Aspects of quantum states asymmetry for the magnetic dipolar interaction dynamics
AAspects of quantum states asymmetry for the magnetic dipolar interaction dynamics
Douglas F. Pinto ∗ and Jonas Maziero † Departamento de Física, Centro de Ciências Naturais e Exatas,Universidade Federal de Santa Maria, Avenida Roraima 1000, 97105-900, Santa Maria, RS, Brazil (Dated: January 5, 2021)We investigate the asymmetry properties of quantum states in relation to the Hamiltonianresponsible for the magnetic dipolar interaction (MDI) dynamics, and we evaluate its relationshipto entanglement production. We consider some classes of pure and mixed quantum states of twoqubits evolved under MDI and, using the asymmetry measure defined via the Wigner-Yanase skewinformation, we describe the asymmetry dependence on the Hamiltonian parameters and initialconditions of the system. In addition, we define and calculate the dynamics of the asymmetry oflocal states, characterizing their temporal and interaction parameters dependence. Finally, becausethe MDI Hamiltonian has a null eigenvalue, the group generator-based asymmetry measure doesnot adequately quantify the state susceptibility with respect to the action of the subspace generatedby the eigenvectors associated with this eigenvalue. For this reason, we also define and study thegroup element-based asymmetry measure with relation to the unitary operator associated with theMDI Hamiltonian.
Keywords: Quantum asymmetry, Quantum entanglement, Quantum coherence, Magnetic dipolar interaction
I. INTRODUCTION
Research on the intrinsically quantum properties present in physical systems allows the gaining of a betterunderstanding of the fundamental characteristics of nature and has promoted the construction of methods thatcontribute to the description, transmission, and manipulation of information contained in its microscopic compounds,leading to developments that have a positive impact on diverse areas of Science [1, 2]. Such properties are primordialfor the execution of certain tasks that are impracticable in the context of classical physics. For instance, entanglementis an indispensable attribute in the composition of quantum communication protocols such as teleportation [3], apromising mechanism for the development of a future quantum internet [4], in the realization of controlled gatesapplicable in quantum computing, and in the implementation of quantum simulations, where well controlled quantumsystems can be used to reproduce the behavior of complex uncontrollable systems [1], in addition to enabling severalother useful applications in the progress of quantum information science technologies [5, 6].Recently, the resource theory of asymmetry has received attention in the area of Quantum Information Science,favoring applications in various contexts involving both single and composite quantum systems [7–18]. For example,in multipartite systems the states which break the dynamic symmetry associated with additive Hamiltonians make itpossible to quantify or to witness entanglement by the Wigner-Yanase skew information [7, 19], a function that has theproperties necessary for characterization of asymmetry measures in the resource theory of asymmetry [16]. Actually,the relationships between skew information measures and quantum correlations have been studied for some time now[20]. And recently, in Ref. [21], the behavior of the non-classicality of spins systems was studied using the measureof Wigner-Yanase skew information as a quantifier. Using the contextual re-interpretation of state asymmetry inrelation to local unitary groups, which are associated with additive Hamiltonian eigenvectors, it was shown that thedifference between the asymmetry of a general global state and the asymmetry of the composition of local states isrelated to total correlations [22].Asymmetry has also been investigated from an open systems perspective, where it is a useful resource in theprocessing of quantum information, which