Universal Quantum Uncertainty Relations Between Non-Ergodicity and Loss of Information

Universal quantum uncertainty relations between non-ergodicity and loss of information

Natasha Awasthi,1, 2 Samyadeb Bhattacharya,2, 3 Aditi Sen(De),2 and Ujjwal Sen2 1College of Basic Sciences and Humanities, G.B. Pant University Of Agriculture and Technology, Pantnagar, Uttarakhand - 263153, India 2Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad - 211019, India 3S. N. Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700 098, India We establish uncertainty relations between information loss in general open quantum systems and the amount of non-ergodicity of the corresponding dynamics. The relations hold for arbitrary quantum systems interacting with an arbitrary quantum environment. The elements of the uncertainty relations are quantiﬁed via distance measures on the space of quantum density matrices. The relations hold for arbitrary distance measures satisfying a set of intuitively satisfactory axioms. The relations show that as the non-ergodicity of the dynamics increases, the lower bound on information loss decreases, which validates the belief that non-ergodicity plays an important role in preserving information of quantum states undergoing lossy evolution. We also consider a model of a central qubit interacting with a fermionic thermal bath and derive its reduced dynamics, to subsequently investigate the information loss and non-ergodicity in such dynamics. We comment on the “minimal” situations that saturate the uncertainty relations.

I. INTRODUCTION We propose a measure of information loss in a quantum system, based on distinguishability of quantum states, which in turn is based on distance measures on the space of density In practical situations, it is arguably impossible to com- operators [29–34]. We quantify the non-ergodicity of the dy- pletely isolate a quantum system from its surroundings and namics based on the distance between the time-averaged state it is subjected to information loss due to dissipation and deco- after sufficiently long processing time and the corresponding herence. In modelling open quantum systems, the simpler ap- thermal equilibrium state. Within this paradigm, we derive an proach is to consider the environment to be memoryless, i.e. uncertainty relation between informaton loss and the amount Markovian [1–5]. The system-environment relation is how- of non-ergodicity for an arbitrary quantum system interact- ever more often than not non-Markovian, and there are pos- ing according to an arbitrary quantum Hamiltonian with an sibilities of information backflow into the system, which can arbitrary environment. The derivation is not for a particular be considered as a resource in information theoretic tasks [6– distance measure, but for all such which satisfies a set of in- 8]. The systems showing such properties are usually associ- tuitively satisfactory axioms. In the illustrations, we mainly ated with various structured environments without the consid- focus on the trace distance, and to a certain extent, also on eration of weak system-environment coupling and the Born- the relative entropy. We find that our relations are compatible Markov approximation [9–17]. In a Markovian evolution, this with Markovian ergodic dynamics, where the system loses all information flow is one-way and quickly leads to an unwanted the information. total loss of coherence and other quantum characteristics. Us- ing structured environments, it may be possible to reduce information loss of the associated quantum system. On the other hand, an important statistical mechanical at- Finally, we have considered a particular structured environ- tribute of a system interacting with an environment, with the ment model, where a central qubit interacts with a collection later being in a thermal state, is the ergodicity of the sys- of mutually non-interacting spins in thermal states at an ar- tem. A physical process is considered to be ergodic, if the bitrary temperature. A spin-bath model of this type, which statistical properties of the process can be realized from a has been considered previously in the literature [10, 11, 16, long-time averaged realization. In the study of the realiza- 17, 35], shows a highly non-Markovian nature. Here we have tion of a thermal relaxation process, ergodicity plays a very derived the reduced dynamics of a particular spin-bath model important role [3, 18, 19]. It also has important applications without the weak coupling and Born-Markov approximations. arXiv:1707.08963v2 [quant-ph] 19 Mar 2018 in quantum control [20–23], quantum communication [24], Subsequently, we investigate the information loss and non- and beyond [25, 26]. Here we intend to capture the notion ergodicity, and find the status of the uncertainty for this sys- of “non-ergodicity” from the perspective of quantum chan- tem. nels, i.e. considering only the reduced dynamics of a quantum system interacting with an environment. In the framework of open quantum systems, a rigorous study on ergodic quantum The organization of the paper is as follows. In Section II, channels can be found in [27]. Ergodic quantum channels are we present the definitions for loss of information and non- channels having a unique fixed point in the space of density ergodicity. We derive the uncertainty relations between infor- matrices [28]. Non-ergodicity of a dynamical process can then mation loss and non-ergodicity in Section III. In Section IV, be quantified as the amount of deviation from a ergodic pro- we consider the central spin model, derive the reduced dy- cess in open system dynamics. namics of the central qubit, and analyze the corresponding in- In this work, we find a connection between information loss formation loss and non-ergodicity. We conclude in Section of a general open quantum system and non-ergodicity therein. V. 2

