the eigenvalues of f ρ is less than .Let V ≡ | φ ih φ | − I . Then the inequality (13) can be re-expressed as T r ( V f ρ ) < (14)Next we investigate few properties of the operator V . P1.
The expectation value of V for all separable state f ρ sep isnon-negative i.e. T r ( V f ρ sep ) ≥ .Proof: The separable state f ρ sep is given by f ρ sep = E s E s E s E s ( E s ) ∗ E s E E ( E s ) ∗ ( E s ) ∗ E s E s ( E s ) ∗ ( E s ) ∗ ( E s ) ∗ E s (15)where E s = (2 + t s ) , E s = ( − it s + ( t s ) ∗ ) , E s = ( t s − i (( t s ) ∗ +( t s ) ∗ ) , E = ( − it s + t s ) , E s = (2+ t s ) , E s = ( t s + it s ) , E s = − i (( t s ) ∗ + ( t s ) ∗ ) , E s = (2+ t s ) , E s = ( − it s +( t s ) ∗ ) , E s = (2+ t s ) . Here, t sij denote the elements of a separable state ρ s .(12) can be re-expressed for any separable state ρ s as T r ( W opt f ρ sep ) = T r ( V f ρ sep ) (16) Since W opt is a witness operator so the expectation value of W opt over all separable state f ρ sep is non-negative. Thus T r ( W opt f ρ sep ) ≥ (17)Using (16) and (17), we have T r ( V f ρ sep ) ≥ (18) P2.
It can be easily shown that V has at least one negativeeigenvalues.Thus, the operator V possess all the properties of a witnessoperator and hence it detect whether the eigenvalue of f ρ isless than or not. If V detect that the eigenvalue of f ρ is lessthan then we can say that the state described by the densityoperator ρ is entangled.Since V is a hermitian operator so it is an observable and canbe implemented experimentally. Therefore the inequality (13)is useful to detect entangled state experimentally. III. NUMBER OF MEASUREMENTS NEEDED TODETERMINE THE VALUE OF
T r ( V f ρ ) The operator V can be expressed in terms of local Paulimatrices as V = 928 [ 79 I ⊗ I + ( α − β )( I ⊗ σ z + σ z ⊗ I )+ 2 αβ ( σ x ⊗ σ x + σ y ⊗ σ y ) + σ z ⊗ σ z ] (19)We find that the decomposition of the operator V in terms oflocal Pauli observables need more than two measurements torealize it. So in this section, our task is to show that it is pos-sible to realize the operator V with just two measurements.To achieve our goal, we approximate the entanglement wit-ness operator V in a way we approximate the positive but notcompletely positive operator. Therefore, approximate entan-glement witness operator e V can be expressed as e V = p V + (1 − p ) I, ≤ p ≤ (20)Choose the minimum value of p in such a way that the op-erator e V will become a positive semi-definite operator. TheHermitian operator e V is positive semi-definite if p min = .Therefore, e V can be re-expressed as e V = 815 V + 715 I, (21)We can observe that the operator e V is not a normalized opera-tor so it can be expressed after normalization as e V = 29 V + 736 I, (22)Since e V is positive semi-definite and T r ( e V ) = 1 so the oper-ator e V can be treated as a quantum state.Again, T r ( V f ρ ) can be written in terms of T r ( e V f ρ ) as T r ( V f ρ ) = 158 T r ( e V f ρ ) − (23)It can be shown that T r ( e V f ρ ) is equal to the average fidelityfor two mixed quantum states e V and f ρ [18], T r ( e V f ρ ) = F avg ( e V , f ρ ) (24)Using (23) and (24), we have T r ( V f ρ ) = 158 F avg ( e V , f ρ ) − (25)C. J. Kwong et.al. [18] have shown that the average fidelitybetween two mixed quantum states can be estimated experi-mentally by Hong-Ou-Mandel interferometry with only twodetectors. Thus the quantity T r ( V f ρ ) needs only two mea-surements to estimate it in experiment.Again, the minimum eigenvalue can be expressed in terms of F avg ( e V , f ρ ) as λ min = 158 F avg ( e V , f ρ ) − (26)Since the minimum eigenvalue of the quantum state f ρ canbe determined using two measurements and minimum eigen-value is responsible for the detection of entanglement so wecan say that the presence of entanglement in ρ can be de-tected using two measurements only. IV. REALIZATION OF OPTIMAL SINGLET FRACTIONWITH ONLY TWO MEASUREMENTS
