Measure of quantum Fisher information flow in multi-parameter scenario
MMeasure of quantum Fisher information flow in multi-parameter scenario
Haijun Xing ∗ and Libin Fu † Graduate School of China Academy of Engineering Physics, Beijing, 100193, China
We generalize the quantum Fisher information flow proposed by Lu et al . [Phys. Rev. A ,042103 (2010)] to the multi-parameter scenario from the information geometry perspective. Ameasure named the intrinsic density flow (IDF) is defined with the time-variation of the intrinsicdensity of quantum states (IDQS). IDQS measures the local distinguishability of quantum states instate manifolds. The validity of IDF is clarified with its vanishing under the parameter-independentunitary evolution and outward-flow (negativity) under the completely positive-divisible map. Thetemporary backflow (positivity) of IDF is thus an essential signature of the non-Markovian dynamics.Specific for the time-local master equation, the IDF decomposes according to the channels, andthe positive decay rate indicates the inwards-flow of the sub-IDF. As time-dependent scalar fieldsequipped on the state space, the distribution of IDQS and IDF comprehensively illustrates thedistortion of state space induced by its environment. As example, a typical qubit model is given. I. INTRODUCTION
Information interchange between the open system andits environment is a critical viewpoint in studying thedynamics of open quantum systems. Memory effect, i.e.,the temporary revival of previously leaked information,is one of the fascinating topics in this fields [1]. It is con-sidered a signature of the non-Markovianity firstly [2–10],then became a vital method for manipulating quantumresources [7, 8] such as entanglement [11–16], quantuminterferometric power [17], temporal steering [18], quan-tum coherence, correlations [10], and quantum Fisher in-formation (QFI) [9, 19]. With the booming of technolo-gies of control and manipulation open quantum systems[20–28], applications in ultracold atomic gases [24, 25],quantum speed limit [29–31], algorithms [32, 33], andthermal machines [34, 35] are under intensive studies inrecent years.Specifically, in the conventional memory-free dynam-ics, the information leaks outwards from the system tothe environment continuously, then dissolved. If the en-vironment has nontrivial structures, the information maybe memorized by the environment, then partially sendingback into the system subsequently. Though the rigorousquantification of these memory effects highly depends onthe interpretation of “information,” the distinguishabil-ity of quantum states is one of the primary choices. Itcan be captured by the trace distance [7, 8] that measuresthe distinguishability of a pair of states, and quantumFisher information (QFI) [9, 19] that we focus on in thismanuscript.The QFI is an intrinsic measure of the local distin-guishability of quantum states via estimating a given pa-rameter. It has tight connections with the “distance”measures between quantum states, such as the Fubini-Study metric [36], quantum geometric tensor [37], Bu-res distance [38], quantum fidelity [38, 39], and relative ∗ Electronic address: [email protected] † Electronic address: [email protected] entropy [40]. Lu et al. [9] define the quantum Fisherinformation flow (QFIF) as a measure of the memoryeffect with the time-variation of QFI with respect to aparameter previously encoded into the probe state. Itdecomposes to sub-QFIF according to the channels ofthe time local master equation. The temporal appear-ance of inwards (positive) sub-QFIF is identified withthe positive decay rates, hence becoming a signature ofnon-Markovian dynamics. It performs well in single pa-rameter cases. Furthermore, its applications in quantummetrology and quantum speed limits are fruitful: lots ofachievements, both theoretical and experimental, havebeen made.However, the practical systems are intrinsically multi-ple dimensional: 1) its states generally locate in multi-dimensional state space and thus is characterized by morethan one parameter; 2) its dynamical evolution