in turn suffers deterioration due to interactions between system andenvironment. However, the freezing of asymmetry in scenarios where open systems are restricted by superselectionrules have been studied, proving useful mainly in occasions where noise is described by invariant operations, whichenables asymmetry conservation mechanisms in the presence of noise described by such operations [12], an advantagethat can be employed in noisy quantum channels.Alike to quantum coherence, state asymmetry properties in relation to the generator of a quantum evolution canplay important roles in the entanglement produced due to interactions between subsystems [23]. Moreover, in a more ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ qu a n t - ph ] J a n general context, asymmetry properties have useful characteristics in applications such as in: Alignment of referenceframes [24, 25], quantum speed limits [11], quantum metrology [13, 16, 26], quantum thermodynamics [15, 27, 28],and quantum communication [14, 29]. The Wigner-Yanase asymmetry, associated with the Hamiltonian, measureshow far the state of the system is from sharing the same base of eigenstates with the observable, established bythe commutation relationship between the state of the system and the Hamiltonian associated with its dynamics.Therefore, if the quantum system breaks the dynamic symmetry generated by the Hamiltonian, the commutator isdifferent from zero and, consequently, we say that the state of the system presents non-zero asymmetry.The presence of the MDI in several physical systems that have properties of coherence and quantum correlations,such as spins systems [30–32], nitrogen vacancy centers in diamond [33, 34], and rotational states of molecules [35], hasmotivated research in the context of Quantum Information Science in order to analyze the usefulness of the MDI inthe application of resources for the execution of quantum gates for quantum computing [35, 36], quantum simulationexecution [37], realization of quantum channels for quantum communication, besides stimulating the investigationof quantum properties of Gibbs thermal states [38–40] and the dynamics of quantum correlations and entanglement[23, 41–45]. More recently, it has been shown that, for dipole-dipole interaction and two-photon resonance between twoqubits and a coherent cavity field, the dipolar interaction can contribute to robustness against intrinsic decoherenceand preserve a higher entanglement rate [46]. In addition, the MDI plays the role of noise source in several physicalsystems, leading to the decay of the quantum properties of the system [47–49]. It has also been shown that for specificconfigurations of systems interacting via the MDI, they can preserve the entanglement properties and quantumcoherence along the dynamics [23].So, it is relevant to investigate the asymmetry properties of the system configurations in relation to MDI Hamiltonianand to analyze the role that asymmetry plays in respect to the entanglement produced along MDI unitary dynamics.In this article, we study the asymmetry of bipartite quantum states described by two magnetic dipoles that evolve dueto the MDI, whose observable associated with the generator of the unitary dynamics is non additive (non local). Aswe showed recently [23], the existence of local quantum coherence in the initial states (before MDI occurs) is generallysufficient for the production of entanglement during the course of the MDI dynamics. However, we also showed thatit is not a necessary condition, since some initial states with null local coherence are also able to produce maximumentanglement. In this particular case, the peculiar property present in the initial global configuration of the system isthe anti-alignment of the dipoles, which, in a way, indicates the presence of asymmetry in relation to the Hamiltoniangenerating the unitary dynamics.The structure of the remainder of this article is the following. In Sec. II, we present the Hamiltonian and the systemstates that we will evaluate from the point of view of the asymmetry measure associated with the Wigner-Yanaseskew information, which is described in Sec. III. In Sec. IV, we examine the behavior of the asymmetry function inrelation to the Hamiltonian of the MDI for pure and mixed product states. In Sec. V, we deal with the asymmetryof local states, and, in Sec. VI, we define and study the asymmetry in relation to the unitary evolution operator. Wepresent our conclusions in Sec. VII. II. OBSERVABLES AND STATES