The singlet fraction is defined as F ( ρ ) = max [ h φ + | ρ | φ + i h φ − | ρ | φ − i (27) h ψ + | ρ | ψ + i h ψ − | ρ | ψ − i ] (28)where {| φ + i , | φ − i , | ψ + i , | ψ − i} are the maximally entangledBell states.Verstraete and Vershelde [17] suggested the optimal trace pre-serving protocol for maximizing the singlet fraction of a givenstate. The optimal singlet fraction is given by F opt ( ρ ) = 12 − T r ( X opt ρ T B ) (29) X opt is given by X opt = ( A ⊗ I ) | ψ + ih ψ + | ( A † ⊗ I ) , (30)where I represent an identity matrix of order 2, | ψ + i = √ ( | i + | i ) and A = (cid:18) a
00 1 (cid:19) , − ≤ a ≤ .We note that if the state ρ is entangled then ρ T B has at leastone negative eigenvalue and hence F opt ( ρ ) is greater than . Thus every entangled two qubit state is useful for telepor-tation. But there are two problems in estimating the quantity F opt ( ρ ) given in (29): (i) ρ T B cannot be realized in the lab-oratory and (ii) the parameter in X opt is state dependent andhence to construct X opt , we need to know the state under in-vestigation. In this section, we obtain the optimal singlet fraction in termsof minimum eigenvalue λ min of SPA-PT of the state ρ us-ing local filtering operation but in this scenario the parameterof the local filtering operation does not depend on the state un-der investigation. Since we found in the previous section that λ min can be estimated for any arbitrary state with two mea-surements so we can say that optimal singlet fraction of anyarbitrary states can be calculated using two measurements. A. Optimal singlet fraction in terms of λ min To start, let us consider the operator X opt ( f ρ − ρ T ) ,where X opt is given in (30). Calculate the trace of it and aftersome simple algebra, (29) reduces to F opt ( ρ ) = 12 − [9 T r ( X opt f ρ ) − ( a + 1)] (31)It can be easily seen from (31) that F opt ( ρ ) > if and onlyif T r ( X opt f ρ ) − ( a + 1) < . This condition leads to T r ( X opt f ρ ) < a + 19 , − ≤ a ≤ (32)In {| i , | i} subspace, the operator a +1 X can be ex-pressed as a + 1 X opt = | χ ih χ | (33)where | χ i = √ a +1 ( a | i + | i ) .We note that the vector | χ i and the eigenvector | φ i of the op-erator f ρ are parallel vectors and thus there exist a real scalar k such that | χ i = k | φ i (34)Using (34), it can be easily shown that the vector | χ i is also aeigenvector corresponding to the minimum eigenvalue λ min .Hence, the resource state ρ is useful for teleportation iff λ min < (35)The singlet fraction F opt ( ρ ) given by (31) can be re-expressed in terms of the minimum eigenvalue λ min as F opt ( a,λ min ) ( ρ ) = 12 − a + 1)2 [ λ min −
29 ] , − ≤ a ≤ (36)Further, we find that if the minimum eigenvalue λ min is re-stricted to lie in the interval [ , ) then the singlet fraction F opt ( a,λ min ) ( ρ ) lies in the interval < F opt ( a,λ min ) ( ρ ) <
12 + a + 14 , − ≤ a ≤ (37)The optimal singlet fraction can be achieved by putting a = ± in (36) and it is given by F opt ( ± ,λ min ) ( ρ ) = 12 − λ min −
29 ] , ≤ λ min < (38)Without any loss of generality, we can take a = 1 andthus we have X opt = | ψ + ih ψ + | . Afterward, we denote F opt ( ± ,λ min ) ( ρ ) as simply F optλ min ( ρ ) . Therefore, F optλ min ( ρ ) = 12 − λ min −