typicallyinvolves the variation of more than one parameter. Thesingle-parameter scheme is therefore inadequate in thor-oughly characterizing the dynamics of open systems. Thegeneralization of QFIF to multi-parameter cases is essen-tial.In this article, we generalize the QFIF to the multi-parameter scenario from the information geometry per-spective. We propose a measure named the intrinsicdensity flow (IDF) with the time-variation of the in-trinsic density of quantum states (IDQS), a fundamen-tal information-theoretic state-distinguishability mea-sure. The validity of IDF will be shown with its vanishingunder the parameter-independent unitary evolution andnegativity (outwards-flow) under the CP-divisible map.It indicates the positive (inwards) IDF is an essential sig-nature of non-Markovian dynamics. Specific to the dy-namics generated by the time-local master equation, theIDF is decomposable according to the channels. The di-rection (sign) of the sub-flow is determined by the decayrates: the temporary appearance of the positive decayrates indicates the backward sub-flow in the correspond-ing channel. It violates the CP-divisible condition andserves as a sufficient signature of the non-Markovian dy-namics.Furthermore, as time-dependent scalar fields equipped a r X i v : . [ qu a n t - ph ] F e b on the state space, the distribution of IDF and IDQSare potent tools to exhibit the detailed picture of thestate space’s distorsion under the open dynamics. Wewill exemplify it with the typical model of a two-levelsystem under the non-Markovian dissipative channels.This manuscript is arranged as follows: In Sec. II, wereview the QFI and QFIF from the information geometryperspective. In Sec. III, the IDF is introduced togetherwith the IDQS. Specifically, a form of IDQS in the time-dependent coordinates is given in Sec. III C, for its tightconnections with the equation of motion. In Sec. IV, thedynamics generated by the time local master equationare studied with the IDF. In Sec. V, the time-variationof IDQS and IDF are studied with a two-level systemunder the dissipative channels. At last, we conclude inSec. VI. II. REVIEW OF QFI AND QFIF FROMINFORMATION GEOMETRY PERSPECTIVE
In the formation geometry, a given state ˆ ρ ( x ) is equiv-alent to a point x = ( x , x , . . . , x d ) in the d -dimensionalparameter space Ω via the model S = { ˆ ρ ( x ) | x ∈ Ω } .A Riemannian metric g F named as the quantum Fishermetric (QFM) with d × d entries g Fµν ( x ) = 18 Tr (cid:104) { ˆ L µ , ˆ L ν } ˆ ρ (cid:105) , (1)1 (cid:54) µ, ν (cid:54) d , is equipped on the space Ω , where {· , ·} is the anti-commutator, and the symmetric logarithmicderivative ˆ L µ is defined via ∂ µ ˆ ρ ( x ) ≡ { ˆ ρ, ˆ L µ } implic-itly [41, 42], with ∂ µ ≡ ∂/∂x µ . The QFI F is four timesof the QFM g F .In the single-parameter metrology, the probe state ˆ ρ ( θ )sketches a curve x ( θ ) in the initial parameter space Ω with the shift of parameter θ . The estimation of θ is thusequivalent to identifying a point x ( θ ) on the curve. Thenumber of states distinguishable in a segment of the curve x ( θ ) is measured by the segment’s length. The lengthis acquired by integrating the line element ds with [39] ds = (cid:88) µν g Fµν ˙ x µ ˙ x ν dθ = g F ( θ ) dθ , (2)where ˙ x ≡ d x ( θ ) /dθ is the derivative of the curve x ( θ ). Hence g F ( θ ) / is an intrinsic measure of the localdensity of states distinguishable on the curve x ( θ ). Asthe square of this density, the QFM itself is also a mea-sure of the distinguishability of ˆ ρ ( θ ) from its neighboringquantum states. Furthermore, it is directly applicable inthe quantum metrology as the upper bound of the esti-mator’s precision [43–45].When the dynamics Φ t is applied, the states corre-sponding to the initial coordinates x is changed toˆ ρ ( x ; t ) ≡ Φ t ˆ ρ ( x ), with ˆ ρ ( x ; 0) ≡ ρ ( x ). It forms anew state space S t = { ˆ ρ ( x ; t ) = Φ t ˆ ρ ( x ) | x ∈ Ω } atthe given time t . Actually, ( x ; t ) is a coordinates of the ( d + 1)-dimensional manifolds (cid:83) t S t with t as an addi-tional dimension.Under the dynamics Φ t , we also have ˆ ρ ( θ ) → ˆ ρ ( θ ; t ).The corresponding QFI F ( θ ; t ) is lost (revival) with thevariation of the state space S t . It can be accounted asthe effects of outflow (inflow) of the information. Thus,Lu et al. define the QFIF as [9] I LWS ( θ ; t ) ≡ d F ( θ ; t ) dt = 4 dg F ( θ ; t ) dt , (3)i.e., the time variation of the QFI. From the informationgeometry perspective, it measures the time-variation ofthe square of the density of distinguishable states alongthe given curve x ( θ ). III. INTRINSIC DENSITY OF QUANTUMSTATES AND INTRINSIC DENSITY FLOWA. Intrinsic density of quantum states
