GGeometric quantum speed limits: a case for Wignerphase space
Sebastian Deffner
Department of Physics, University of Maryland Baltimore County, Baltimore,MD 21250, USAE-mail: [email protected]
Abstract.
The quantum speed limit is a fundamental upper bound on thespeed of quantum evolution. However, the actual mathematical expression ofthis fundamental limit depends on the choice of a measure of distinguishability ofquantum states. We show that quantum speed limits are universally characterizedby the Schatten- p -norm of the generator of quantum dynamics. Since computingSchatten- p -norms can be mathematically involved, we then develop an alternativeapproach in Wigner phase space. We find that the quantum speed limit in Wignerspace is fully equivalent to expressions in density operator space, but that thenew bound is significantly easier to compute. Our results are illustrated for theparametric harmonic oscillator and for quantum Brownian motion.
1. Introduction
It has recently been argued that already the first generation of real-life quantumcomputers will be able to perform certain tasks exponentially faster than classicalcomputers [1]. This so-called “quantum supremacy” [2] rests in the fact that looselyspeaking quantum state space is exponentially larger than the classical computationalspace, and hence significantly less operations are necessary to perform the samecomputation. However, the working principles of quantum computers and classicalcomputers are fundamentally different, which makes it not immediately clear how toquantify the “quantum speed-up” [3]. To make matters even more involved, in thetheory of quantum computation “time” is not actually a physical time, but rathera synonym for the “number of computational operations” [3]. The more practicalquestion is, however, how fast a quantum computer could actually operate.To address this issue a somewhat opposite approach has been developed inquantum dynamics, where the notion of a quantum speed limit has found wide-spreadprominence. Whereas in the theory of quantum computation one is after characterizingquantum speed-ups – the quest for faster and faster computations with less and lesssingle operations – the quantum speed limit sets the ultimate, maximal speed withwhich any quantum system can evolve. This means, in particular, that every singlequantum operation takes a finite, minimal time to be accomplished – and thus evenquantum computers will not be able to achieve any arbitrary speed-ups. This quantumspeed limit originates in the Heisenberg indeterminacy principle [4, 5], ∆ E ∆ t (cid:38) (cid:126) / a r X i v : . [ qu a n t - ph ] A p r eometric quantum speed limits: a case for Wigner phase space τ QSL = π (cid:126) / E , where∆ E = ( (cid:10) H (cid:11) − (cid:10) H (cid:11) ) / . Since, however, the variance of an operator is not necessarilya good quantifier for dynamics [8], Margolus and Levitin [9] revisited the problem andderived a second bound on the quantum evolution time in terms of the average energy E = (cid:104) H (cid:105) − E g over the ground state with energy E g , τ QSL = π (cid:126) / E . It was eventuallyrealized that these two bounds are not independent, and that only the unified boundis tight [10].Nowadays, it has been established in virtually all areas of quantum physics [11–16]that the quantum speed limit [17–20], sets a fundamental upper bound on the speedof any quantum dynamics. In particular in recent years, the quantum speed limit hasbeen extensively studied and generalized for isolated [21–26] and open [27–39] quantumsystems. The renewed and concentrated interest was inspired by three letters [27–29],which broadened the scope of the quantum speed limit beyond unitary dynamics. Inparticular, Ref. [27] showed that the maximal speed of quantum evolution is given bythe time averaged norm of the generator of the dynamics [27], which only in the caseof unitary dynamics and for orthogonal states reduces to the average energy E [25].All treatments of the quantum speed limit have in common that the analyses startwith a choice of a measure of the distinguishability of quantum states. For instance,Ref. [28] studies the relative purity, Ref. [27] starts with the Bures angle betweeninitially pure states and time-evolved states, and Ref. [40] even defines an entirelynew metric. Finally, Pires et al [41] derived a whole family of quantum speed limits,which is based on a family of contractive Riemannian metrics.Thus, two natural questions arise: (i) Are all of these treatments of quantumspeed independent, or can the quantum speed limit be universally characterized –independently of the chosen measure of distinguishability? (ii) Most of the expressionsfor the quantum speed limit are mathematically rather involved. Hence, can one finda mathematically simple bound that is computable and nevertheless captures theuniversal behavior?In the following we will answer both questions. First, we will show that thequantum speed is universally characterized by any Schatten- p -norm of the generatorof the quantum dynamics. In the second part of the analysis, we will derive a newquantum speed limit in terms of the Wasserstein norm of the rate of change of theWigner function. We will argue that the quantum speed limit in Wigner phase spacecaptures the same qualitative behavior as the speed limits derived in density operatorspace. However, we will also see that the new quantum speed limit is significantlyeasier to compute, for pure as well as mixed states, and for isolated as well asopen dynamics. As an illustrative example we will discuss the semi-classical, hightemperature limit, and we will confirm that the quantum speed limit is a pure quantumfeature, i.e., that classical systems do not experience a fundamental bound on theirrates of change.
