Quantum scrambling and the growth of mutual information
QQuantum scrambling and the growth of mutualinformation
Akram Touil and Sebastian Deffner Department of Physics, University of Maryland, Baltimore County, Baltimore,MD 21250, USAE-mail: [email protected] E-mail: [email protected] Abstract.
Quantum information scrambling refers to the loss of localrecoverability of quantum information, which has found widespread attention fromhigh energy physics to quantum computing. In the present analysis we proposea possible starting point for the development of a comprehensive framework forthe thermodynamics of scrambling. To this end, we prove that the growth ofentanglement as quantified by the mutual information is lower bounded by thetime-dependent change of Out-Of-Time-Ordered Correlator. We further showthat the rate of increase of the mutual information can be upper bounded by thesum of local entropy productions, and the exchange entropy arising from the flowof information between separate partitions of a quantum system. Our results areillustrated for the ion trap system, that was recently used to verify informationscrambling in an experiment, and for the Sachdev-Ye-Kitaev model.
1. Introduction
One of the most intriguing problems in theoretical physics is the information paradox[1–3], which suggests that physical information crossing the event horizon couldpermanently disappear in a black hole. In essence, the paradox originates in theunresolved incompatibility of current formulations of quantum mechanics and generalrelativity. To date, many possible solutions have been proposed, some as esoteric asthe many-worlds interpretation of reality [2, 4], whereas others are rooted in quantuminformation theory [5, 6].A particularly fruitful concept has been dubbed quantum information scrambling [7]. Within this paradigm, information that passes the event horizon is quickly andchaotically “scrambled” across the entirety of the horizon. Thus, the information onlyappears lost, as no local measurement allows to fully reconstruct the original quantumstate [8–10]. In recent years, the study of quantum information scrambling has led tonew physical concepts, such as the black hole complementarity and the holographicalprinciple [11, 12]. In addition, information scrambling has found attention in highenergy physics [13,14], quantum information [15,16], condensed matter physics [17,18],and quantum thermodynamics [19–21].Remarkably, it has also been recognized that exploiting the AdS/CFT duality [22]and the “ER=EPR-conjecture” [23], the information scrambling dynamics of blackholes can be studied with analog quantum systems. Loosely speaking, the dynamicsof two black holes connected through an Einstein-Rosen bridge can be mathematically a r X i v : . [ qu a n t - ph ] M a y uantum scrambling and the growth of mutual information ‡ .Therefore, in the present analysis we prove upper and lower bounds on thequantum mutual information. As main results, we find (i) that the time-dependentmutual information is lower bounded by the change of the OTOC, and (ii) that therate of change of the mutual information is upper bounded by the sum of the stochasticentropy productions [43] in the separate partitions of a quantum system. Our findingsare illustrated for the experimental system described in Ref. [24] and an example of aquantum chaotic system, the SYK model.
2. Time-dependent mutual information and the change of the OTOC
Imagine a quantum system S that can be separated into two partitions, A and B . Thetotal system S evolves under unitary dynamics, and the quantum state is initiallyprepared as a product, ρ S (0) = ρ A (0) ⊗ ρ B (0). Typically, quantum informationscrambling then occurs in situations, in which ρ S (0) is chosen to be pure, and theunitary dynamics of S yields the growth of entanglement between A and B . Thismeans, in particular, state tomography on only A is not longer sufficient to reconstruct ρ A (0) for any time t > O ( t ) = (cid:68) O † A O † B ( t ) O A O B ( t ) (cid:69) . (1)Here, O A and O B are local operators acting only on A and B , respectively. Morespecifically, we have O A ≡ o A ⊗ I B , O B ≡ I A ⊗ o B , and O B ( t ) = U † ( t ) O B (0) U ( t ),where U ( t ) denotes the unitary time evolution operator of S . It has been argued,that O ( t ) characterizes the spread of the operator O B ( t ) as it evolves in time, whichtracks how information is scrambled from A to B [7–10]. The average in Eq. (1) isoften taken over a thermal state in S [7, 37], which is, however, not necessarily aninstrumental choice [37]. ‡ For obvious reasons, the projective measurements considered in Ref. [19] are neither feasible norpractical in complex many body systems uantum scrambling and the growth of mutual information Figure 1.
