Local Quantum Uncertainty in Two-Qubit Separable States: A Case Study
NNoname manuscript No. (will be inserted by the editor)
Local Quantum Uncertainty in Two-Qubit SeparableStates: A Case Study
Ajoy Sen · Debasis Sarkar · Amit Bhar
Received: date / Accepted: date
Abstract
Recent findings suggest, separable states, which are otherwise of nouse in entanglement dependent tasks, can also be used in information process-ing tasks that depend upon the discord type general non classical correlations.In this work, we explore the nature of uncertainty in separable states as mea-sured by local quantum uncertainty. Particularly in two-qubit system, we findseparable X-state which has maximum local quantum uncertainty. Interest-ingly, this separable state coincides with the separable state, having maximumgeometric discord. We also search for the maximum amount of local quantumuncertainty in separable Bell diagonal states. We indicate an interesting con-nection to the tightness of entropic uncertainty with the state of maximumuncertainty.
Keywords
Local Quantum uncertainty · non classical correlations anddiscord PACS
First AuthorDepartment of Applied Mathematics, University of Calcutta, 92, A.P.C. Road, Kolkata-700009, IndiaE-mail: [email protected] AuthorDepartment of Applied Mathematics, University of Calcutta, 92, A.P.C. Road, Kolkata-700009, IndiaE-mail: [email protected] AuthorDepartment of Mathematics, Jogesh Chandra Chaudhuri College, 30, Prince Anwar ShahRoad, Kolkata-700033, IndiaE-mail: [email protected] a r X i v : . [ qu a n t - ph ] A ug Ajoy Sen et al.
As a measure of non classical correlation beyond entanglement, discord[1,2,3] has generated lot of interest in recent years. Numerous literature hasbeen engaged in understanding its precise role in both the quantum com-puting and information theoretic tasks, e.g., DQC1 model [4,5,6], Groversearch algorithm[7], remote state preparation[8,9], state merging[10,11] entan-glement distribution[12,13], state discrimination[14,15,16,17], two-qubit stateordering[18], quantum cryptography[19]. Recently, an operational method ofusing discord, as a resource, has been experimentally established [20] and aninteresting connection between discord and interferometric power of quantumstate has been established [21]. Several other versions of discord have also beenproposed[3] including geometric and relative entropic discord.Measurement, in general, disturbs a quantum state. Classically, we canmeasure any two observable with arbitrary accuracy. However, such kind ofmeasurement is not possible in quantum theory even if we use flawless mea-surement device. Heisenberg uncertainty principal provides the precession insuch kind of measurement. No quantum state shows uncertainty under themeasurement of single global observable. However, measurement of a singlelocal observable can manifest uncertainty in a quantum state. Uncertaintyin a quantum state can arise due to its classical mixing or due to its non-commutativity with the measuring observable. Girolami et al. [22] have intro-duced the concept of local quantum uncertainty (LQU, in short) as a measureof minimum uncertainty by measurement of a single local observable on aquantum state. This quantity identifies the true quantum part of error whicharises due to non-commutativity between state and observable and it does notchange under classical mixing. Zero uncertainty implies the existence of quan-tum certain (commutative) local observable corresponding to the state. Everyentangled state possesses this kind of uncertainty, i.e., there is no quantumcertain local observable for any entangled state. Even, mixed separable statescan show the same characteristic. The only class of states which remain invari-ant under such local measurement is the states with zero quantum discord[1].Thus the non-zero discord state show uncertainty under the measurement ofa single local observable.For a bi-partite quantum state ρ AB , local quantum uncertainty(LQU) isdefined as, U ΛA ( ρ AB ) := min K Λ I ( ρ AB , K Λ ) (1)The quantity I denotes skew Information, defined by Wigner and Yanase[23]as, I ( ρ, K ) := −