2. Quantum speed and the geometric approach
We begin by briefly reviewing the main results of Ref. [27] and by establishing notionsand notations. Consider a quantum master equation,˙ ρ t = L t ( ρ t ) , (1) eometric quantum speed limits: a case for Wigner phase space L t , reduces to the von-Neumann equation, L t ( ρ ) =[ H t , ρ t ] /i (cid:126) , but we explicitly allow for any open systems dynamics – Markovian aswell as non-Markovian. Note that Ref. [27] theoretically predicted that non-Markovianenvironments can speed up quantum dynamics. This was experimentally verified incavity QED [35].In geometric quantum mechanics [42] it has proven useful to quantify thedistinguishability of quantum states in terms of the Bures angle [43], L ( ρ , ρ t ) = arccos (cid:16)(cid:112) F ( ρ , ρ t ) (cid:17) = arccos (cid:18) tr (cid:26)(cid:113) √ ρ ρ t √ ρ (cid:27)(cid:19) (2)where we further introduced the quantum fidelity F ( ρ , ρ t ) [44]. To obtain an upperbound on the speed of evolution one then considers the magnitude of the geometricspeed, | ˙ L| , and it is easy to see that we have [27]2 cos ( L ) sin ( L ) ˙ L ≤ (cid:12)(cid:12)(cid:12) ˙ F ( ρ , ρ t ) (cid:12)(cid:12)(cid:12) . (3)For initially pure states, ρ = | ψ (cid:105) (cid:104) ψ | , Eq. (2) can be further simplified and it canbe shown that [27]2 cos ( L ) sin ( L ) ˙ L ≤ |(cid:104) ψ | ˙ ρ t | ψ (cid:105)| ≤ min {(cid:107) ˙ ρ t (cid:107) p for p ∈ { , , ∞}} . (4)Here (cid:107) A (cid:107) p denotes the Schatten- p -norm of an operator, O , which is defined as (cid:107) O (cid:107) p ≡ (tr {| O | p } ) /p = (cid:32)(cid:88) k o pk (cid:33) /p (5)and the o k are the singular values of O , i.e. the eigenvalues of the Hermitianoperator | O | ≡ √ O † O . The more familiar trace, Hilbert-Schmidt, and operator normscorrespond respectively to p = 1 , , and ∞ . Equation (4) can then be used to definethe quantum speed limit v QSL ,˙ L ≤ v QSL L ) sin ( L ) ≡
12 cos ( L ) sin ( L ) min {(cid:107) ˙ ρ t (cid:107) p for p ∈ { , , ∞}} . (6)Note that in contrast to Ref. [16] we did not include the denominator into the definitionof v QSL . The reason for this choice will become obvious shortly.Although useful for theoretcial predictions of experimental outcomes [35] Eq. (4)also left several questions unaddressed. Probably the most immediate one is, how theabove treatment would have to be generalized to initially mixed states. Generallythis is a mathematically involved problem, since the quantum fidelity, F ( ρ , ρ t ), andits derivatives are non-trivial to handle. A comprehensive analysis of this issue wasproposed by Pires et al [41]. They showed that if one considers the angle defined interms of the Wigner-Yanase information, A ( ρ , ρ t ), L WY ( ρ , ρ t ) = arccos ( A ( ρ , ρ t )) = arccos (tr {√ ρ √ ρ t } ) , (7)instead of the Bures angle (2), then an infinite family of bounds of the quantum speedcan be derived. However, similarly to Eq. (6) the quantum speed is bounded by a eometric quantum speed limits: a case for Wigner phase space (cid:107) ˙ ρ t (cid:107) , and only the “prefactor” depends on thechoice of the starting point, whether it be Eq. (2) or Eq. (7). Thus, the analysis ofRef. [41] makes the second question even more obvious: Namely, formulating quantumspeed limits seems to be somewhat arbitrary, since every single treatment starts witha choice of a measure of distinguishability in order to define the geometric speed.Hence, it would be desirable to find a universal