To identify a state ˆ ρ ( x ; t ) ∈ S t in general cases, oneshould acknowledge all of the components of x at a giventime t . It is equivalent to localizing the point x in the d -dimensional initial parameter space Ω , where ˆ ρ ( x ; t )is surrounded by states in all of the “directions.” Further-more, the dynamical evolutions generally affect the dis-tinguishability of quantum states in the multi-directionof the state space S t . A single element of the QFM, i.e., g F ( θ ), is thus inadequate for characterizing the distin-guishability of ˆ ρ ( x ; t ) out of its neighborhood.Theoretically, the local statistical distinguishability ofquantum states in Ω can be measured with the intrinsicdensity of quantum states (IDQS) [46] D Q ( x ; t ) = (cid:113) | g F ( x ; t ) | = dV ( S t ) d d x , (4)with | A | denoting the determinant of matrix A , wherethe invariant volume element dV ( S t ) quantifies the num-ber of quantum states locating in the element d d x . Wemention that for the pure state, IDQS is the measure thatdefines the completeness relationship of (sub-manifoldsof) the projective Hilbert spaces [40, 46]. B. Intrinsic density flow
Although the state ˆ ρ ( x ; t ) is evolving under the mapΦ t , the corresponding point x is stationary in the ini-tial parameter space Ω . On the contrary, the QFM,thereby IDQS, is time-dependent and capable of char-acterizing the distortion of state space under the map.Specifically, IDQS is a qualified “information” measurethat meets the essential criteria [3, 7, 47] satisfied bythe trace distance: IDQS is non-negative, invariant forparameter-independent unitary dynamics, and contrac-tion under the parameter-independent completely pos-itive and trace-preserving (CPTP) maps. In a conciseform, we have (for proof, see Appendix A, B, and C) D Q ( x ; t ) (cid:62) D Q ( x ; t (cid:48) ) (cid:62) , (5)for arbitrary given state ˆ ρ ( x ; t ) and ˆ ρ ( x ; t (cid:48) ) ≡ Λ t (cid:48) ,t [ˆ ρ ( x ; t )] with Λ t (cid:48) ,t denoting an arbitrary parameter-independent CPTP map, where the first equality isreached by the unitary channel Λ t (cid:48) ,t [ · ] = ˆ U ( t, t (cid:48) ) · ˆ U † ( t, t (cid:48) )with ∂ µ ˆ U ( t, t (cid:48) ) = 0. The unitary invariance of D ( x ; t )indicates IDQS measuring an information which conser-vative in the composite of system and environment. Thecontraction of IDQS indicates the revival information isalways smaller than the previous leaking information.Based on the above discussions, we define the intrinsicdensity flow (IDF) as I ( x ; t ) ≡ ddt D Q ( x ; t ) , (6)i.e., the time-variation of the IDQS. Its negative valueindicates the leaking of information from system to envi-ronment. The positive value indicates the backflow of theprevious leaking information, which is a signature of thenon-Markovianity. Specifically, the IDF has the followingproperties: a1 IDF is not positive under the parameter-independentCP-divisible dynamics and vanishes under theparameter-independent unitary dynamics. It di-rectly results from Eq. (5) and makes I ( x ; t ) >
0a sufficient condition for the non-CP divisible dy-namics. a2 IDF is a linear function of d ˆ ρ/dt and D ( x ; t ) as (fordetails, see Appendix C) I ( x ; t ) = I R ( x ; t ) D ( x ; t ) , (7)with the relative intrinsic density flow (RIDF) I R ( x ; t ) ≡