29 ] , ≤ λ min < (39)We note here that the problem of finding the optimal singletfraction reduces to finding the minimum eigenvalue of SPA-PT of any arbitrary state ρ . B. Optimal singlet fraction in terms of average fidelity F avg ( e V , f ρ ) Now, we are in a position to give the expression for optimalsinglet fraction in terms of average fidelity F avg ( e V , f ρ ) .The optimal singlet fraction in terms of λ min can be re-writtenas F optλ min ( ρ ) = 12 − λ min −
29 ] , ≤ λ min < (40)Using (26) in (40), we get F optλ min ( ρ ) = 12 − F avg ( e V , f ρ ) −
715 ] , ≤ F avg ( e V , f ρ ) < (41)Since F avg ( e V , f ρ ) can be determined experimentally [18] so F optλ min ( ρ ) can be realized using Hong-Ou-Mandel interfer-ometry with only two detectors. C. Optimal Teleportation Fidelity for Two Qubit System
For two qubit system, optimal teleportation fidelity f opt ( ρ ) and optimal singlet fraction F optλ min ( ρ ) are relatedby [19] f opt ( ρ ) = 2 F optλ min ( ρ ) + 13= 23 − F avg ( e V , f ρ ) −
715 ]59135 ≤ F avg ( e V , f ρ ) < (42)We can say a teleportation scheme is quantum if teleportationfidelity is greater than . We can find that the teleportationfidelity given in (42) is always greater than . Since f opt ( ρ ) depends only on F avg ( e V , f ρ ) so again the optimal teleporta-tion fidelity can be realized by Hong-Ou-Mandel interferom-etry with only two detectors. V. APPLICATION
In this section, we will study a particular type of hybrid en-tangled system prepared with qubit and binary coherent state (BCS) in the context of quantum teleportation. Although co-herent state is described by infinite dimensional Hilbert spacebut binary coherent state can be described by two dimensionalHilbert space [20]. The states | + α i and | − α i are called BCSand the set {| + α i , |− α i} forms a non-orthogonal BCS basis.The state | α i is given by | α i = exp ( −| α | X n α n n ! | n i (43)The BCS basis can be expressed in terms of computationalbasis as [20] | + α i = cos ( θ ) | i + sin ( θ ) | i , | − α i = sin ( θ ) | i + cos ( θ ) | i , < θ ≤ π (44)The parameter θ can be determined by the overlapping be-tween the non-orthogonal states | + α i and | − α i and it isgiven by h + α | − α i = sin (2 θ ) (45) A. Generation of Hybrid Entangled State between a qubit andBCS
We now describe the method suggested in [21] for the gen-eration of hybrid entangled state between a qubit and BCS.We should note that strong Kerr nonlinear media can be usedto generate hybrid entanglement but this nonlinear effects inexisting media are extremely weak. Thus it would be bet-ter to use weak Kerr nonlinearity to generate entanglement.Weak Kerr nonlinearity interaction Hamiltonian is given by H k = ~ χa † a a † a . The interaction between a single-photonqubit | ψ i = c | i + d | i , | c | + | d | = 1 and a coherentstate | α i under interaction Hamiltonian H k is described as | Ψ ( ϑ ) i = exp ( iH k t ~ ) | ψ i | α i = c | i | α i + d | i | αexp ( iϑ ) i (46)Taking ϑ = π in (46), the state | Ψ ( ϑ ) i reduces to | Ψ ϑ = π i = c | i | α i + d | i | − α i (47)(47) represent a hybrid entangled system between a qubit andBCS.We may note here that an optical fibre of about 3000 km isrequired for ϑ = π for an optical frequency of ω = 5 × rad/sec using currently available Kerr nonlinearity [22]. B. Qubit-BCS Hybrid Entangled State as a Non-MaximallyTwo Qubit entangled State