Sketch of a black hole scrambling quantum information. A characterizes the information falling into the black hole, A determines the initialstate of the black hole, A the Hawking radiation, and A the remaining blackhole. Note, however, that (to the best of our knowledge) there is no rigorous proof ofthe existence of operators correctly tracking information scrambling in any physicalscenario. For instance, taking operators that commute with the Hamiltonian of S results in a time-invariant OTOC. Thus, one often takes an average over operators,rather then working with O ( t ) directly [44, 45]. For the following analysis, we will be motivated by the conceptual framework thatmaps notions from high energy physics onto a quantum information theoretic language[23]. To this end, imagine a situation in which some quantum information falls acrossthe event horizon of a black hole, and we can describe the dynamics of that blackhole by a scrambling (entangling) unitary map, U ( t ). The set-up is depicted in Fig. 1,which is similar in spirit to Ref. [46]. Hosur et al consider that ( A , A ) and ( A , A )describe EPR pairs, connected through the scrambling unitary U . More specifically, A determines the initial information thrown into the black hole, A encodes the initialstate of the black hole, A the Hawking radiation and A the remainder of the blackhole.For such scenarios, it has been shown [47–49] that (cid:104) O A O A ( t ) O A O A ( t ) (cid:105) avg = 2 −I (2) A ,A A , (2)where I (2) i,j is the R´enyi-2 mutual information between the partitions i and j . Noticethat A appears in the subscript of I (2) , which is related to why knowing A and A specifies A . In the case of a unitary map modeling the scrambling of information in ablack hole the Hawking radiation ( A ) and knowledge of the initial state of the blackhole ( A ) is enough to reconstruct the quantum state described by A . This can be uantum scrambling and the growth of mutual information § , we can write I (2) i,j = S i (2) + S j (2) = − ln(tr (cid:8) ρ i (cid:9) ) − ln(tr (cid:8) ρ j (cid:9) ) . (3)Furthermore, the average in Eq. (2) is taken over the Haar measure on the unitarygroup U(d) with (cid:90) Haar dU = 1 , (4)and where we have for an arbitrary function f and ∀ V ∈ U ( d ), (cid:90) Haar dU f ( U ) = (cid:90) Haar dU f ( V U ) = (cid:90)
Haar dU f ( U V ) . (5)It is interesting to note that for quantum systems comprised of qubits, such as theexperimental system analyzed in Ref. [24], the Haar average is equivalent to an averageover the Pauli group for each operator [45].Equation (2) can now be used to relate the OTOC with a thermodynamicallyrelevant quantity, the quantum mutual information. Adopted to our current purposes A and A are operators that live on subsystem A , and A and A live on B . Therefore,we consider 1 − (cid:104) O A O B ( t ) O A O B ( t ) (cid:105) avg = 1 − −I (2) A,B ≤ I (2)
A,B , (6)where we identified A ≡ O A and A A ≡ O B . To justify this identification, wenote that knowledge about the degrees of freedom A and A is enough to infer theinformation encoded in A . The same argument holds for any closed quantum systemand any unitary evolution U ( t ). In this case, the analog of “Hawking radiation”, is asubset of the degrees of freedom that is enough to reconstruct the initial information.Note that the R´enyi-2 mutual information is upper bounded by the quantummutual information I [50,51]. This is a direct consequence of the strong subadditivityof the von Neumann entropy [50, 51]. We have I ( t ) = S A ( t ) + S B ( t ) − S S ( t ) , (7)where S i = − tr { ρ i ln( ρ i ) } is the von Neumann entropy of system i with density matrix ρ i . Note that S S ( t ) = S S (0) for unitary dynamics. Thus, we immediately obtain1 − (cid:104) O A O B ( t ) O A O B ( t ) (cid:105) avg ≤ I ( t ) . (8)Now introducing the notation ¯ O ( t ) ≡ (cid:104) O A O B ( t ) O A O B ( t ) (cid:105) avg and noticing that bydefinition ¯ O (0) = 1, we can write I ( t ) ≥ ¯ O (0) − ¯ O ( t ) , (9)which is true for all times t >