12 tr (cid:20) g F Tr( L d ˆ ρdt ) (cid:21) , (8)where Tr (tr) denotes the trace operation in Hilbertspace (parameter space), L is a d × d -dimensionalmatrix with the entryˆ L µν = 12 (cid:104) ˆ L ν (2 ˆ ∂ µ − ˆ L µ ) + ˆ L µ (2 ˆ ∂ ν − ˆ L ν ) (cid:105) . (9)IDF thus inherits the linear structures of the masterequation, as shown in Sec. IV. a3 By choosing x as a complete basis, i.e., a coordi-nate system of the state space concerned, the IDQS D ( x ; 0) captures all the local information of theinitial state ρ ( x ; 0). Then effects of the dynami-cal evolution Φ t on all of the components of x areaccounted in I ( x ; t ). [a][b] Figure 1: (color online). Two coordinates (parameterizationschemes) form the quantum metrology perspective. (a) Theinitial parameter x are encoded into the probe ρ firstly, thensent to the channel Φ t . The precision of x ’s estimation isupper bounded by the QFM g F ( x ; t ). (b) The parameter x are encoded after the state through the channel Φ t . Theprecision of x ’s estimation is upper bounded by the QFM g F ( x ). a4 The RIDF is independent of the parameterizationmodel. For two time-independent coordinates x and y of the initial parameter space Ω , we have I R ( x ; t ) = I R ( y ; t ) . (10)The corresponding IDFs only differ in a time-independent constant as I ( x ; t ) = (cid:12)(cid:12)(cid:12) ∂ y ∂ x (cid:12)(cid:12)(cid:12) I ( y ; t ).Hence, the direction of IDF is also independent ofthe parameterization model.These properties make the IDF be a qualified measureof the local information flow. Before further studies, wewill introduce another form of IDF. C. State space S and IDF In practical studies, researchers favor to identify thestate space S t at different times with ( x ; t ) ∼ ( x (cid:48) ; t (cid:48) )if ˆ ρ ( x ; t ) = ˆ ρ ( x (cid:48) , t (cid:48) ). We denote it as the state space S ≡ { ˆ ρ ( x ) = ˆ ρ ( x ; t ) | x ∈ Ω , t } , where x serves as thecoordinates of the state space S . A dynamical evolutionis thus depicted by the movement in S : the initial stateˆ ρ ( x ) sketches an orbit illustrated by the equations ofmotion x ( t ) with x (0) = x . One can define an alterna-tive IDQS in space S as D Q ( x ) = (cid:112) | g F ( x ) | with respectto the coordinates x .For state ˆ ρ ( x ) with the equation of motion x ( t ), wehave the IDQS D Q ( x ; t ) = (cid:12)(cid:12)(cid:12) ∂ x ∂ x (cid:12)(cid:12)(cid:12) D Q ( x ) , and the corre-sponding IDF decomposes as I ( x ; t ) = D Q ( x ; t ) × (cid:26)
12 tr (cid:20) ddt log g F ( x ) (cid:21) + tr (cid:20) ddt log (cid:0) ∂ x ∂ x (cid:1)(cid:21)(cid:27) , (11)where the first term describes shift of the point x ( t ) alongits orbit in the state space S . The second term is con-tributed by the Jacobian which connect the state space S and initial parameter space Ω . From the initial infor-mation perspective, it depicts the variation of the frame x , i.e., the background geometry of S .We mention that, from the quantum metrology per-spective, the coordinates x and x are parameters en-coded into the probe and awaiting estimation. As shownin Fig. 1a (b), g F ( x ; t ) ( g F ( x )) depicts the QFI acquiredvia parameterizing states before (after) the dynamicalevolution Φ t . In case Fig. 1 b, the channel Φ t is actuallypart of the state preparation. IV. INTRINSIC DENSITY FLOW WITHTIME-LOCAL MASTER EQUATION
In this section, we study the dynamics of open quan-tum systems with the IDF. Specifically, we focus on thestate ˆ ρ ( x ; t ) whose evolution is governed by the time-local master equation [3, 48–50] ddt ˆ ρ ( x ; t ) = K ( t )ˆ ρ ( x ; t ) , (12)with the generator K ( t ) acting on the state ˆ ρ as K ( t )ˆ ρ = − i[ ˆ H, ˆ ρ ] + (cid:88) i γ i (cid:20) ˆ A i ˆ ρ ˆ A † i − { ˆ A † i ˆ A i , ˆ ρ } (cid:21) , (13)where all of the ˆ H , γ i , and ˆ A i are generally time-dependent. It is a generalization of the conventionalLindblad master equation that all ˆ A i and γ i are time-independent, and γ i are non-negative. Eq. (13) leads aCP-divisible dynamic, if and only if γ i is non-negativefor all channel ˆ A i at all of the time [3, 48, 49]. Hence,the temporary appearance of negative γ i is taken as thesignature of the non-Markovian (non CP-divisible) dy-namics. It is also necessary for the memory effects andbackflow of the information [51, 52]. We