12 tr { [ √ ρ, K ] } (2)and clearly it is a measure of non-commutativity between a quantum state andan observable. The minimization in (1) is performed over all local maximallyinformative observable (or non-degenerate spectrum Λ ) K Λ = K ΛA ⊗ I . ocal Quantum Uncertainty in Two-Qubit Separable States: A Case Study 3 Local quantum uncertainty is inherently an asymmetric quantity. It is in-variant under local unitary operations. It vanishes for states with zero discord.For pure bi-partite states, it reduces to a entanglement monotone(linear en-tropy of reduced subsystems). In, two-qubit system all Λ dependent quantitiesbecome proportional and the dependency on Λ can be dropped. So, LQU canbe taken as a measure of bi-partite quantumness in two-quit system.Local quantum uncertainty has been calculated in DQC1 model and itcan explain quantum advantages by separable states like discord. LQU alsohave significant application in quantum metrology. It has close connection toquantum Fisher information[24,25] due to the link between Fisher informationmetric and skew information[26,27]. It provides a upper bound to the varianceof best estimator of the parameter in parameter estimation protocol. LQU hasgeometrical significance in terms of Hellinger distance[28] between the stateand its least disturbed counterpart after root of unity local unitary operation.More specifically, for a quantum state ρ , U A ( ρ ) = D H ( ρ, K A ρK A ), where D H ( ρ, χ ) := tr { ( √ ρ − √ χ ) } is the squared Hellinger distance and K A isroot of unity operation on party A.Explicit closed form of LQU has been derived only for some symmetricclass of states and for simple systems [22,29]. For a quantum state ρ of 2 ⊗ n system, U A ( ρ ) = 1 − λ max ( W ) (3)where λ max is the maximum eigenvalue of the matrix W = ( w ij ) × , w ij =tr {√ ρ ( σ i ⊗ I ) √ ρ ( σ j ⊗ I ) } and σ i ’s are standard Pauli matrices in this case.In two-qubit system, Bell states achieve maximum uncertainty value 1, whichshows that the definition of LQU is normalized. Since, LQU indicates non zerovalue for separable state, so it will be interesting to investigate on “ how muchwe can do with separable state ?” i.e., maximum uncertainty we can achievewith separable state. LQU provides a guaranteed upper bound to the varianceof best estimator of parameter in parameter estimation protocol. The maximalvalue will provide the limit of precession which can be achieved using separablestates in such metrological task. Our next curiosity will be whether there is anyconnection between such separable states with maximum LQU and maximumgeometric discord. The problem is in fact a mathematical optimization problemover all separable states ( S ) as,max ρ ∈S U A ( ρ ) = max ρ ∈S (1 − λ max ( W ))= 1 − min ρ ∈S ( λ max ( W )) (4)i.e., we need to find out the minimum of λ max ( W ) over all separable states.We will begin form separable X class of states. This is an important subclassof states as it contains Bell diagonal states and Werner states. This class ofstates are frequently encountered in studying quantum dynamics, condensedmatter systems, etc [30,31,32,33,34]. Ajoy Sen et al.