0, and which constitutes our first main result. § For the sake of simplicity we only consider initially pure states. If the composite quantum statewas initially mixed, we would obtain the same results up to additive constants. uantum scrambling and the growth of mutual information O (0) − ¯ O ( t ) is a monotonicallygrowing function, and Eq. (9) asserts that also I ( t ) has to be growing. This isconsistent with intuitive understanding of “scrambling”, which should be equivalentto the growth of entanglement between A and B . We will now continue theanalysis by illustrating Eq. (9) with two important examples, before we discuss thethermodynamic significance of I ( t ). The experimentalverification of quantum information scrambling [24] was conducted with a 7-qubitfully-connected quantum computer with a family of 3-qubit entangling unitaries U ( t ) (cid:107) .These entangling (scrambling) unitaries were constructed from a combination of 1-qubit and 2-qubit gates. Due to the experimental specifics, the observables O A and O B had to be of special form. Therefore, Ref. [24] considered a modified version ofthe OTOC, namely MO ( t ) = (cid:88) φ,O p (cid:68) O † O † P ( t ) O O P ( t ) (cid:69) , (10)where O ≡ | ψ (cid:105) (cid:104) φ | acts on the first qubit, | ψ (cid:105) denotes the state of the qubit, and | φ (cid:105) is the teleported state (the last qubit of the experiment). Moreover, O P ( t ) are Paulimatrices evolved by a scrambling unitary in the Heisenberg picture, and the averageis taken over all Pauli matrices and state vectors | φ (cid:105) .It is relatively easy to see that ∆ MO (cid:39) ∆ O at times close to zero, since O P ( t )has a simple form (in terms of operator complexity). However, in general we have ∆ MO ≤ ∆ O , since the specific average taken in Ref. [24] to compute MO is only anapproximate 1-design, which does not capture the complex dynamics of O P ( t ) as theoperator spreads to the other support (becomes non-local) with time.In Fig. 2 we plot the mutual information (7), together with ∆ O (9) and ∆ MO (10) for the scrambling dynamics of Ref. [24]. As in Ref. [24], B = { , , , , , } refers to the qubits in the experiment, and A = { } is the first qubit. We observe thatall quantities are montonically increasing functions of time t .Furthermore, I (0) = 0 indicates the absence of any scrambling in the system,while I ( t ) = 2 ln(2) indicates maximal scrambling: the maximum value of theinformation that B can know about A (or that A can know about B) is reached.This accentuates an important advantage of I as a measure of scrambling over MO .In general, I max ( t ∗ ) = min { d A , d B } ln(2) is the maximum value of I at the scramblingtime, t ∗ , whereas the maximal value of MO depends on the specifics of the performedexperiment. As a second example to illustrateEq. (9) we choose the Sachdev-Ye-Kitaev (SYK) model [52–54]. This is an exactlysolvable, chaotic many-body system consisting of N interacting Majorana fermionswith random interactions between q of these fermions ( q taken as an even number).The SYK model has found important applications, for instance, as a quantum gravitymodel of a 1 + 1-dimensional black hole [54] in the limit of large N . (cid:107) The exact and rather lengthy expressions for U ( t ) can be found in the methods section of Ref. [24]. uantum scrambling and the growth of mutual information ℐΔΔℳ
Figure 2.