In this section, we describe the dynamics generator, the Hamiltonian, and states we use to study asymmetrysubsequently. Let us start by introducing the general shape of the Magnetic Dipolar Interaction (MDI) Hamiltonian,expressed in the same form as in Ref. [23]: H = D [( (cid:126)σ ⊗ σ ) · ( σ ⊗ (cid:126)σ ) − n · (cid:126)σ ⊗ ˆ n · (cid:126)σ ] , where D = µ γ a γ b (cid:126) / πr isthe parameter that describes the magnitude of magnetic dipole interaction, r is the distance between the dipoles, µ symbolizes the permeability of vacuum, γ a,b represents the gyromagnetic factor of the subsystems a and b , respectively, (cid:126)σ is the vector of Pauli matrices, σ is the identity matrix of dimension and ˆ n ∈ R is the unit vector pointing in thedirection of the line that connects the dipole centers. Here, we consider Planck’s constant (cid:126) = 1 and we fix D = 1 .For simplicity, but without loss of generality, we assume that the dipoles centers lie along the z -axis (ˆ n = (0 , , .Thus H = 2 − ( σ ⊗ σ + σ ⊗ σ − σ ⊗ σ ) = 0 | Ψ − (cid:105)(cid:104) Ψ − | + 2 | Ψ + (cid:105)(cid:104) Ψ + | − ( | Φ − (cid:105)(cid:104) Φ − | + | Φ + (cid:105)(cid:104) Φ + | ) , where | Ψ ± (cid:105) = 2 − / ( | (cid:105) ± | (cid:105) ) , | Φ ± (cid:105) = 2 − / ( | (cid:105) ± | (cid:105) ) form the Bell basis of maximally entangled states, with {| (cid:105) , | (cid:105)} being the standard basis, and we use the notation | xy (cid:105) ≡ | x (cid:105) ⊗ | y (cid:105) for the tensor product.We will first consider that the dipoles are prepared in the configuration of pure-product states, so that | ψ ab (cid:105) := | ψ a (cid:105) ⊗ | ψ b (cid:105) = ( α a | (cid:105) + β a | (cid:105) ) ⊗ ( α b | (cid:105) + β b | (cid:105) ) , (1)where α a,b = cos( θ a,b / and β a,b = sin( θ a,b / are the probability amplitudes associated with each particle, and θ a,b ∈ [0 , π ] , i.e, we consider the dipoles’ configurations along coaxial rings in the Bloch’s sphere representation [23].The dynamics of states evolving under the MDI Hamiltonian is given by the unitary operator U = exp ( − i H t ) . Forthe pure initial states above, ignoring a global phase, the evolved quantum state reads: | ψ abt (cid:105) = e − i H t | ψ ab (cid:105) = ( α a β b cos t − iβ a α b sin t ) | (cid:105) + ( β a α b cos t − iα a β b sin t ) | (cid:105) + e it ( α a α b | (cid:105) + β a β b | (cid:105) ) . (2)In order to evaluate the consequences of increased entropy of the states in the asymmetry relationship with theMDI Hamiltonian, we also consider two classes of mixed states: ρ abj := ρ ja ⊗ ρ jb = 2 − ( σ + r ja σ j ) ⊗ − ( σ + r jb σ j ) , (3)where r js = tr ( ρ js σ j ) ∈ [ − , for s = a, b . This states correspond to local Bloch’s vectors in the j axis for eachsubsystem, and we shall take j = 1 or j = 3 . In these cases, the evolution provided by MDI is given by ρ abj ( t ) = U ρ abj U † ,so that ρ ab ( t ) = (1 + r a r b ) [ | Ψ + (cid:105)(cid:104) Ψ + | + | Φ + (cid:105)(cid:104) Φ + | ] + (1 − r a r b ) [ | Ψ − (cid:105)(cid:104) Ψ − | + | Φ − (cid:105)(cid:104) Φ − | ]+ ( r b + r a ) (cid:2) e it | Φ + (cid:105)(cid:104) Ψ + | + e − it | Ψ + (cid:105)(cid:104) Φ + | (cid:3) + ( r b − r a ) (cid:2) e it | Φ − (cid:105)(cid:104) Ψ − | + e − it | Ψ − (cid:105)(cid:104) Φ − | (cid:3) (4)and ρ ab ( t ) = (1 + r a ) (1 + r b ) | (cid:105)(cid:104) | + (1 − r a r b + ( r a − r b ) cos (2 t )) | (cid:105)(cid:104) | + i ( r a − r b ) sin (2 t ) | (cid:105)(cid:104) | − i ( r a − r b ) sin (2 t ) | (cid:105)(cid:104) | + (1 − r a r b − ( r a − r b ) cos (2 t )) | (cid:105)(cid:104) | + (1 − r a ) (1 − r b ) | (cid:105)(cid:104) | . (5) III. WIGNER-YANASE ASYMMETRY MEASURE