II. DEFINITIONS: MEASURES FOR LOSS OF of initial state, the entire information D(ρ1(0), ρ2(0)) is lost INFORMATION AND NON-ERGODICITY for arbitrary inputs ρ1(0), ρ2(0). Later in the paper, we draw a connection between average loss of information and non- Before proceeding to the main results, let us deﬁne the two ergodicity of the underlying dynamics, with the later being primary quantities under present investigation, i.e. loss of in- deﬁned in the succeeding subsection. formation and a measure of non-ergodicity, based on distance measures. B. Non-ergodicity

A. Loss of information Ergodicity plays an important role in statistical mechanics, to describe the realization of relaxation of the system to the We quantify the loss of information in quantum systems due thermal equilibrium. The ergodic hypothesis states that if to environmental interaction, in terms of distinguishability a system evolves over a long period of time, the long time- measures for quantum states. The loss of information, denoted averaged state of the system is equal to its thermal state cor- by I∆(t), at any instant of time, can be quantified by the max- responding to the temperature of the environment with which imal difference between the initial distinguishability between the system is interacting. Ergodicity can also be defined in a pair of states, ρ1(0), ρ2(0), and that for the corresponding terms of observables. For any observable f , if its long time time evolved states ρ1(t) = Φ(ρ1(0)), ρ2(t) = Φ(ρ2(0)) at time average, h f iT is equal to its ensemble average, h f ien, the dy- t, where Φ denotes the open quantum evolution of the initial namics is considered to be ergodic for the observables. Here states. Mathematically, it is given by the time and ensemble averages of the observable are respectively defined as I∆(t) = max (D(ρ1(0), ρ2(0)) − D(ρ1(t), ρ2(t))) , (1) ρ1(0), ρ2(0) 1 Z τ h f¯i = lim Tr[ f ρ(t)] = Tr[ f ρ¯]; h f ien = Tr[ f ρth]. where the distance measure D(ρ, σ) must satisfy the following τ→∞ τ 0 conditions: Ergodicity further assumes the equality of h f¯i and h f ien, in- P1. D(ρ, σ) ≥ 0 ∀ density matrices ρ, σ. dependent of the initial state of the evolution. Therefore, non- P2. D(ρ, ρ) = 0 ∀ ρ and D(ρ, σ) = 0 ⇐⇒ ρ = ergodicity of the dynamics for the observable can be quanti- σ, ∀ ρ, σ. fied by the difference between the time average and ensemble average, i.e. by |h f¯i − h f ien| = |Tr[ f (¯ρ − ρth)]|. Based on P3. D(Φ(ρ), Φ(σ)) ≤ D(ρ, σ) ∀ ρ, σ and ∀ completely these understandings of ergodicity of a dynamics, we define positive trace preserving maps Φ(·), on the space of den- a measure of non-ergodicity as the distance between the long sity operators, B(H), on the Hilbert space H. time-averaged state (¯ρ) of the system and its corresponding thermal state (ρ ), and so is given by The class of distance measures satisfying these conditions, in- th clude trace distance, Bures distance, Hellinger distance [36– N (¯ρ) = D(¯ρ , ρth). (4) 38]. Though the von Neumann relative entropy and Jensen- Shannon divergence also satisfy the aforementioned condi- Here we impose two further conditions on the allowed dis- tions, they are not generally considered as geometric distance measures: tances, since they certain other metric properties. But also note here that the square root of Jensen-Shannon divergence P4. The measure must be symmetric, i.e D(ρ, σ) = does satisfy metric properties [39–41] and can be considered D(σ, ρ), ∀ρ, σ. as a valid distance measure. It is also important to men- P5. The measure must satisfy the triangle in- tion that all the aforementioned valid distance measures are equality, given by D(ρ, σ) ≤ D(ρ, κ) + bounded. D(κ, σ), ∀ density matrices ρ, σ, κ. Loss of information for time-averaged states: To draw the connection with non-ergodicity, discussed below, we now The conditions P1-P5 are satisfied by the geometric distance define the long time-averaged state as measures like trace distance, Bures distance, and Hellinger distance. Note the von Neumann relative entropy neither sat- 1 Z τ ρ¯ = lim ρ(t)dt. (2) isfies the symmetry property nor the triangle inequality and τ→∞ τ 0 hence we cannot use it directly for our investigation. However, The information loss for the time-averaged state, which we we will later show the possibility of overcoming such “short- call “ average loss of information”, can then be defined as comings” of the relative entropy distance. Interestingly, it has been shown [40] that Jenson-Shannon divergence satisfies the I¯∆ = max (D(ρ1(0), ρ2(0)) − D(¯ρ1, ρ¯2)) , (3) symmetry property and for its square root, the triangle in- ρ (0), ρ (0) 1 2 equality holds. Therefore, the square root of Jensen-Shannon which is lower and upper bounded by 0 and 1 respectively. divergence can also be taken as a proper distance measure for From Eq. (3) we can infer that, when the open system dy- our investigation. Note that the measure of non-ergodicity, namics has a unique steady state or fixed point, independent given in (4), depends on the initial state. Hence, to obtain a 3 measure of non-ergodicity which is state-independent, we in- entropy. As before, we can define a state-independent mea- troduce sure of non-ergodicity as