Mutual information I (blue, top line), change of the OTOC ∆ O ( t )(red, middle line), and change of the modified OTOC ∆ MO (orange, bottomline) as function of time. ℐΔ
Figure 3.
Mutual information I ( t ) (blue, upper line) and ∆ O (red, lower line)for the SYK model with N = 10 Majorana fermions, q = 4, J =2, and averagedover 300 realizations, where in each realization we generate a new Hamiltonianwith different random interaction terms. The Hamiltonian can be written as H = ( i ) q (cid:88) ≤ i
3. Stochastic entropy production in quantum scrambling
Quantum information scrambling is an inherently dynamical phenomenon. Especiallyin chaotic quantum systems it is, thus, important to understand the rate with whichinformation is lost to local observation [42]. Moreover, from a thermodynamic pointof view, it appears appealing to relate the rate with which the quantum mutualinformation, I ( t ), grows to the local entropy production in subsystems A and B .Therefore, motivated by analyses of the rate of information production [57–59], wenow seek to upper bound the rate of change of I ( t ) in terms of the thermodynamicresources (locally) consumed while scrambling information. We start by considering the continuity equation for S as expressed in continuousvariables ∂ t ρ S ( x, y ; t ) = − ∇ · j S ( x, y ; t ) . (13)Here, ρ S ( x, y ; t ) = (cid:104) xy | ρ S ( t ) | xy (cid:105) is the density function of the system evaluated invariables x and y . Without loss of generality and for the sake of simplicity, we choose x and y as the coordinates in which ρ A and ρ B are diagonal, respectively, cf. Fig. 4,and j S ( x, y ; t ) denotes the probability current. Note that this choice is made purelyout of mathematical convenience with the sole purpose to be able to relate the rate ofchange of the mutual information to stochastic entropy production: quantities that arebasis independent (see below). It has proven useful in quantum stochastic dynamics uantum scrambling and the growth of mutual information Figure 4.
Sketch of the two subsystems A and B as represented in continuousvariables. to analyze such abstract and “non-experimental” quantities to gain insight into theuniversal behavior, for instance see also Refs. [60, 61]. From an experimental pointof view this choice would be highly impractical, since it requires the instantaneousdiagonalization of ρ A and ρ B . However, rephrasing the current treatment in a moregeneral choice of coordinates, would require to write all expressions in terms of fourvariables (instead of only two) to account for the off-diagonal terms. For the presentpurposes we expect no additional physical insight, and such a choice would only makethe mathematical expressions messier.The corresponding, local continuity equations are obtained by tracing out thecorresponding other subsystem. In particular, we have (cid:82) dx ρ S ( x, y ; t ) = ρ B ( x ; t ) and (cid:82) dy ρ S ( x, y ; t ) = ρ A ( y ; t ). Therefore, we can write ∂ t ρ A ( x ; t ) = − ∇ x · j A ( x ; t ) + j S ( x, t ) . (14)and ∂ t ρ B ( y ; t ) = − ∇ y · j B ( y ; t ) − j S (0 , y ; t ) , (15)where j S ( x, t ) is a boundary term that describes the influx of information from A to B , and j S (0 , y ; t ) is the flow from B to A .Now, again using that at t = 0 subsystems A and B are prepared in a productstate, we can write by simply taking the derivative of Eq. (7)˙ I = − (cid:90) dx ( ∂ t ρ A ) ln ( ρ A ) − (cid:90) dy ( ∂ t ρ B ) ln ( ρ B ) . (16)Employing the local continuity equations (14) and (15), we thus have˙ I = (cid:90) dx ( ∇ x · j A ) ln ( ρ A ) − (cid:90) dx j S ( x, t ) ln ( ρ A )+ (cid:90) dy ( ∇ y · j B ) ln ( ρ B ) + (cid:90) dy j S (0 , y ; t ) ln ( ρ B ) . (17)The latter can be further simplified by partial integration, and we obtain˙ I = − (cid:90) dx j A · ∇ x ln ( ρ A ) − (cid:90) dx j S ( x, t ) ln ( ρ A ) − (cid:90) dy j B · ∇ y ln ( ρ B ) + (cid:90) dy j S (0 , y ; t ) ln ( ρ B ) , (18)for which it is now easy to find upper bounds. uantum scrambling and the growth of mutual information I ≤ | ˙ I| , and then bounding the absolute value withthe Cauchy-Schwarz inequality we can write [58]˙ I ≤ α (cid:18)(cid:90) dx j A ρ A (cid:19) / + γ (cid:18)(cid:90) dx j S ρ S ( x, t ) (cid:19) / + β (cid:18)(cid:90) dy j B ρ B (cid:19) / + γ (cid:18)(cid:90) dy j S ρ S (0 , y ; t ) (cid:19) , (19)where we introduced the Frieden integrals [57, 58], α = (cid:90) dx ρ A ( ∇ x ln ( ρ A )) and β = (cid:90) dy ρ B ( ∇ y ln ( ρ B )) . (20)The Frieden integral is related to the Fisher information [58], but generally depends onthe choice of variables, x and y , and the geometry of the quantum system S . Similarly,we have γ = (cid:90) dx ρ S ( x, t ) (ln ( ρ A )) and γ = (cid:90) dy ρ S (0 , y ; t ) (ln ( ρ B )) , (21)which are geometric terms corresponding to the flow of information across theboundary separating A and B .However, we also immediately recognize the stochastic entropy production [62,63]in subsystems A and B , which reads,˙ S A = (cid:90) dx j A ρ A and ˙ S B = (cid:90) dy j B ρ B . (22)In conclusion, we obtain˙ I ≤ α (cid:16) ˙ S A (cid:17) / + β (cid:16) ˙ S B (cid:17) / + γ | ˙ S E | , (23)where we introduced γ | ˙ S E | to denote to the exchange entropy due to flow ofinformation between A and B .Equation (23) provides an intuitive way to think about information scrambling, orinformation flow between any arbitrary partitions A and B . The mutual informationachieves a maximum if and only if the stochastic irreversible entropy productionswithin A and B as well as the entropy flow between A and B vanish. So far wehave only considered scenarios where the dynamics of the quantum system is drivenby information flow. A fullly thermodynamic formalism, where a system can be incontact with an information reservoir as well as the usual heat and work reservoirs [29],will require a thoroughly developed conceptual framework which is beyond the scopeof the present analysis. The above analysis can be generalized to discrete representations of S . To this end,we consider the von Neumann equation describing the unitary dynamics of ρ S ∂ t ρ S = − i (cid:126) [ H, ρ S ] . (24) uantum scrambling and the growth of mutual information I ( t ) (7) can be written as˙ I = i (cid:126) [tr { [ H, ρ S ] (ln ( ρ A ) ⊗ I B ) } + tr { [ H, ρ S ] ( I A ⊗ ln( ρ B )) } ] , (25)which is mathematical a little more tedious than the continuous case. Therefore, werelegate the technical details of the derivation to the appendix.Expressing the quantum states in Fock-Liouville space and after straightforwardmanipulations we again find ˙ I ≤ A ˙ S A + B ˙ S B + C | ˙ S E | , (26)where as before A , B , and C are discrete versions of the Frieden integral [57, 58],that depend only on the geometry of the problem. Furthermore, ˙ S A and ˙ S B are thestochastic irreversible entropy production [43] in A and B , respectively. Finally, ˙ S E is the entropy (or information) flow between A and B .