In this section, we present the asymmetry measure based on the skew information of Wigner and Yanase [19]. Herewe are interested in quantifying the presence of asymmetry of states in relation to the Hamiltonian that generates thedynamics for MDI. The Wigner-Yanase asymmetry of an arbitrary state ρ in relation to H is given by the followingexpression [18]: A ( ρ, H ) := − tr (cid:104) ρ / , H (cid:105) = tr (cid:0) ρ H (cid:1) − tr (cid:16) ρ / H ρ / H (cid:17) , (6)where [ ., . ] represents the commutator. It is easy to verify that the asymmetry of states evolved under a time-independent Hamiltonian is also time-independent, i.e., A ( ρ t , H ) = A ( ρ, H ) , (7)where ρ t = U ρU † and U = e − i H t . Besides, for pure states √ ρ = (cid:112) | ψ (cid:105)(cid:104) ψ | = | ψ (cid:105)(cid:104) ψ | , and the calculation of asymmetryis reduced to computing the variance of the generator of the dynamics: A ( | ψ (cid:105)(cid:104) ψ | , H ) = (cid:104) ψ |H | ψ (cid:105) − (cid:104) ψ |H| ψ (cid:105) . (8)Such expressions, besides of being associated to the quantum coherence related to the generator eigenbasis, due tothe Hamiltonian’s structure considered in this work, they also reveal an important role of their respective eigenvalues,a fact that will become evident in the next sections. Actually, the Hamiltonian of MDI has a null eigenvalue, andthe contribution of the subspace associated with such eigenvalue is not captured by the asymmetry measure definedabove. This fact will be used as an argument to consider a Wigner-Yanase asymmetry defined using the unitaryoperation, rather than using the Hamiltonian, for measuring how much a given initial state changes under the actionof a given transformation. IV. ASYMMETRY OF STATES IN RELATION TO THE MDI HAMILTONIAN
For pure states, using the expression for asymmetry in Eq. (8), we shall have A ( | ψ ab (cid:105) , H ) = 49 (cid:110) α a β b + β a α b ) + (cid:0) α a α b + β a β b (cid:1) − (cid:104) ( α a β b + β a α b ) − (cid:0) α a α b + β a β b (cid:1)(cid:105) (cid:27) . (9) FIG. 1: (Color online) Behavior of the Wigner-Yanase asymmetry of states described by | ψ ab (cid:105) as a function of the initialstate parameters θ a and θ b . It can be seen that the maximum value of the asymmetry function, A = 1 , is reached when theinitial state is | + + (cid:105) or | − −(cid:105) , and its minimum value, A = 0 , is obtained for the states | (cid:105) and | (cid:105) . Besides, we identifythe intermediate values of asymmetry A = 4 / and A = 6 / , which are associated with the initial states {| (cid:105) , | (cid:105)} and (cid:80) j =1 1 √ | E j (cid:105) , respectively, with | E j (cid:105) being the eigenvectors of H . The dependency of the asymmetry function on the parameters θ a and θ b , that define the possible initial stateconfigurations, is illustrated graphically in Fig. 1. We can observe in this figure that the maximum values ofasymmetry are reached in the regions around the following two states | + + (cid:105) = ( | Φ + (cid:105) + | Ψ + (cid:105) ) / √ and | − −(cid:105) = ( | Φ + (cid:105) − | Ψ + (cid:105) ) / √ , (10)that are balanced superpositions of a pair of the Hamiltonian eigenstates corresponding to non-zero and distincteigenvalues. In this case, the maximum asymmetry coincides with the maximum local coherence of each dipole, andcorresponds to initial states for which maximum entanglement is generated in some instant of time along the MDIdynamics (see Ref. [23]). Throughout this article, quantum coherence is measured with relation to the standard basis {| (cid:105) , | (cid:105)} .We observe a similar pattern in the regions of asymmetry values around / ≈ , , which are associated withbalanced