We will now show that the hybrid entangled state (47) canbe used as a resource state for quantum teleportation. To ac-complish our task, we first express BCS in computational ba-sis as in (44) and then treat the hybrid entangled state as anentangled state in four dimensional Hilbert space. Therefore,qubit-BCS hybrid entangled state (47) can be expressed in thecomputational basis as | Ψ i = ccos ( θ ) | i + csin ( θ ) | i + dsin ( θ ) | i + dcos ( θ ) | i (48)Let us assume that Alice have generated the hybrid entan-gled state (48). She attach an ancilla prepared in | i a to thestate | Ψ i . Therefore, the resulting three qubit state | Ψ a i is given by | Ψ a i = [ ccos ( θ ) | i + csin ( θ ) | i + dsin ( θ ) | i + dcos ( θ ) | i ] ⊗ | i a (49)Alice then apply two qubit CNOT-gate on qubit ’2’ and qubit’a’. The state | Ψ a i reduces to | Φ a i = c | i ⊗ ( cos ( θ ) | i a + sin ( θ ) | i a )+ d | i ⊗ ( sin ( θ ) | i a + cos ( θ ) | i a ) (50)She performs a single qubit measurement in {| i , | i } basis.If the measurement result is | i then with probability | c | , sheprepare the state | Φ (1)2 a i = cos ( θ ) | i a + sin ( θ ) | i a (51)Again, if the measurement result is | i then with probability | d | , she prepare the state | Φ (2)2 a i = sin ( θ ) | i a + cos ( θ ) | i a (52) C. Mixed Qubit-BCS Entangled System Shared Between TwoDistant Parties As A Resource State For Quantum Teleportation
Case-I:
Let us assume that Alice have succeeded to gen-erate the non-maximally entangled state | Φ (1)2 a i given by (51).She now want to share a subsystem with her distant partnerBob so that they can use the shared entangled state in sendingthe quantum information. To achieve this, Alice send the sub-system ’2’ to Bob through the memoryless amplitude damp-ing channel. The transformation under memoryless amplitudedamping channel with parameter p (0 ≤ p ≤ that governsthe evolution of the system and the environment is given by | i ⊗ | i E → | i ⊗ | i E | i ⊗ | i E → p − p | i ⊗ | i E + √ p | i ⊗ | i E (53)When the qubit in mode ’2’ of the hybrid system | Φ a i passesthrough the memoryless amplitude damping channel then thesystem evolve as a mixed state and it is given by ρ a = X i =0 ( K i ⊗ I ) | Φ (1)2 a ih Φ (1)2 a | ( K † i ⊗ I )= cos ( θ ) 0 0 √ − p sin (2 θ )0 psin ( θ ) 0 00 0 0 0 √ − p sin (2 θ ) 0 0 (1 − p ) sin ( θ ) (54) where the Kraus operators K and K are given by K = (cid:18) √ − p (cid:19) , K = (cid:18) √ p (cid:19) (55)Now our task is to investigate whether the mixed state ρ a shared between Alice and Bob is still entangled and if it isentangled then under what condition? If we find that the state ρ a is entangled under certain conditions then the state ρ a canbe used as a resource state for quantum teleportation. To probethe above question, we calculate the optimal singlet fractiongiven in (39). The optimal singlet fraction for the state ρ a isgiven by F opt ( p,θ ) ( ρ a ) = 12 + 12 [ p (1 − p ) sin (2 θ ) + p sin ( θ ) − psin ( θ )] , < θ ≤ π (56)For < p < and < θ ≤ π , we have F opt ( p,θ ) ( ρ a ) > (57)Hence, the hybrid system described by the density matrix ρ a is useful as a resource state for the teleportation of a singlequbit. Further we observe the following points:(i) For any value of the non-orthogonal parameter θ (0 < θ < , the value of the quantity F opt ( p,θ ) ( ρ a ) decreasesas the noise parameter p increases from zero to unity.(ii) For any value of the noise parameter p (0 < p < ,the value of the quantity F opt ( p,θ ) ( ρ a ) increases as the non-orthogonal parameter θ increases from zero to π . Case-II:
If Alice have succeeded to generate the non-maximally entangled state | Φ (2)2 a i given by (52) then alsoresult remains the same as in case-I. VI. CONCLUSION
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