4. Concluding remarks
We conclude the analysis with a few remarks on thermodynamic implications. To thisend, note that both quantum systems, A and B , can be considered as open systems,for which the respective other system plays the role of an “environment”. Imaginenow that B is much larger than A in the sense that B becomes a heat reservoir for A . For such a scenario and ultra-weak coupling it was shown in Ref. [63] that thethermodynamic entropy, S S ( t ), is related to the correlation entropy , S cor , and wehave S S ( t ) = S A ( t ) + S B ( t ) + S cor ( t ) . (27)Comparing the latter with the definition of the quantum mutual information (7), weimmediately conclude S cor ( t ) = −I ( t ) for unitary dynamics.This observation indicates that the chaotic spread of quantum information isintimately related to thermalization in quantum systems – a conclusion that canhardly be substantiated by looking only at the OTOC. Note, however, that for theabove identification we had to assume that the heat reservoir B is large comparedto A , and that B remains in equilibrium at all times. More generally, one also hasto account for the entropy that is produced due to the fact that the reservoir B ispushed out of equilibrium through the interaction with A [64, 65]. We leave this moresophisticated analysis for a forthcoming publication [66]. The present analysis provides a possible starting point for the development of acomprehensive framework for the thermodynamics of information scrambling. Inparticular, we related the OTOC with thermodynamically relevant quantities byproving that the change of the OTOC sets a lower bound on the time-dependentquantum mutual information. This bound was demonstrated for two experimentallyrelevant scenarios, namely for a system of trapped ions and the SYK model. Wefurther showed that the rate of increase of the mutual information is upper boundedby the sum of local stochastic entropy productions, and the flow of entropy betweenseparate partitions of the quantum system. uantum scrambling and the growth of mutual information
Acknowledgments
Fruitful discussions with Nathan M. Myers and Bin Yan are gratefully acknowledged.This research was supported by grant number FQXi-RFP-1808 from the FoundationalQuestions Institute and Fetzer Franklin Fund, a donor advised fund of Silicon ValleyCommunity Foundation (S.D).
7. Appendix: Entropy production in Fock-Liouville space
This appendix is dedicated to the technical details that lead to Eq. (26). To this end,we express all quantum states in Fock-Liouville space [67]. In this formalism operatorsare represented as vectors and superoperators as operators, which is convenient tonumerically simplify computations in Hilbert space.We introduce the notation [67], ρ S ≡ | ρ S (cid:105) and (cid:104) ρ | σ (cid:105) ≡ tr (cid:8) ρ † σ (cid:9) , which definesa pre-Hilbert space and completeness is guaranteed by definition. Thus, the vonNeumann equation becomes [68], | ˙ ρ S (cid:105) = W | ρ S (cid:105) and W = 1 i (cid:126) (cid:0) H ⊗ I − I ⊗ H (cid:62) (cid:1) . (B.1)Note, that in contrast to classical stochastic dynamics that are described by ratematrices, the dynamics in Fock-Liouville space is determined by a skew Hermitianmatrix, W .Thus, the rate of change of I ( t ) (7) can be expressed as˙ I = − (cid:88) m,m (cid:48) W m,m (cid:48) ρ m (cid:48) ln ( ρ (cid:48) Am ) − (cid:88) m,m (cid:48) W m,m (cid:48) ρ m (cid:48) ln ( ρ (cid:48) Bm ) , (B.2)where ρ (cid:48) A ≡ ρ A ⊗ I B and ρ (cid:48) B ≡ I A ⊗ ρ B . Using standard tricks from stochasticthermodynamics of adding and