superpositions of all Bell base states. These states correspond to dipole a having null local coherence anddipole b showing maximum coherence, or vice versa: | (cid:105) ⊗ |±(cid:105) = 2 − ( ±| Ψ + (cid:105) ± | Ψ − (cid:105) + | Φ + (cid:105) + | Φ − (cid:105) ) , |±(cid:105) ⊗ | (cid:105) = 2 − ( ±| Ψ + (cid:105) ∓ | Ψ − (cid:105) + | Φ + (cid:105) + | Φ − (cid:105) ) , (11) | (cid:105) ⊗ |±(cid:105) = 2 − ( | Ψ + (cid:105) − | Ψ − (cid:105) ± | Φ + (cid:105) ∓ | Φ − (cid:105) ) , |±(cid:105) ⊗ | (cid:105) = 2 − ( | Ψ + (cid:105) + | Ψ − (cid:105) ± | Φ + (cid:105) ∓ | Φ − (cid:105) ) , (12)and lead to similar intermediate entanglement values ( ≈ , ) at specific times ( t = π/ ) of the MDI dynamics [23].Above, and in what follows, we ignore global phases, once the Wigner-Yanase asymmetry doesn’t depend on them.We notice also the existence of states with asymmetry equal to / , corresponding to the states | (cid:105) = 1 √ | Ψ + (cid:105) + | Ψ − (cid:105) ) and | (cid:105) = 1 √ | Ψ + (cid:105) − | Ψ − (cid:105) ) , (13)leading to the production of maximum entanglement in some period of the MDI dynamics (as e.g. at t = π/ ). Thisobservation leads us to conclude that the amount of initial state asymmetry does not necessarily establish a direct FIG. 2: (Color online) (a) Wigner-Yanase asymmetry of the ρ ab class of states, whose quantum coherence is null in the standardbasis, as a function of the r a and r b parameters. For r a = − r b we have A = 4 r a / . So, the maximum value of asymmetryfor this class of states is obtained when r a = ± and r b = ∓ , which corresponds to the states | (cid:105) and | (cid:105) , respectively.The line of states where r a = r b has asymmetry equal to zero. (b) Asymmetry of the ρ ab class of states as a function of theparameters r a and r b , associated with the Bloch vector of each dipole. For r a = r b we have A = r a . Thus, for such a classof states, the maximum values for asymmetry are obtained for r a = ± and r b = ± , which corresponds to the states | + + (cid:105) and | − −(cid:105) , respectively. The line of states with r a = − r b has A = r a / . and unambiguous relationship with entanglement creation.There are also states with maximum local coherence, suchas | + (cid:105) ⊗ |−(cid:105) = 1 √ | Φ − (cid:105) − | Ψ − (cid:105) ) and |−(cid:105) ⊗ | + (cid:105) = 1 √ | Φ − (cid:105) + | Ψ − (cid:105) ) , (14)that have asymmetry around / and also lead to maximum dynamic entanglement.From the results presented above, we infer that the presence of asymmetry in relation to the Hamiltonian of theMDI does lead to the existence of entanglement at some time. However, in general there is no direct quantitativerelationship between A and E . This fact is more evident for initial state configurations involving a pair of states inwhich one of the elements is the singlet state | Ψ − (cid:105) . This is so because the nullity of the corresponding eigenvalueleads to the non-contribution of this state in the calculation of the asymmetry, reducing its magnitude in these cases.On the other hand, we can observe that the line of states θ b = θ a and θ b = 2 π − θ a produce null asymmetry in allregions whose configurations are θ a = nπ with n ∈ Z , a behavior analogous to what occurs during the entanglementdynamics [23]. So, in the next section, we will calculate the asymmetry of local states, which, in addition to helpin understanding the role of local states in the evolution of asymmetry along the dynamics of the MDI, this kind offunction also contains the partial contributions of the subspace associated with the null eigenvalue of the Hamiltonian.Regarding the mixed states configurations defined in Sec. II, the Wigner-Yanase asymmetry for the class ofincoherent states is given by A ( ρ ab , H ) = 2 (cid:18) − r a r b − (cid:113) (1 − r a ) (1 − r b ) (cid:19) / , (15)while for the class of states with coherence the asymmetry reads A (cid:0) ρ ab , H d (cid:1) = (cid:18) r a r b − (cid:113) (1 − r a ) (1 − r b ) (cid:19) / . (16)These functions are shown in Fig. IV. These results reveal the main aspects already obtained for pure states, but alsoshow that the decrease in purity leads to the diminishing in the asymmetry of the states. V. ASYMMETRY OF LOCAL STATES