N M N M(Rel) = max (¯ρ), (5) N = max S (¯ρ||ρth), (11) ρ(0) ρ(0) where maximization is performed over all initial states (ρ(0)). The above deﬁntion and the inequality (10) leads to another uncertainty relation

III. CONNECTING INFORMATION LOSS WITH q ¯T M(Rel) T NON-ERGODICITY I∆ + 2N ≥ max D (ρ1(0), ρ2(0)) , (12) ρ1(0), ρ2(0)

With the definitions given in the preceding section, we now in terms of trace and relative entropy distances. But it is to be establish a connection between loss of information and non- noted that there is a certain limitation in this relation, because ergodicity. For the distance measures, which satisfy P1-P5, of the fact that the relative entropy is not a bounded function. we obtain When supp ρ * supp ρth, the relative entropy diverges. One such example is obtained for the zero temperature bath, where D(¯ρ , ρ¯ ) ≤ N (¯ρ ) + N (¯ρ ). 1 2 1 2 ρth = |0ih0| is pure. In that case, the relation (12) becomes trivial. But in that case, we can find state- dependent uncertainty Using Eq. (3), we therefore have the inequality relations by defining state- dependent information loss as

I¯∆ ≥ max (D(ρ1(0), ρ2(0)) − (N (¯ρ1) + N (¯ρ2))) . (6) ρ1(0), ρ2(0) I¯∆(¯ρ1, ρ¯2) = (D(ρ1(0), ρ2(0)) − D(¯ρ1, ρ¯2)) . (13)