subtracting terms we write˙ I = (cid:88) m,m (cid:48) ρ m (cid:48) W m,m (cid:48) ln (cid:18) W m,m (cid:48) ρ (cid:48) Am (cid:48) W m (cid:48) ,m ρ (cid:48) Am (cid:19) + (cid:88) m,m (cid:48) ρ m (cid:48) W m,m (cid:48) ln (cid:18) W m,m (cid:48) ρ (cid:48) Bm (cid:48) W m (cid:48) ,m ρ (cid:48) Bm (cid:19) + 2 (cid:88) m,m (cid:48) ρ m (cid:48) W m,m (cid:48) ln (cid:18) W m (cid:48) ,m W m,m (cid:48) (cid:19) − (cid:88) m (cid:48) ln ( ρ (cid:48) Am (cid:48) ) · (cid:32)(cid:88) m W m,m (cid:48) (cid:33) · ρ m (cid:48) − (cid:88) m (cid:48) ln ( ρ (cid:48) Bm (cid:48) ) · (cid:32)(cid:88) m W m,m (cid:48) (cid:33) · ρ m (cid:48) , (B.3)which is not as involved as it looks. In particular, note that we have independent of uantum scrambling and the growth of mutual information (cid:80) m,m (cid:48) W m,m (cid:48) = 0, and hence˙ I = (cid:88) m,m (cid:48) ρ m (cid:48) W m,m (cid:48) ln (cid:18) W m,m (cid:48) ρ (cid:48) Am (cid:48) W m (cid:48) ,m ρ (cid:48) Am (cid:19) + (cid:88) m,m (cid:48) ρ m (cid:48) W m,m (cid:48) ln (cid:18) W m,m (cid:48) ρ (cid:48) Bm (cid:48) W m (cid:48) ,m ρ (cid:48) Bm (cid:19) + 2 (cid:88) m,m (cid:48) ρ m (cid:48) W m,m (cid:48) ln (cid:18) W m (cid:48) ,m W m,m (cid:48) (cid:19) . (B.4)In complete analogy to the continuous case (18) we can now upper bound ˙ I as˙ I ≤ (cid:88) m,m (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) ρ m (cid:48) ρ (cid:48)− Am (cid:48) · ρ (cid:48) Am (cid:48) W m,m (cid:48) ln (cid:18) W m,m (cid:48) ρ (cid:48) Am (cid:48) W m (cid:48) ,m ρ (cid:48) Am (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:88) m,m (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) ρ m (cid:48) ρ (cid:48)− Bm (cid:48) · ρ (cid:48) Bm (cid:48) W m,m (cid:48) ln (cid:18) W m,m (cid:48) ρ (cid:48) Bm (cid:48) W m (cid:48) ,m ρ (cid:48) Bm (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + C | ˙ S E | , (B.5)where we already introduced the exchange entropy production C | ˙ S E | = A (cid:88) m,m (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) ρ (cid:48) Am (cid:48) W m,m (cid:48) ln (cid:18) W m,m (cid:48) W m (cid:48) ,m (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + B (cid:88) m,m (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) ρ (cid:48) Bm (cid:48) W m,m (cid:48) ln (cid:18) W m,m (cid:48) W m (cid:48) ,m (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (B.6)with A = (cid:80) m,m (cid:48) (cid:12)(cid:12) ρ m (cid:48) ρ (cid:48)− Am (cid:48) (cid:12)(cid:12) and B = (cid:80) m,m (cid:48) (cid:12)(cid:12) ρ m (cid:48) ρ (cid:48)− Bm (cid:48) (cid:12)(cid:12) . Note that this is anidentification only by analogy, as W is not a proper rate matrix. Since S is closed andevolves under unitary dynamics, the exchange entropy describes the flow of informationbetween A and B .The first two terms in Eq. (B.5) can be further simplified to read˙ I ≤ A ˙ S A + B ˙ S B + C | ˙ S E | , (B.7)where we finally introduced the local , stochastic entropy production [43]˙ S A = (cid:88) m,m (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) ρ (cid:48) Am (cid:48) W m,m (cid:48) ln (cid:18) W m,m (cid:48) ρ (cid:48) Am (cid:48) W m (cid:48) ,m ρ (cid:48) Am (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (B.8)and ˙ S B = (cid:88) m,m (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) ρ (cid:48) Bm (cid:48) W m,m (cid:48) ln (cid:18) W m,m (cid:48) ρ (cid:48) Bm (cid:48) W m (cid:48) ,m ρ (cid:48) Bm (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . (B.9) References [1] Hawking S W 1976
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