With the study of asymmetry associated with the product state classes developed so far, it was possible to observethat despite positive asymmetry being a necessary condition for the production of entanglement under MDI dynamics,there is no direct relationship between them. Thus, as the asymmetry is not affected by the unitary dynamics ofglobal states, i.e., A ( ρ abt , H ) = A ( ρ ab , H ) , in order to understand the mechanism of temporal evolution of asymmetryand their local contributions, we will evaluate the asymmetry of the local evolved states, i.e., we want to quantify thesusceptibility of local states under the action of the dynamics generator H . For defining this quantifier, we take thepartial trace [50] over one of the dipoles, e.g., ρ at = tr b ( ρ abt ) , and, in order to preserve the correspondence with the Hamiltonian’s dimensionality, we compose this reduced statewith the maximally mixed state for the other dipole, i.e., ˜ ρ at := ρ at ⊗ σ . (17)Once constructed this state, we defined the local Wigner-Yanase asymmetry of subsystem A as A l ( ρ at , H ) := A (˜ ρ at , H ) := − tr [˜ ρ at , H ] . (18)The local asymmetry A l ( ρ bt , H ) is defined in a similar way. A. Local asymmetry for pure-product initial states
For the evolved pure states shown in Sec. II, using the state ˜ ρ at = tr b | Ψ abt (cid:105)(cid:104) Ψ abt | ⊗ σ (19) = (cid:8)(cid:0) α a α b + α a β b cos ( t ) + β a α b sin ( t ) (cid:1) | (cid:105)(cid:104) | + (cid:0) β a β b + β a α b cos ( t ) + α a β b sin ( t ) (cid:1) | (cid:105)(cid:104) | + (cid:2) α a β a cos ( t ) (cid:0) α b exp (2 it ) + β b exp ( − it ) (cid:1) − iα b β b sin ( t ) (cid:0) α a exp (2 it ) + β a exp ( − it ) (cid:1)(cid:3) | (cid:105)(cid:104) | + (cid:2) α a β a cos ( t ) (cid:0) β b exp (2 it ) + α b exp ( − it ) (cid:1) + iα b β b sin ( t ) (cid:0) β a exp (2 it ) + α a exp ( − it ) (cid:1)(cid:3) | (cid:105)(cid:104) | (cid:9) ⊗ σ (20)we compute the local asymmetry of subsystem a . The expression for A l ( ρ at , H ) is too cumbersome to be shown here.So, this local asymmetry is shown graphically in Fig. 3 as a function of dipole a initial state and time for some initialstates of dipole b . From these plots, it is possible to determine the favorable time bands for obtaining the highestvalues of local asymmetry, that occurs for t = kπ , with k = 0 , , , · · · , and for regions where θ a = π/ or θ a = 3 π/ ,regardless of θ b , and for t = kπ/ in the region where θ a = θ b = π/ . Besides, we can see that A l ( ρ at , H ) has a periodequal to π. In Fig. 4, the local asymmetry is shown as a function of the initial state parameters for some instants of time.The maximum values of A l ( ρ at , H ) are around . , and are obtained in the regions of θ a = π/ or θ a = 3 π/ with t = kπ for k = 0 , , , · · · . In addition, for t = π/ or t = 2 π/ , in the regions θ a = θ b = π/ and θ a = θ b = 3 π/ , respectively, the value . is also obtained for A l ( ρ at , H ) .We observe that the general dependence of local asymmetry with the initial state and time is intricate and quitesimilar with the dependence of local quantum coherence given by the l -norm coherence. Besides, there is even lessdirect relationship between local asymmetry and entanglement than what was observed for global asymmetry. So,in the next section, we shall study the system state global asymmetry with regard to the element of the group oftime transformations, i.e., we shall look at the quantum state susceptibility in relation to the time evolution operatorassociated with the MDI Hamiltonian. FIG. 3: (Color online) Dynamics of local asymmetry, A l ( ρ at , H ) , of dipole a as a function of time and of the pure-productinitial states configurations parameters θ a and θ b . For these plots, we consider the parameter θ b fixed and allow the parameters t and θ a to vary “continuously” in their respective intervals. FIG. 4: (Color online) These plots illustrate the behavior of the local asymmetry of subsystem a , A l ( ρ at , H ) , as a function oftime and of the parameters of the initial pure-product state configurations represented by θ a and θ b . B. Local asymmetry for mixed-product initial states