further assume ∂ µ ˆ H = 0 and ∂ µ ˆ A i = 0 . It indicates the linearity ofthe von Neumann equation and inconsistent with thequantum no-cloning theorem [9, 53]. It also makes stateˆ ρ ( x ; t ) a stationary point in the initial parameter space.Firstly, we focus on a special case with γ i = 0 ∀ i ,where Eq. (13) reduces to the unitary evolution. Thecorresponding IDF vanishes as I ( x ; t ) = 0, with theadditional footnote 0 denoting the unitarity. It directlyresults from the time-invariance of the metric under uni-tary dynamics with (for proof, see Appendix A) ddt g Fµν ( x ; t ) = 0 . (14)It indicates that the parameter space Ω is frozen. It istremendously different from the picture in state space S .In the coordinates x , the system may demonstrate verycomplicated dynamics.In the general cases of Eq. (13), the system exchangesinformation with the environment through each of thechannel ˆ A i with γ i (cid:54) = 0. Specifically, the IDF decomposesas I ( x ; t ) = (cid:80) i I i ( x ; t ), where I i ( x ; t ) ≡ γ i (cid:20) g F (cid:18) ddt g F (cid:19) i (cid:21) D Q ( x ; t ) (15) denotes the sub-IDF through the channel ˆ A i with thederivatives (for details, see Appendix D)( ddt g Fµν ) i = 12 tr (cid:110) ([ ˆ A † i , ˆ L ν ][ ˆ A i , ˆ L µ ] + [ ˆ A † i , ˆ L µ ][ ˆ A i , ˆ L ν ])ˆ ρ (cid:111) . (16)For the matrix ( d g F /dt ) i is negative semidefinite, wehave the direction (sign) of sub-flows I i ( x ; t ) (cid:62) , γ i ( t ) <
0= 0 , γ i ( t ) = 0 (cid:54) , γ i ( t ) > . (17)These results are full of physical implications: (1)The decomposition of IDF according to the channel re-sults from the time-local master equation’s linearity to d ˆ ρ/dt . (2) The direction of the sub-flow I i is controlledby γ i . If there exist a channel such that γ i < t , the corresponding sub-flow will flow backto the system. It is consistent with the CP-divisiblecondition given by the Gorimi-Kossakowski-Sudarshan-Lindblad theorem [48, 49]. We mention that these re-sults are the natural generalization of the propositionEq. (6) in [9]. However, it is now valid in the multi-parameter scenario and independent of the parameteri-zation scheme. V. TWO-LEVEL SYSTEMS
For two-level system with basis {| (cid:105) , | (cid:105)} , we parame-terize the general mixed state ˆ ρ asˆ ρ ( n ) = 12 (cid:16) ˆ I + n · ˆ σ (cid:17) , (18)with the Pauli matrices ˆ σ = (ˆ σ , ˆ σ , ˆ σ ), where S is theBloch sphere, and coordinates x is the Bloch vector n =( n , n , n ) , with | n | ≡ (cid:80) µ ( n µ ) (cid:54)
1. We have thedensity D Q ( n ) ≡ (cid:113) | g F ( n ) | = 18 (cid:112) − | n | . (19)This density only depends on the radius | n | , i.e., thestate’s purity. It results from the unitary invariance ofIDQS. D Q ( n ) is divergent if | n | = 1. It is induced bythe radial element g Fnn ( n ) = 1 / [4(1 − | n | )], which de-picts the distinct statistical difference between pure andmixed states. However, the statistical distance betweenany pair of pure and mixed states are still finite, so is thevolume given by this density. The minimum of D Q ( n ) isreached by the completely mixed state with n = 0. Sur-prisingly, this minimum is non-zero. It depicts the statis-tical difference between ˆ ρ ( ) and its neighboring states,although ˆ ρ ( ) itself is usually termed as information-free. -40040 [a] [c]00.20.4 020 [d]-3-11[b] 0 1 2 30 1 2 3 0 1 2 30 1 2 3 -1.0 0.0 1.0 - - - [a1][a2][a3] [b1][b2][b3] [c1][c2][c3] [d1][d2][d3] -20 ----- ----0.040.060.080.10 ----------1.751.801.851.901.952.00 -1.0 0.0 1.0 - Figure 2: (color online). Two-level systems in dissipative channel ( W = 3 λ ). (a) γ as a function of time; (b) The IDQS D Q ( n ; t ) as a function of time, with r = √ .
9; (c) the IDF I ( n ; t ) as a function of time; (d) the RIDF I R ( n ; t ) as afunction of time. (a1)-(a3) D Q ( n ) in n - n plane with n = 0 at λt = 0 .
02, 0 .
5, and 1 .
0; (b1)-(b3) The IDQS D Q ( n ; t ) in n - n plane with n = 0 at λt = 0 .
02, 0 .
5, and 1 .
0. (c1)-(c3) I ( n ; t ) in n - n plane with n = 0 at λt = 0 .
1, 0 .
5, and 1 . I R ( n ; t ) in n - n plane with n = 0 at λt = 0 .
1, 0 .