It draws a direct connection between non-ergodicity and loss This will lead us to the state-dependent uncertainty relation of information in open system dynamics. Using the state- r independent measure of non-ergodicity (Eq. (5)), we can ar- X NRel(¯ρ ) rive at an uncertainty relation between information loss and a I¯T (¯ρ , ρ¯ ) + i ≥ DT (ρ (0), ρ (0)). (14) ∆ 1 2 2 1 2 measure of non-ergodicity, given by i=1,2 ¯ M I∆ + 2N ≥ max (D(ρ1(0), ρ2(0))) . (7) But other than these extreme cases, the relation (12) works ρ1(0), ρ2(0) perfectly. The above relation is valid for any distance measure which Note that the distinguishability measures like trace dis- satisﬁes the conditions P1-P5, and for any quantum system, tance, Bures distance and Jensen-Shanon divergence, men- interacting with an arbitrary environment. tioned earlier, not only satisﬁes all the conditions P1-P5, but In this paper, we will mainly work on the uncertainty re- they are also bounded. But in the cases of some unbounded lation based on the distance measure given by DT (ρ, σ) = distance measure, to avoid the triviality of the uncertainty re- 1 lation (7), we can use the state-dependent uncertainty relation 2 Tr|ρ−σ| for pairs of states ρ and σ. The importance of quantum relative entropy [42–44] as a “distance-type” measure, notwithstanding its inability in satisfying symmetry and other X ¯ relations, from the perspective of quantum thermodynamics is I∆(¯ρ1, ρ¯2) + N (¯ρi) ≥ D(ρ1(0), ρ2(0)). (15) unquestionable, and hence obtaining uncertainity relation in i=1,2 terms of quantum relative entropy can be interesting. Towards this aim, we use a relation between relative entropy and trace A. Qubits distance [45], given by

S (ρ||σ) ≡ Tr[ρ(log ρ − log σ)] ≥ 2(DT (ρ, σ))2. (8) Upto now, we have considered an arbitrary density matrix of arbitrary dimension. Let us now restrict to the case of a two- The above inequality helps us to overcome the drawbacks of level system (TLS) as a simple example to further understand relative entropy for not satisfying P4 and P5. Let us ﬁrst the connection between non-ergodicity and information loss. rewrite (6) in terms of trace distance as For a TLS, the pair of states maximizing the trace distance is located on the antipodes of the Bloch sphere i.e., the pair ¯T ≥ T − NT NT I∆ max D (ρ1(0), ρ2(0)) (¯ρ1) + (¯ρ2) . (9) of states consists of pure and mutually orthogonal states [46]. ρ1(0), ρ2(0) Therefore in the case of trace distance, the uncertainty relation Using inequalities (8) and (9), we arrive at (7), for a qubit, reads as   r r    NRel(¯ρ ) NRel(¯ρ ) I¯T + 2N M(T) ≥ 1. (16) ¯T ≥  T −  1 2  ∆ I∆ max D (ρ1(0), ρ2(0))  +  , ρ1(0), ρ2(0)   2 2  Similarly, the uncertainty relation given in (12) reduces to (10) Rel where N (¯ρi) = S (¯ρi||ρth) denotes the measure of non- q ¯T M(Rel) ergodicity for the time-averaged stateρ ¯i in terms of relative I∆ + 2N ≥ 1. (17) 4

Let us now consider a simple Markovian model, where a qubit IV. NON-ERGODICITY AND INFORMATION is weakly coupled with a thermal bosonic environment. In ab- BACK-FLOW IN A CENTRAL SPIN MODEL sence of any external driving Hamiltonian, the qubit eventu- ally thermally equilibrates with the environment. Under Born- In this section, we consider a speciﬁc non-Markovian Markov approximation, the master equation for this model is model and study the status of uncertainty relation derived in given by Sec III. The system here consists of a single qubit, interacting with N number of non-interacting spins. The total Hamilto- i 1 ρ˙(t˜) = [ρ(t˜), H0] + γ(n + 1) σ−ρ(t)σ+ − {σ+σ−, ρ(t˜)} nian of the system, governing the dynamics, is given by ~ 2 + γn σ ρ(t˜)σ − 1 {σ σ , ρ(t˜)} , + − 2 − + H˜ = H˜ + H˜ + H˜ , (21) (18) S B I where H0 = ~Ω0|1ih1| is the Hamiltonian of the system, γ is where the system Hamiltonian H˜S , bath Hamiltonian H˜ B, and a constant parameter and n = 1/(exp(~Ω0/KT˜m) − 1) is the interaction Hamiltonian H˜ I are respectively given by Planck number. Here σ+ and σ− are respectively the raising and lowering operators of the TLS, with |1i being the excited H˜S = ~gω0σz, state of the same. The solution of the Markovian master equa- ˜ ω PN i HB = ~g N i=1 σz, (22) tion in (18) is given by PN H˜ = ~g √α σ σi + σ σi + σ σi . I N i=1 x x y y z z