Here we will start with the standard way of parameterizing a two-qubit Xstate. Let us, first, consider any two-qubit state ρ in the form, ρ = a a a a a a a a (5)The elements of the matrix satisfy, a = a † , a = a † (Complex Conjugation) (cid:88) i =1 a ii = 1 , (Normalization) a a ≥ a , a a ≥ a , (Positivity) (6)The state contains seven independent parameter. However, LQU is local uni-tary invariant and we can easily drive out the phases from off diagonal elementby mere local unitary operation. Hence, we are left with only five positive realparameters and henceforth with out loss of generality we will consider all a ij ’sas real and non negative. This X state has four real eigenvalues λ , λ , λ , λ and corresponding eigenvectors are | v (cid:48) (cid:105) , | v (cid:48) (cid:105) , | v (cid:48) (cid:105) , | v (cid:48) (cid:105) . Here, λ i ’s and | v (cid:48) i (cid:105) ’shave dependence on a ij ’s. Thus, ρ can be decomposed as ρ = (cid:80) i =0 λ i | v i (cid:105)(cid:104) v i | with | v i (cid:105) being the normalized form of | v (cid:48) i (cid:105) . The state vectors | v i (cid:105) ’s are mutu-ally orthonormal and we can easily write √ ρ = (cid:80) i =0 √ λ i | v i (cid:105)(cid:104) v i | . After a bitsimplification it reads, √ ρ = α α α α a α α α (7)with α = (cid:18) √ λ ω ω + 1 + √ λ ω ω + 1 (cid:19) α = (cid:18) √ λ ω ω + 1 + √ λ ω ω + 1 (cid:19) α = (cid:18) √ λ ω + 1 + √ λ ω + 1 (cid:19) α = (cid:18) √ λ ω + 1 + √ λ ω + 1 (cid:19) α = (cid:18) √ λ ω ω + 1 + √ λ ω ω + 1 (cid:19) α = (cid:18) √ λ ω ω + 1 + √ λ ω ω + 1 (cid:19) (8) ocal Quantum Uncertainty in Two-Qubit Separable States: A Case Study 5 and ω = (cid:18) a − a + λ − λ a (cid:19) ω = (cid:18) a − a − λ + λ a (cid:19) ω = (cid:18) a − a + λ − λ a (cid:19) ω = (cid:18) a − a − λ + λ a (cid:19) (9)We also have the relation (cid:104) v (cid:48) i | v (cid:48) i (cid:105) = ω i + 1. According to the definition of LQU,and it turns out that, W = Diag(2( α α + α α + α α ) , α α + α α − α α ) , (cid:80) i =1 α i − α − α ). Let us define, w = 2 ( α α + α α + α α ) w = 2 ( α α + α α − α α ) w = (cid:32) (cid:88) i =1 α i − α − α (cid:33) (10)Till now, we have not taken into account the separability condition. PPT(Positive partial transpose) criteria works as a necessary and sufficient condi-tion of separability in two-qubit system. PPT criteria gives the following twoseparability conditions, a a ≥ a and a a ≥ a (11)These two conditions are dual to the original positivity constraints. We canwrite an optimization problem of LQU over all separable X-states as,Minimize λ max ( W ) = max { w , w , w } Subject to : a ≤ min( √ a a , √ a a ) a ≤ min( √ a a , √ a a ) (cid:88) a ii = 1 a ij ≥ , ∀ i, j (12)If λ ∗ max is the solution of the minimization problem, we will have maximumLQU, U ∗ A = max(1 − λ max ) = 1 − λ ∗ max . Now let us consider the case α α ≥ w ≥ w . Hence λ max ( W ) = max { w , w } . Depending upon Ajoy Sen et al. the sign of w − w we formulate two optimization problems from (12) as,Minimize λ max ( W ) = w Subject to : w ≤ w α α ≥ a ≤ min( √ a a , √ a a ) a ≤ min( √ a a , √ a a ) (cid:88) a ii = 1 a ij ≥ , ∀ i, j (13)and Minimize λ max ( W ) = w Subject to : w ≤ w α α ≥ a ≤ min( √ a a , √ a a ) a ≤ min( √ a a , √ a a ) (cid:88) a ii = 1 a ij ≥ , ∀ i, j (14)In the first case (13), after simplifying, we get U A = 4 max { α + α } and inthe second case (14), U A = max { ( α − α ) + ( α − α ) + 2( α − α ) } .A little simple calculation and the form of constraints in the optimizationproblem suggest us to choose a = a , a = a and a = a = √ a a for the sake of minimization purpose and we get w = 4( a − a ) and w = 16 a a . We also need to consider normalization condition a + a = . Under these constraints, after solving, we get a = √ √ , a = √ − √ .The regions corresponding to the two optimization problem are shown in (1).Exactly similar analysis follows if we choose α α <