5, and 1 . A. Dissipative channels
In this sub-section, we study a typical model wherea two-level system is immersed in a dissipative environ-ment [3, 9, 54]. In the interaction picture, the systemundergoes a dynamic generated by the master equation ddt ˆ ρ ( t ) = γ ( t ) (cid:20) ˆ σ − ˆ ρ ˆ σ + − { ˆ σ + ˆ σ − , ˆ ρ } (cid:21) , (20)with the raising (lowering) operator ˆ σ + (ˆ σ − ). By assum-ing the environment has a Lorentzian spectral densitywith vanishing detuning, we have γ ( t ) = − h ( t ) /h ( t )with the characteristic function h ( t ) = (cid:40) e − λt/ (cid:2) cosh( dt ) + λ sinh( dt ) (cid:3) , W < λ e − λt/ (cid:2) cos( dt ) + λ sin( dt ) (cid:3) , W (cid:62) λ , (21) d = (cid:112) | λ − W | , where W measures the system-environment coupling strength, λ defines the width ofthe spectral density. In the weak coupling regime with W < λ/ γ ( t ) is non-negative. The system undergoes aMarkovian dynamics, where the information—measured by both of the trace distance [8] and QFI [9]—is lostcontinuously. For conciseness, we mainly focus on thestrong coupling regime with W (cid:62) λ/
2, where h ( t ) dis-plays an oscillatory behavior and the non-Markovian dy-namics emergent with the negative γ ( t ).We begin with the equations of motion n µ ( t ) = h ( t ) n µ , µ = 1 , , (22a) n ( t ) = h ( t ) (1 + n ) − . (22b)Under this equation, the Bloch sphere contracts toground state | (cid:105) firstly with the positive γ ( t ), then par-tially swell back corresponding to the negative γ ( t ), asshown by Fig. 2 a1-a3. Together with Eq. (11) and (19),we have the IDF I ( n ; t ) = − γ (cid:20) n ) − | n | (cid:21) D Q ( n ; t ) (23)with the IDQS D Q ( n ; t ) = h ( t ) (cid:112) − | n | . (24)As shown by Fig. 2a, γ ( t ) oscillates between the pos-itive and negative values. It induces the informationflow I ( n ; t ) to propagate outwards and inwards, andthe IDQS D Q ( n ; t ) increases and decreases correspond-ingly. The time-local extremum of the IDQS is given atthe transition times of inwards and outwards flow, whenthe IDQS vanishes with the Bloch sphere shrinks to thepoint | (cid:105) . In Fig. 2b (2c), we illustrate the IDQS (IDF) offour initial states: ˆ ρ (0 , , r ), ˆ ρ ( r , , ρ (0 , , − r ),and ˆ ρ (0 , , t = 0, ˆ ρ , ˆ ρ , and ˆ ρ locate in the sameshell with radius r = √ .
9, the maximally mixed stateˆ ρ locates in the center of the space. They show the sametime-dependent non-Markovian behavior. The state ˆ ρ near the stationary point | (cid:105) has the biggest IDQS overthe four states. We mention that the distinguishabilityof the “information free” state ˆ ρ from its neighboringstates are still captured by IDQS and IDF, which exhib-ited the same dynamical characteristics as other states.For both the IQDS and IDF are scalar fields equippedon the parameter space Ω , the variation of their distri-butions provides us an interesting viewpoint to study thedynamics of the open system. As shown by Fig. 2 c2 andc3, the IDF takes its maximum at point | (cid:105) in the begin-ning of the first contraction. It induces the tremendousdecrease of the IDQS for states in its neighborhood, asshown by Fig. 2 b2 and exemplified by state ˆ ρ . TheIDQS always takes its maximum (minimum) in the point | (cid:105) ( | (cid:105) ) in the subsequent evolutions. It is induced by thefact that the inward flow’s magnitude is always smallerthan the previous outwards flow, although the flow takesits maximum at state | (cid:105) in the following stages.Furthermore, the RIDF captures a parameterization-independent signature of the dynamics. As a time-dependent scalar field equipped on the parameter space,its distribution is an overall description of the relativestrength of state space’s distortion. Specific for this dis-sipative model, we have I R ( n ; t ) = − γ (cid:20) n ) − | n | (cid:21) . (25)Its distribution is shown by Fig. 2 d1-d3. It indicatesthe IDQS near point | (cid:105) is lost and acquired with a moresignificant relative strength, not only in the first contrac-tion but also in the full evolutions. It consists of the in-sights that the excited states component | (cid:105)(cid:104) | is mostlyinfluenced by the dissipative channel. Furthermore, thegradient and range of I R ( n ; t ) indicate the uniformityof the time-variation of IDQS fields. As exemplified byFig. 2 d and d1, the IDQS leaks with a larger gradientin the first contraction. In the following stage, as shownby Fig. 2 d2 and d3, the gradient and range are tremen-dously decreased. The IDQS at all of the points are lostand revival with roughly the same RIDF. It results fromthat the Bloch sphere shrinks to an ellipsoid highly lo-calized around the point | (cid:105) after the first contraction. VI. CONCLUSIONS
In conclusion, we have generalized the quantum Fisherinformation flow (QFIF) [9] to the multi-parameter sce-nario from the information geometry perspective. Wepropose a measure named the intrinsic density flow (IDF)with the time-variation of quantum states’ local distin-guishability, quantified by the intrinsic density of quan-tum states. The validity of IDF has been shown withits vanishing under the parameter-independent unitaryevolution and negativity (propagate outwards) under thecompletely positive and trace-preserving map. It makesthe positive (inwards) IDF be a vital signature of non-Markovian dynamics.Specific for dynamics generated by the time local mas-ter equation, we have shown the IDF is decomposableaccording to the channels. The direction (sign) of thesub-flow is determined by the decay rates: the tempo-rary appearance of the positive decay rates indicates thesub-flow in the corresponding channel flowing backward.It violates the CP-divisible condition and serves as anessential signature of the non-Markovian dynamics.Not only having tight connections with the non-Markovian dynamics, but the IDQS and IDF themselvesare also significant for the studies of open quantum sys-tems. As time-dependent scalar fields equipped on thestate space, the distribution of IDF and IDQS are potenttools to exhibit the global picture of the state space’svariation under the open dynamics. We have exempli-fied it with the qubit system under the non-Markoviandissipative channel.Via ten years of productive studies, the QFIF exhibitsits values in both theoretical and experimental aspects.Our research and the IDF provide a path to generalizethese studies to the multi-parameter cases. Furthermore,its tight connections with the information geometry havebeen built, which gives us the systematic methods tostudy the dynamics of the open quantum systems. Wemention that IDF is one of many measures provided bythe quantum Fisher metric, though possibly the most im-portant one. Other explorations are in processing. Thepotential value of this field is promising. We expect thisarticle can catalyze more studies of the open quantumsystem from the information geometrical perspective.