ρ(t˜) = ρ11(t˜)|1ih1| + ρ22(t˜)|0ih0| + ρ12(t˜)|1ih0| + ρ21(t˜)|0ih1|, Here σk, k = x, y, z are the Pauli spin matrices, the superscript ‘i’ represents the ith spin of the bath, g is a constant factor with with the dimension of frequency, ω0 and ω are the dimensionless parameters characterizing the energy level differences of the ρ (t˜) = ρ (0)e−γ(2n+1)t˜ + n 1 − e−γ(2n+1)t˜ , 11 11 2n+1 system and the bath respectively and α denotes the coupling ρ22(t˜) = 1 − ρ11(t˜), constant of the system-bath interaction. By using the total (2n+1)t˜ ρ (t˜) = ρ (0) exp −γ − 2iΩ t˜ . N 12 12 2 0 P i angular momentum operators Jk = σk, and the Holstein- i=1 One can find from the solution given above that the long Primakoff transformation, given by time-averaged state for this evolution is independent of ini- √ † !1/2 √ † !1/2 tial states and equal to the thermal state corresponding to † b b b b ˜ J+ = Nb 1 − , J− = N 1 − b, the temperature of the bath Tm, which can be expressed as 2N 2N p|0ih0| + (1 − p)|1ih1|, with p = 1/(1 + exp(−~Ω0/KT˜m)). Hence the dynamics is ergodic and we find that the informa- the bath and interaction Hamiltonians can now be rewritten as ¯T tion loss I∆ = 1; i.e. the system loses all its information. It is also noteworthy that Markovianity of a quantum evolution b†b does not mean it will be ergodic. An example of such Marko- H˜ B = −~gω 1 − , N vian non-ergodic evolution is the dephasing channel expressed b†b 1/2 † b†b 1/2 H˜ I = 2~gα σ+ 1 − b + σ−b 1 − (23) by the master equation √ 2N 2N b†b − ~gα Nσz 1 − N . ρ˙ = iΩ0[σz, ρ] + γd (σzρσz − ρ) (19) We consider the initial (uncorrelated) system-bath state as Here the Lindblad operator is in the same basis as the system ρS (0) ⊗ ρB(0). Let us take the initial system qubit as ρS (0) = Hamiltonian σ . A system interacting with a bosonic environ- ρ11(0)|1ih1| + ρ22(0)|0ih0| + ρ12(0)|1ih0| + ρ21(0)|0ih1| and the z ˜ ˜ ment can lead to such an evolution [3]. The solution of this initial bath state to be a thermal state ρB(0) = exp(−HB/KT) ˜ equation is given by in an arbitrary temperature T with K being the Boltzmann constant. The reduced dynamics of the system state can then be calculated by tracing out the bath degrees of freedom and ρ11(t˜) = ρ11(0), is given by ρS (t) = TrB exp (−iHt) ρS (0) ⊗ ρB(0) exp (iHt) . ρ22(t˜) = ρ22(0), (20) −2(iΩ+γd)t Where ρ12(t˜) = ρ12(0)e . H˜ KT˜ We realize from Eq. (20) that under this particular evolu- H = , t = gt˜, and T = , ~g ~g tion, the system will decohere, but the digonal elements of the density matrix will remain invariant, leading to infinitely are dimensionless, specifying Hamiltonian, time and temper- many fixed points for the dynamics. So this particular evolu- ature respectively. After solving the global Schrodinger¨ evo- tion will certainly be non-ergodic, since there exists infinitely lution, the reduced dynamics can be exactly obtained [17, 47] many fixed points and the time averaged state will depend on as the initial state of the system. This gives a definite example ! which proves that Markovianity does not imply ergodicity of ρ11(t) ρ12(t) ρS (t) = , (24) the dynamics. ρ21(t) ρ22(t) 5 where 0.2 ρ (t) = ρ (0)(1 − Θ (t)) + ρ (0)Θ (t), α = 0.1 , T = 1 11 11 1 22 2 (25) ρ12(t) = ρ12(0)∆(t), 0.15 N=500 with