0. In this case we get a = √ − √ , a = √ √ . The sates corresponding to both the solutions aremerely connected by local unitary σ x ⊗ σ x . Hence we obtain a unique (up tolocal unitaries) rank-2 separable X-state ρ ∗ with U ∗ A = . ρ ∗ = 14 √ √ √ √ √ √ √ √ − √ √ √ − √ (15)Interestingly, exactly same state (up to local unitaries) was shown[35] to havemaximum geometric discord among separable X-states. ocal Quantum Uncertainty in Two-Qubit Separable States: A Case Study 7 Fig. 1 (color online) The figure shows the region corresponding to w ≥ w and w ≥ w . The dotted upper boundary curve indicates the value of λ max . The two marked redpoints indicate the minimum value of λ max in those regions and hence the points correspondsto the solution of the optimization problem, i.e., maximum LQU Any two-qubit quantum state, under local unitary equivalence, can betaken as, ρ = ( I ⊗ I + x t σ ⊗ I + I ⊗ y t σ + (cid:80) i =1 t i σ i ⊗ σ i ) where I n denotes the identity matrix of order n , σ = ( σ , σ , σ ) where σ i ’s are usualPauli matrices, T = Diag[ t , t , t ] is the correlation matrix with components t i = tr( ρσ i ⊗ σ i ). x = ( x , x , x ), y = ( y , y , y ) are Bloch vectors with x i = tr( ρσ i ⊗ I ), y i = tr( ρ I ⊗ σ i ). Our numerical simulation with separablestates did not reveal any state with LQU greater than . This tempted us toconjecture that maximum value of LQU for two-qubit separable states is .This is in same spirit to the similar conjecture on discord, made in [36]. Start-ing from a different measure (discriminating strength) Farace et al. [37] find arelation between their correlation measure and LQU. For two-qubit system,they considered rank 2, 3 ,4 separable states and performed similar optimiza-tion. Their numerical result shows that rank-2 states achieve the maximumLQU for B-92 states but the analytical proof is still absent. However they pre-sented analytical proof in other dimensions. Maximum LQU for separable Bell Diagonal states : This class of statesbelongs to X class of states and have the form, ρ = p I | φ + (cid:105)(cid:104) φ + | + p x | φ − (cid:105)(cid:104) φ − | + p y | ψ + (cid:105)(cid:104) ψ + | + p z | ψ − (cid:105)(cid:104) φ − | These states are entangled if any one parameter among p I , p x , p y , p z is greaterthan . In terms of Bloch representation, the state can also be written as, ρ = 14 [ I ⊗ I + (cid:88) i =1 t ii σ i ⊗ σ i ]Bloch vectors corresponding to this class can be obtained as x = , y = .Whenever t = t = t = t (say) i.e., T = t I , LQU corresponding to this Ajoy Sen et al. class is, U A ( ρ ) = 1 − √ t (cid:0) √ t + √ − t (cid:1) Maximum value of LQU can reach in separable domain ( − ≤ t ≤ ). If weconsider any two of the t ii are equal or when all t ii ’s are unequal, our numeri-cal suggest that we can’t reach more than by separable Bell diagonal states. Dissonance : ρ ∗ ≡ ρ ∗ AB is a rank-2 separable state and it can be written as, ρ ∗ AB = λ | φ (cid:105)(cid:104) φ | + λ | φ (cid:105)(cid:104) φ | where | φ (cid:105) = a | (cid:105) + b | (cid:105) , | φ (cid:105) = a | (cid:105) + b | (cid:105) are orthogonal states and the parameters are λ = λ = , a = b = √ √ √ , a = b = √ √ . Its dissonance ( D A ) can be easily obtained[38]form the purified Koashi-Winter relation, D A ( ρ ∗ AB ) = S ( ρ ∗ A ) − S ( ρ ∗ AB ) + E F ( ρ ∗ BC ) (16) E F denotes the entanglement of formation, S denotes von Neumann entropyand dissonance is considered w.r.t. the measurement on party A. Let | Ψ ∗ ABC (cid:105) = √ λ | φ (cid:105)| (cid:105) + √ λ | φ (cid:105)| (cid:105) be a purification of the state ρ ∗ AB . We can easilyevaluate the reduced states ρ ∗ AB , ρ ∗ BC , ρ ∗ A and obtain E F ( ρ ∗ BC ) = S ( ρ ∗ A ) ≈ . S ( ρ ∗ AB ) = 1. Hence, D A ( ρ ∗ AB ) = 0 . The role of maximally uncertain separable state can be investigated in con-nection to the tightness of entropic uncertainty relation. In the presence ofquantum memory, entropic uncertainty relation was proposed by Berta et al. [39] as, S ( P | B ) + S ( Q | B ) ≥ − c ( P, Q ) + S ( A | B ) (17)and later it was improved by