Acknowledgments
This work is supported by the National Natural Sci-ence Foundation of China (NSFC) (Grant No. 12088101,No. 11725417, and No. U1930403) and Science Chal-lenge Project (Grant No. TZ2018005).
Appendix A: Proof of Eq. (14): The IDF vanishesunder the parameter-independent unitary channel.
We will prove the invariance of QFM metric ddt g F ( x ; t ) = 0 , (A1)under the unitary evolution U generated by ddt ρ ( x ; t ) = − i[ H, ρ ] , (A2)where H is assumed Hermitian and parameter-independent with H = H † and ∂ µ H = 0. Proof.
We denote the symmetric logarithmic derivativeof ρ ( x ; 0) as L µ ( x ; 0), which satisfying ∂ µ ρ ( x ; 0) = { ρ ( x ; 0) , L ( x ; 0) } , and L † µ ( x ; 0) = L µ ( x ; 0). Then L µ ( x ; t ) = U ( t, L ( x ; 0) U † ( t,
0) (A3)is a valid symmetric logarithmic derivative of thestate ρ ( x ; t ) = U ( t, ρ ( x ; 0) U † ( t, ∂ µ ρ ( x ; t ) = { L µ ( x ; t ) , ρ ( x ; t ) } and L † µ ( x ; t ) = L µ ( x ; t ). Insert it into the definition of QFM Eq. (1),we have g Fµν ( x ; t ) = 18 tr [ { L µ ( x ; t ) , L ν ( x ; t ) } ρ ( x ; t )]= g Fµν ( x ; 0) . (A4)Eq. (A1), i.e., Eq. (14) is thus proved. Appendix B: Proof of Eq. (5): The IDQS is notincreased under parameter-independent CPTP map.
Firstly, we prove that the IDQS is not increased underparameter-independent CPTP map Λ t (cid:48) ,t which reads D Q ( x ; t ) (cid:62) D Q ( x ; t (cid:48) ) , (B1)with ρ ( x ; t (cid:48) ) = Λ t (cid:48) ,t [ ρ ( x ; t )] and t (cid:48) (cid:62) t . Proof.