2 − ω (n/N−1) t PN n α2 − n/ N sin(ηt/2) e T , T(t) Θ1( ) = n=0( + 1) (1 2 ) η/2 Z IΔ

0 2 − ω (n/N−1) N=100 PN 2 sin(η t/2) e T Θ2(t) = n=0 nα (1 − (n − 1)/2N) η0/2 Z , 0.05 N=50 PN −i(Λ−Λ0)t/2 θ ∆(t) = n=0 e cos(ηt/2) − i η sin(ηt/2) − ω − 0 θ0 0 e T (n/N 1) × cos(η t/2) + i 0 sin(η t/2) , η Z 0 0 5 15 20 N ω t P − T (n/N−1) Z = n=0 e , FIG. 1. (Color online) Time-dynamics of instantaneous information q √ 2 ω 2n+1 2 n loss. We plot IT (t) on the vertical axis against t on the horizontal axis, η = 2 ω0 − 2N − α N 1 − 2N + 4α (n + 1)(1 − 2N ), ∆ for different values of the total number of bath spins N, where the system-environment duo governed by the Hamiltonian in Eq. (21) q √ 2 0 ω 2n−1 2 (n−1) is being considered. We set α = 0.1 and T = 1. All quantities are η = 2 ω0 − 2N − α N 1 − 2N + 4α n(1 − 2N ), dimensionless. √ 2n+1 θ = 2 ω0 − ω/2N + α N 1 − , 2N 0.3 √ 0 2n−1 θ = −2 ω0 − ω/2N − α N 1 − , α = 0.1 , N = 200 2N 0.25 Λ = −2ω 1 − 2n+1 − √α , 2N N Λ0 = −2ω 1 − 2n−1 − √α . 0.2 T = 10 2N N The time-averaged state for this system can then be calculated T(t) IΔ as T = 1 ρ¯ = ρ (0)(1 − Θ¯ ) + ρ (0)Θ¯ , 0.1 11 11 1 22 2 (26) ρ¯12 = ρ12(0)∆¯ , 0.05 with T = 0.5 − ω (n/N−1) ¯ PN 2 1 e T 0 Θ1 = 2(n + 1)α (1 − n/2N) 2 , n=0 η Z 0 5 t 15 20 − ω (n/N−1) ¯ PN 2 − − 1 e T Θ2 = n=0 2nα (1 (n 1)/2N) η02 Z , T FIG. 2. (Color online) I∆ (t) vs t for various temperatures. We set N = 200 and α = 0.1. The physical system is the same as in Fig.1. ∆¯ = 0. All quantities are dimensionless. Note that in general the coherence of the time-averaged state will vanish as ∆¯ = 0. But there are specific resonance conditions under which there can be non-zero coherence present In Figs.1,2,3, the instantaneous loss of information is in the time-averaged state [17]. But in this work, we will not depicted with time for different values of the number of ˜ consider such situations. bath-spins (N), temperature (T) and system-bath interaction Before investigating the uncertainty relation in terms of strength (α) respectively, by keeping other parameters fixed. trace distance given in (7), we explore the behavior of loss From the figures, we deduce the following: of information at instantaneous time with different parameters Observation 1: The instantaneous loss of information shows involved in this dynamics. For such study, let us restrict our- oscillatory behavior whose amplitude decreases with time. selves to the set of pure initial qubits over which the optimiza- Observation 2: The increase of number of spins of the bath, tion involved in (7) is performed. In particular, we take the ini- in temperature, as well as in the interaction strength can be θ θ −iφ seen as increase of influence of bath on the system. Hence, tial pair of orthogonal pure states to be cos 2 |1i + sin 2 e |0i θ θ −iφ expectantly in all cases, the loss of information increases with and sin 2 |1i − cos 2 e |0i, with 0 ≤ θ ≤ π, 0 ≤ φ < 2π. The instantaneous and average information losses in this case are increase of the above system parameters. given respectively by Let us now check the uncertainty relation given in (16) for the qubit case, taking the same initial pair of pure or- T ¯(T) ¯ ¯ I∆ (t) = Θ1(t) + Θ2(t), I∆ = Θ1 + Θ2. (27) thogonal states and the thermal state at arbitrary temperature 6