Pati et al. [40] as, S ( P | B ) + S ( Q | B ) ≥ − c ( P, Q ) + S ( A | B )+max { , D A ( ρ AB ) − J A ( ρ AB ) } (18)where ρ AB is the initial state between quantum system A and quantum mem-ory B . S ( P | B ) and S ( Q | B ) are the conditional von Neumann entropies of thestate ρ AB after measurement of the observable P and Q on A respectivelyand S ( A | B ) := S ( ρ AB ) − S ( ρ B ) is the quantum conditional entropy withoutmeasurement. c ( P, Q ) ≡ max i,j |(cid:104) p i | q j (cid:105)| and { p i } , { q j } are the eigenvectors ofthe observables P and Q . D A ( ρ AB ), J A ( ρ AB ) denotes quantum discord andclassical correlation of the state ρ AB respectively and they are defined as, J A ( ρ AB ) = max { Π Aj } (cid:2) S ( ρ B ) − S ( ρ B | A ) (cid:3) (19) D A ( ρ AB ) = I ( ρ AB ) − J A ( ρ AB ) (20) ocal Quantum Uncertainty in Two-Qubit Separable States: A Case Study 9 Fig. 2
Nature of entanglement as measured by negativity N and uncertainty gap ∆ forthe states χ . Both curve shows monotonic behavior. In fact maximum uncertainty gap isachieved at (cid:15) = 1. Uncertainty gap increases as (cid:15) → (cid:15) ≈ . ∆ increases up to itsmaximum. maximum is taken over all projective measurement { Π Aj } on party A, S ( ρ B | A )denotes the standard conditional entropy of the state obtained by the projec-tive measurement on A[1].We define the sum of uncertainty S ( P | B ) + S ( Q | B ) as U P,QB and the lowerbound − c ( P, Q ) + S ( A | B ) + max { , D A ( ρ AB ) − J A ( ρ AB ) } as L P,QB . Thedifference ∆ P,Q := U P,QB − L P,QB denotes a kind of uncertainty gap in a quan-tum state corresponding to the pair of observables P and Q . This quantity isobviously non-negative and may characterizes the discrepancy between uncer-tainty of the measurement outcomes of P and Q [41]. Maximally entangledstates have ∆ σ x ,σ z = 0 corresponding to the maximally unbiased spin observ-ables σ x and σ z . We are interested in checking the status of this discrepancyaround the state of maximal uncertainty. We will consider both separableand entangled states around ρ ∗ . We consider the mixed entangled state χ = (cid:15)ρ ∗ + (1 − (cid:15) ) | φ + (cid:105)(cid:104) φ + | , where | φ + (cid:105) = | (cid:105) + | (cid:105)√ . For this state, we have the con-ditional entropies S ( Q | B ) = H ( { (2 −√ (cid:15) , (2+ √ (cid:15) , − (2+ √ (cid:15) , − (2 −√ (cid:15) } ) − S ( P | B ) = H ( { (2 −√ (cid:15) , (2 −√ (cid:15) , − (2 −√ (cid:15) , − (2 −√ (cid:15) } ) − σ x and σ z . H ( { p i } ) is the usual Shannon entropy of theprobability distribution { p i } . The uncertainty gap is monotonic as evidentfrom the FIG. 2. ∆ σ x ,σ z obtains its highest value corresponding to (cid:15) = 1, i.e., ρ ∗ has maximum uncertainty gap from this class. Even if we consider the noisymixed separable state pρ ∗ + − p I , the state ρ ∗ shows the highest discrepancy. We have thus shown that among separable X states there is a unique statewhich attains the maximum value of LQU. The same state also attains the maximum value of geometric discord among similar class of states. We believethat this result can be extended (based on our numerical exploration andalso from the work of Farace et. al.) to whole separable class of states andin that case this value will be a good indicator of entanglement since theamount of uncertainty beyond the value necessarily imply the existence ofentanglement. We hope, our result will provide a limit that we can achieve byseparable states in some other quantum information theoretic tasks or someprotocols in near future. Acknowledgements
The author A. Sen acknowledges the financial support from Univer-sity Grants Commission, New Delhi, India. The author D. Sarkar also acknowledges DSTSERB for financial support.
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