measure of quantum speed, and definethe quantum speed limit exclusively in terms of this measure.Both, Ref. [27] and Ref. [41] make the strong case that the choice has to be acontractive, Riemannian metric on quantum state space. This choice is justified if onewould like to find tight bounds on the quantum speed [27]. If one is only interested inthe qualitative behavior, however, other measures might be more convenient to workwith. For instance, instead of choosing the Bures angle, L ( ρ , ρ t ), we also could haveworked with the Bures distance [42], L D ( ρ , ρ t ) = (cid:114) (cid:16) − (cid:112) F ( ρ , ρ t ) (cid:17) . (8)One easily convinces oneself that a such defined geometric speed, ˙ L D , leads to aninequality similar to Eq. (3). Therefore, we observe already here that all of thesetreatments have in common that eventually the quantum speed is characterized by aSchatten- p -norm of the generator of the dynamics, (cid:107) ˙ ρ t (cid:107) p = (cid:107) L ( ρ t ) (cid:107) p .
3. Quantum speed in density operator space
We have seen above that typically quantum speed is characterized by the dynamicalbehavior of the quantum fidelity, F ( ρ , ρ t ). Since F ( ρ , ρ t ) is mathematically ratherinvolved several continuity bounds have been derived. For instance, we have [42]1 − (cid:112) F ( ρ , ρ t ) ≤ (cid:96) ( ρ t , ρ ) ≤ (cid:112) − F ( ρ , ρ t ) , (9)where (cid:96) denotes the trace distance, i.e., the Schatten-1-distance (cid:96) ( ρ t , ρ ) = (cid:107) ρ t − ρ (cid:107) ≡ tr {| ρ t − ρ |} . (10)The obvious question is whether a quantum speed limit can be derived starting withthe trace distance (cid:96) . To this end, we consider the geometric speed˙ (cid:96) ( ρ t , ρ ) = tr (cid:110) | ρ t − ρ | − ( ρ t − ρ ) ˙ ρ t (cid:111) , (11)which can be bounded from above by with the triangle inequality for operators, | tr { O } | ≤ tr {| O |} , as˙ (cid:96) ( ρ t , ρ ) ≤ (cid:12)(cid:12)(cid:12) ˙ (cid:96) ( ρ t , ρ ) (cid:12)(cid:12)(cid:12) ≤ tr {| ˙ ρ t |} = (cid:107) ˙ ρ t (cid:107) . (12)We immediately conclude that whether we choose the Bures angle (2) or the tracedistance (9) only determines the functional dependence of the geometric speed on thechoice of the metric (4). The dynamics and, hence, the actual quantum speed limit,however, is fully characterized by the trace norm of the rate with which the quantumstate changes. eometric quantum speed limits: a case for Wigner phase space p -norm as a startingpoint of the derivation, (cid:96) p ( ρ t , ρ ) = (cid:107) ρ t − ρ (cid:107) p ≡ (tr {| ρ t − ρ | p } ) p , (13)for which the geometric speed is bounded from above by˙ (cid:96) p ( ρ t , ρ ) ≤ (cid:107) ˙ ρ t (cid:107) p . (14)A proof of the latter result can be found in Appendix A. In conclusion we have thatthe actual quantum speed limit is given by v QSL ≡ min {(cid:107) ˙ ρ t (cid:107) p for p ∈ [0 , ∞ ) } . (15)Equation (15) constitutes our first main result. Derivations of geometric quantumspeed limits depend on a rather arbitrary, although well-motivated choice of a measureof distinguishability of quantum states. The final expression will be functionallydepended on this choice, see for instance Ref. [28] for the relative purity, Ref. [27]for the Bures angle, and Ref. [41] for the Wigner-Yanase information. However, sinceall of these measures fulfill continuity inequalities [42] (see also Eq. (9)), the actualquantum speed limit v QSL is given by the smallest Schatten- p -norm of the generatorof the dynamics.