We begin with the coordinates y in which g F ( y ; t ) is diagonal, i.e., g Fµν ( y ; t ) = 0 for µ (cid:54) = ν . Forthe QFI is not increased under CPTP map Λ t (cid:48) ,t in thesingle parameter cases, we have g Fµµ ( y ; t ) (cid:62) g Fµµ ( y ; t (cid:48) ) , (B2)for all µ . For g F ( y ; t (cid:48) ) is positive semidefinite, we have (cid:89) µ g Fµµ ( y ; t (cid:48) ) (cid:62) (cid:12)(cid:12) g F ( y ; t (cid:48) ) (cid:12)(cid:12) (B3) with the Hadamard’s inequality. Together with Eq. (B2),we have (cid:12)(cid:12) g F ( y ; t ) (cid:12)(cid:12) (cid:62) (cid:12)(cid:12) g F ( y ; t (cid:48) ) (cid:12)(cid:12) . (B4)Multiplying | ∂ y /∂ x | on both sides of Eq. (B4), we havethus proved Eq. (B1).The equality in Eq. (B1) is reached by the unitarychannel. It is the direct results of Eq. (A1). Furthermore,we have D Q ( x ; t ) (cid:62) Appendix C: IDF with the symmetric logarithmicderivative
Firstly, we expand the IDF as I ( x ; t ) ≡ ddt (cid:12)(cid:12) g F (cid:12)(cid:12) = 12 | g F | tr (cid:20) ddt log g F (cid:21) , (C1)with tr[ ddt log g F ] = tr[ ddt g F / g F ], g F ≡ g F ( x ; t ) for sim-plicity, and the element ddt g Fµν = 12 tr[ L ν { ρ, ˙ L µ } + L µ { ρ, ˙ L ν } + { L µ , L ν } ˙ ρ ] , (C2)with ˙ A ≡ ddt A for sccinctness. For the derivative ddt ∂ µ ρ = 12 tr (cid:104) { ˙ ρ, L µ } + { ρ, ˙ L µ } (cid:105) , (C3)we have the anti-commutaor { ρ, ˙ L µ } = 2 ∂ µ ˙ ρ − { ˙ ρ, L µ } = (2 ∂ µ − L µ ) ˙ ρ − ˙ ρL µ . (C4)Insert it into Eq. (C3), we have ddt g Fµν ( x ; t ) = Tr [ L µν ( ˙ ρ )] , (C5)with the operator L µν = 12 [ L ν (2 ∂ µ − L µ ) + L µ (2 ∂ ν − L ν )] . (C6) Appendix D: Fisher information flow with time-localmaster equation
In this appendix, we give the form of IDF under thetime-local master equation Eq. (13). We begin with thetracetr (cid:20) L ν (2 ∂ µ − L µ ) ddt ρ (cid:21) i =tr (cid:20) L ν (2 ∂ µ − L µ ) (cid:18) A i ρA † i − { A † i A i , ρ } (cid:19)(cid:21) =tr (cid:20) L ν A i ( L µ ρ + ρL µ ) A † i − L ν L µ A i ρA † i − L ν { A † i A i , L µ ρ + ρL µ } + 12 L ν L µ { A † i A i , ρ } (cid:21) =tr (cid:104) A † i L ν A i L µ ρ + L µ A † i L ν A i ρ − A † i L ν L µ A i ρ (cid:105) −
12 tr (cid:104) L ν A † i A i L µ ρ + L µ L ν A † i A i ρ + A † i A i L ν L µ ρ + L µ A † i A i L ν ρ − L ν L µ A † i A i ρ − A † i A i L ν L µ ρ (cid:105) =tr (cid:104) A † i L ν A i L µ ρ + L µ A † i L ν A i ρ − A † i L ν L µ A i ρ (cid:105) −
12 tr (cid:104) L ν A † i A i L µ ρ + L µ A † i A i L ν ρ + L µ L ν A † i A i ρ − L ν L µ A † i A i ρ (cid:105) . (D1)Insert it into Eq. (C5), we have the derivatives ddt g Fµν ( x ; t ) i = 12 tr (cid:26) [ L ν (2 ∂ µ − L µ ) + L µ (2 ∂ ν − L ν )] ddt ρ (cid:27) i = 12 tr (cid:110) A † i L ν A i L µ ρ + L µ A † i L ν A i ρ + A † i L µ A i L ν ρ + L ν A † i L µ A i ρ − L ν A † i A i L µ ρ − L µ A † i A i L ν ρ − A † i L ν L µ A i ρ − A † i L µ L ν A i ρ (cid:111) = −
12 tr (cid:8)(cid:0) [ A i , L ν ] † [ A i , L µ ] + [ A i , L µ ] † [ A i , L ν ] (cid:1) ρ (cid:9) . (D2)It indicates the derivative matrix ( d g F ( x , t ) /dt ) i is neg-ative semi-definite. We can diagonalize it with a realorthonormal matrix O as O (cid:18) ddt g F (cid:19) i O T = diag (cid:104) λ ( i )1 , λ ( i )2 , . . . , λ ( i ) d (cid:105) , (D3)with the element λ ( i ) k = (cid:88) µ,ν O kµ (cid:18) ddt g Fµν (cid:19) i O Tνk = − tr (cid:26)(cid:104) A i , (cid:88) ν O kν L ν (cid:105) † (cid:104) A i , (cid:88) µ O kµ L µ (cid:105) ρ (cid:27) (cid:54) . (D4) Furthermore, 1 / g F ( x ; t ) is positive definite, it indicates α µ ≡ (cid:18) O g F O T (cid:19) µµ > . (D5)Hence, we have the tracetr (cid:20) g F (cid:18) ddt g F (cid:19) i (cid:21) = (cid:88) µ λ ( i ) µ α µ (cid:54) . (D6)Insert it into Eq. (15), we have thus proved Eq. (17)together with D ( x ; t ) (cid:62) [1] H.-P. Breuer and F. 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