For the evolved mixed states shown in Sec. II, we have the reduced states: ρ aj ( t ) = tr b (cid:0) ρ abj ( t ) (cid:1) . So, the localasymmetry for dipole a is computed using the density matrices: ρ a ( t ) =4 ρ aj ( t ) ⊗ σ (21) = (( r b + r a ) cos (3 t ) − ( r b − r a ) cos ( t )) ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | + | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) /
8+ ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | + | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) and ˜ ρ a ( t ) = ρ a ( t ) ⊗ σ / (22) = (2 + ( r b + r a ) + ( r a − r b ) cos (2 t )) ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) /
8+ (2 − ( r b + r a ) − ( r a − r b ) cos (2 t )) ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) / . (23)Thereby, the respective asymmetries of these local states are given by the expressions: (cid:114) A l ( ρ a ( t ) , H )5 = (cid:112) ( r a − r b ) cos ( t ) + ( r a + r b ) cos (3 t ) + 2 − (cid:112) ( r b − r a ) cos ( t ) − ( r a + r b ) cos (3 t ) + 2 (24)and (cid:113) A l ( ρ a ( t ) , H ) = (cid:113) − r a + ( r a − r b ) sin ( t ) − (cid:113) r a + ( r b − r a ) sin ( t ) . (25)These functions are shown graphically in Fig. 5. The same observations made in the last subsection for pure-productinitial states also hold here. FIG. 5: (Color online) Local asymmetry of dipole a , A l ( ρ aj ( t ) , H ) , as a function of time and r ja for some values of r jb , for theinitial states ρ abj = ρ aj ⊗ ρ bj evolved under the MDI Hamiltonian. We see that the local asymmetry has period equal to π . VI. ASYMMETRY IN RELATION TO THE UNITARY OPERATOR
In this section, we study the Wigner-Yanase skew information in relation to the unitary operator generated by themagnetic dipolar interaction Hamiltonian. We do this expecting to obtain a more sensitive measure regarding thetemporal evolution of the states and which includes the contribution of the phases corresponding to the eigenvaluesof the observable responsible for the dynamics. So, we define the unitary asymmetry as the Wigner-Yanase skewinformation of a state ρ with respect to the non-Hermitian unitary time evolution operator: A U ( ρ, U t ) := 12 tr (cid:110) [ √ ρ, U t ] † [ √ ρ, U t ] (cid:111) (26) = 1 − tr (cid:110) √ ρU t √ ρU † t (cid:111) , (27)For pure states, this function can be recast as A U ( | ψ (cid:105) , U t ) = 1 − tr (cid:110) | ψ (cid:105)(cid:104) ψ | U t | ψ (cid:105)(cid:104) ψ | U † t (cid:111) (28) = 1 − |(cid:104) ψ | ψ t (cid:105)| , (29)where | ψ t (cid:105) = U t | ψ (cid:105) . Such a function determines the degree of dissimilarity between the initial prepared state and thestate obtained from the evolution dictated by the unitary operator.To assess the unitary asymmetry in the context of the MDI, we consider the spectral decomposition of the unitaryoperator given by the following Bell-diagonal matrix: U t = | Ψ − (cid:105)(cid:104) Ψ − | + exp ( − it ) | Ψ + (cid:105)(cid:104) Ψ + | + exp ( it ) ( | Φ − (cid:105)(cid:104) Φ − | + | Φ + (cid:105)(cid:104) Φ + | ) . (30)Considering the same class of pure states of the previous sections, | ψ (cid:105) = | ψ ab (cid:105) ≡ c | Ψ − (cid:105) + c | Ψ + (cid:105) + c | Φ − (cid:105) + c | Φ + (cid:105) , with c = 2 − / (cid:2) cos (cid:0) θ a (cid:1) sin (cid:0) θ b (cid:1) − sin (cid:0) θ a (cid:1) cos (cid:0) θ b (cid:1)(cid:3) ,c = 2 − / (cid:2) cos (cid:0) θ a (cid:1) sin (cid:0) θ b (cid:1) + sin (cid:0) θ a (cid:1) cos (cid:0) θ b (cid:1)(cid:3) ,c = 2 − / (cid:2) cos (cid:0) θ a (cid:1) cos (cid:0) θ b (cid:1) − sin (cid:0) θ a (cid:1) sin (cid:0) θ b (cid:1)(cid:3) ,c = 2 − / (cid:2) cos (cid:0) θ a (cid:1) cos (cid:0) θ b (cid:1) + sin (cid:0) θ a (cid:1) sin (cid:0) θ b (cid:1)(cid:3) , (31)the unitary asymmetry is given by A U ( | ψ ab (cid:105) , U t ) = 1 − (cid:2) c + c cos (2 t ) + (cid:0) c + c (cid:1) cos ( t ) (cid:3) − (cid:2)(cid:0) c + c (cid:1) sin ( t ) − c sin (2 t ) (cid:3) . (32)Analyzing the behavior of the unitary asymmetry dynamics illustrated in Figs. 6 and 7, and comparing with theevolution of entanglement under the MDI [23], we emphasize that the regions of maximum entanglement during thedynamics of the MDI are contained in the regions of states of maximum unitary asymmetry A U , although they mayoccur in different periods of their temporal dynamics. In addition, the regions of null entanglement coincide withregions of null unitary asymmetry. On the other hand, there are also regions with maximum unitary asymmetry butwith partial entanglement.1 FIG. 6: (Color online)Behavior of the unitary asymmetry as a function of dipole a initial state parameter θ a and of the timeparameter t , for some fixed values of dipole b initial state parameter θ b . This sequence of plots allow us to identify the periodsof time resulting in a greater unitary asymmetry A U . FIG. 7: (Color online) These graphs illustrate the behavior of asymmetry of states with respect to the MDI unitary operatoras a function of the parameters determining the initial conditions, θ a and θ b , for some values fixed for the time parameter t . Theonly region of states that remain constant throughout the MDI dynamics are those where the parameters θ a and θ b correspondsto the states | (cid:105) and | (cid:105) , whose asymmetry A U is null, and which coincides with the regions where the global asymmetry A ( | ψ ab (cid:105) , H ) is null. However, in general the state asymmetry A U behaves similarly to the global asymmetry A ( | ψ ab (cid:105) , H ) whenthe time parameter is fixed around π/ . In this figure, we can see that the maximum value of the unitary asymmetry A U isobtained in three periods of the A ’s dynamics, with regions of different state configurations. For the period t = π/ , we havemaximum asymmetry A U in the region of states with θ a = θ b = π/ or θ a = θ b = 3 π/ . For the case where t = π/ , themaximum unitary asymmetry A U is reached in regions where θ a = π and θ b = 0 or θ b = 2 π and vice versa. In addition, for theperiod t = π , any line of states where θ a = π/ or θ a = 3 π/ for any regions of θ b , and vice versa, lead to the maximum valueof the unitary asymmetry A U . VII. CONCLUSIONS
In this work, we analyzed the quantum state Wigner-Yanase asymmetry in relation to the Magnetic DipolarInteraction (MDI) Hamiltonian as the generator of the temporal evolution. We described the dependence of asymmetryin terms of the parameters that define the Hamiltonian and in terms of the initial state configurations of the establishedbipartite system, where we considered classes of pure and mixed initial states separable and restricted to real localphases. We obtained analytical expressions for the asymmetry of pure and mixed states, from which it was possible toobserve the regions that admit maximum asymmetry and thus establish relations with the purity and entanglementproduction during the dynamics under the MDI. We also defined the local asymmetry, a quantity that reveals thelocal states susceptibility under the action of the Hamiltonian generator of the global dynamics under the MDI.Furthermore, in order to quantify the role of Hamiltonian eigenvalues for MDI dynamics, we defined the Wigner-Yanase skew information measure in relation to the MDI unitary operator, obtaining thus a better agreement betweenstates with greater skew-information and states capable of producing entanglement along the MDI dynamics.
Acknowledgments
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