0.175 1.9997 2.006 (a) 2.005 (b) 0.15 T = 1 , N = 200 1.99965 2.004 α = 0.1 0.125 T = 0.0001 2.003 T = 0.01 1.9996 2.002 1.99955 2.001 2 T(t) IΔ α = 0.07 1.9995 1.999 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0.05 1.795 1.04 α = 0.05 1.79 (c) 1.035 (d) 0.025 1.785 1.03 1.78 T = 1 1.025 T = 100 0 1.775 0 5 t 15 20 1.77 1.02 1.765 1.015 FIG. 3. (Color online) IT (t) with t for three different values of 1.76 1.01 ∆ 0 5 10 15 20 25 30 0 5 10 15 20 25 30 system-bath coupling (α). We set T = 1 and N = 200. The physical system is the same as in Fig.1. All quantities are dimensionless. FIG. 4. (Color online) Behavior of the uncertainty for different T M(T) system-bath interaction strength. We denote the values of I∆ +2N ~gω for different α, as vertical bars on the horizontal axis that represents ρ = p |0ih0| + (1 − p )|1ih1|, where p = 1 1 + tanh 0 . th 1 1 1 2 KT˜ α. Here, N = 1000, and the different panels are for different T. The After performing the maximization, we find system considered is the one given by the Hamiltonian in Eq. (21). At already a moderate interaction strength, the quantity converges to ¯T M(T) ¯ ¯ ¯ I∆ + 2N = Θ1 + Θ2 + 2|p1 − Θ1|. (28) a certain value, which is higher than unity. The converged value goes close to unity with the increase of temperature. All quantities are We now examine the conditions for which the uncertainty re- dimensionless. lation (16) saturates. Note that for the ergodic situations, i.e. if the steady state is unique and is equal to the thermal state, the information loss is equals to unity, leading to a trivial equality 1.15 (a) 1.15 (b) in (16). Keeping N fixed to 1000 and fixing the temperature to 1.14 1.14 T M(T) ¯ N=100 N=200 different values, we investigate the values of I∆ + 2N for 1.13 1.13 increasing interacting strength. We observe that the sum goes 1.12 1.12 close to unity for a strong interaction strength as depicted in 1.11 1.11 Fig.4 for high temperature. T M(T) 0 5 10 15 20 25 30 0 5 10 15 20 25 30 In Figs.5 and6, we analyze the sum I∆ + 2N as the number of bath spins are ramped up from 100 to 1000. Scru- 1.14 1.13 (c) (d) tinizing these figures, we can safely conclude that with the 1.13 increase in number of spins of the bath, or in the bath tem- 1.12 N = 500 perature, or in the system-bath interaction strength, the sum 1.12 N = 1000 (T) M(T) N 1.11 I∆ + 2 goes very close to unity in this qubit case, provided the optimization involved is restricted to pure qubits. 1.11 However, numerical evidence strongly suggests that for this 1.1 non-Markovian model, given in Eq. (21), there will be no 0 5 10 15 20 25 30 0 5 10 15 20 25 30 non-trivial situation when the uncertanity relation in Eq. (16) saturates to unity, provided the maximization is carried out T M(T) over pure state. FIG. 5. (Color online) Bar diagram for I∆ + 2N against α. The situation here is the same as in Fig.4, except that the di fferent panels We find that the saturation value of Θ¯ 1 and Θ¯ 2, when α −→ ∞, are respectively given by are for different N, for fixed T which is set to be 10. All quantities are dimensionless.

− ~ω (n/N−1) Θ¯ sat = 1 Pn=N 1 e KT , 1 2 n=0 4+N (1−(2n+1)/2N) Z (n+1)(1−n/2N)2 − ~ω (n/N−1) (29) leads to the equality in (16). It is interesting that in the men- Θ¯ sat = 1 Pn=N 1 e KT . 2 2 n=0 (1−(2n−1)/2N)2 Z tioned limit, the non-ergodicity measure is finite and equals 4+N n(1−(n−1)/2N) to 3/8. Therefore, we find a non-ergodic situation, where If the interaction Hamiltonian given in Eq. (22), is considered the equality of the uncertainty relation holds.When the equal- in absence of the z − z interaction, the saturated values of Θ¯ 1 ity of the mentioned relations hold for a non-ergodic evolu- ¯ ¯ sat ¯ sat and Θ2 in the limit N → ∞, α → ∞ will be Θ1 = Θ2 = 1/8. tion, these relations imply that when the non-ergodicity of the In the infinite temperature limit, we have p1 = 1/2, which dynamics increases, the information loss in the system de- 7

2.025 V. CONCLUSION 2.04 (a) 2.02 (b) 2.03 2.015 In open quantum dynamics, the information exchange be- 2.02 N = 100 2.01 N = 200 tween the system and bath plays an important role, while the time-evolved state’s correspondence with the Gibb’s ensem- 2.01 2.005 ble conspire to imply the ergodic nature of the system. In 2 2 this article, we establish a relation between loss of informa- 1.99 1.995 tion and a measure of non-ergodicity. Both the definitions are 0 5 10 15 20 25 30 0 5 10 15 20 25 30 2.01 2.006 given in terms of distinguishability, which can be measured by 2.008 (c) 2.005 (d) a suitably chosen distance measure. We have shown that the 2.006 2.004 information loss and the quantifier of non-ergodicity follow an 2.004 N = 500 2.003 N = 1000 uncertainty relation, valid for a broad class of distinguishabil- 2.002 ity measures, which includes trace distance, Bures distance, 2.002 2.001 Hilbert-Schmidt distance, Hellinger distance, and square root 2 2 of Jensen-Shannon divergence. We have further considered 1.998 1.999 0 5 10 15 20 25 30 0 5 10 15 20 25 30 trace distance between a pair of quantum states as a specific distinguishability measure and connected the corresponding information loss with non-ergodicity, which is now defined in FIG. 6. (Color online) The panels here are the same as in Fig.5, terms of relative entropy between the time-averaged state and except that T = 0.01. All quantities are dimensionless. the thermal state, maximized over all possible initial states. We have shown that in a Markovian model, the uncertainty relation saturates and shows a complete information loss. We also considered a structured environment model of a central quantum spin interacting, according to Heisenberg interaction, with a collection of mutually non-interacting quantum spin-half particles, leading to non-Markovian dynamics. In this case, we observed that with the increase of temperature, number of spins in the bath, and the system-bath interaction strength, there is increase in information loss at instantaneous time. In this scenario, we found that the uncertainty relation shows a nonmonotonic behavior with the increase of temperature for small values of interaction strength, provided the op- creases. Nonergodic dynamics are, in general, good for in- timization is performed over pure qubits. Moreover, we found formation processing as they have less chance of leakage of that although the uncertainty relation in this model goes close information compared to ergodic dynamics. In particular, for to the saturation value, it fails to saturate exactly. Interestingly non-ergodic evolution for which the uncertainty relations dis- however, we found that in absence of z − z system-bath inter- cussed in this paper are equalities, the loss of information can action and in the limit of large bath size, high bath tempera- be quantified by and attributed to the nonergodicity in the evo- ture, and strong system bath interaction, uncertainty relation lution. It is also important to mention that spin bath models between information loss and non-ergodicity, based on trace do not always indicate a shows non-ergodic dynamics. In a re- distance measure, is saturated, providing a non-ergodic situ- cent work [48], such dynamics hase been considered with the ation that saturates the uncertainty. The uncertainty relations Born-Markov approximation region to find the effective re- have been obtained by using the usual notion of the ergodicity duced dynamics. It is shown in the mentioned work that there where we require to have the unique fixed point, of the dy- are situations where a unique fixed point (stationary state) can namics, to be thermal. We note that the entire analysis goes exist for the evolution, and hence in those situations, the dy- through for a more general definition, where a single fixed namics is ergodic [27]. point is sufficient to imply ergodicity.

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