4. Quantum speed in Wigner phase space
The universal expression of the quantum speed limit, v QSL in Eq. (15), is a powerfulexpression that can be used to obtain physical insight into the dynamical propertiesof quantum systems [35]. However, computing the tightest bound, i.e., the operatornorm [27, 45] is far from being a trivial task. Imagine, for instance, we want to studya driven, open quantum system such as in quantum Brownian motion [46]. In thiscase, the dynamics is typically solved in a computationally convenient and continuousbasis [47]. Extracting the singular values from such a representation of the time-dependent density operator is computationally expensive, if it is at all feasible. Thus,it would be desirable to find an alternative expression for v QSL which gives the samequalitative information, but which is also much easier to compute.Especially in the treatment of open quantum systems [47] as well as to study thesemi-classical limit [48] it has proven useful to express quantum states in their Wignerrepresentation W ( x, p ) = 1 π (cid:126) (cid:90) dy (cid:104) x + y | ρ | x − y (cid:105) exp (cid:18) − ip y (cid:126) (cid:19) . (16)If we want to derive a quantum speed limit in Wigner phase space, we now needto choose a measure of distinguishability. To this end, consider the total variationdistance , which is given by the Wasserstein-1-distance [49], D ( W t , W ) = (cid:107) W t − W (cid:107) ≡ (cid:90) d Γ | W (Γ , t ) − W (Γ) | , (17)with Γ = ( x, p ). The Wasserstein-1-distance can be regarded as a generalization ofthe trace distance to (semi-)probability distributions. eometric quantum speed limits: a case for Wigner phase space D ( W t , W ) = (cid:90) d Γ W (Γ , t ) − W (Γ) | W (Γ , t ) − W (Γ) | ˙ W (Γ , t ) (18)which can again be bounded with the help of the triangle inequality˙ D ( W t , W ) ≤ (cid:12)(cid:12)(cid:12) ˙ D ( W t , W ) (cid:12)(cid:12)(cid:12) ≤ (cid:90) d Γ (cid:12)(cid:12)(cid:12) ˙ W (Γ , t ) (cid:12)(cid:12)(cid:12) . (19)Comparing Eqs. (12) and (19) we immediately see that we have obtained an analogousexpression for the quantum speed limit, v QSL , in Wigner space. In Appendix B weshow that if we consider the more general case of any Wasserstein- p -distance [49], D p ( W t , W ) = (cid:107) W t − W (cid:107) p ≡ (cid:18)(cid:90) d Γ | W (Γ , t ) − W (Γ) | p (cid:19) /p , (20)we find ˙ D p ( W t , W ) ≤ (cid:107) ˙ W t (cid:107) p . (21)Hence we conclude for the quantum speed limit in phase space, v W QSL , that we have v W QSL ≡ min (cid:110) (cid:107) ˙ W t (cid:107) p for p ∈ [0 , ∞ ) (cid:111) . (22)Equation (22) has the same functional form as the quantum speed limit derived indensity operator space (15). However, Eq. (22) is significantly easier to compute, sinceit only involves the absolute value of a real valued function, instead of the singularvalues of a high-dimensional operator.What remains to verify is that v QSL (15) and v W QSL (22) contain the same physicalinformation and that v QSL and v W QSL behave qualitatively similarly.
5. Qualitative comparison of the two approaches
In a mathematical sense, the Weyl-Wigner transform (16) is a well-defined, invertibleintegral transform between the phase-space and operator representations of quantumstates [50]. Therefore, one would expect v QSL (15) and v W QSL (22) to be fully equivalent.That this is, indeed, the case we will now illustrate by computing v QSL (15) and v W QSL (22) for a solvable example. For the sake of simplicity we restrict ourselves tounitary dynamics induced by the parametric harmonic oscillator with Hamiltonian, H = P M + 12 M ω t x . (23)It can be shown that the dynamics is fully analytically solvable [51]. For systemsinitially starting in the ground state, ρ = | ψ (cid:105) (cid:104) ψ | , with ψ ( x ) = (cid:18) M ω π (cid:126) (cid:19) / exp (cid:18) − mω x (cid:126) (cid:19) (24) eometric quantum speed limits: a case for Wigner phase space ρ ( x, y, t ) = (cid:115) M ω π (cid:126) Y t + ω X t × exp (cid:18) − M ω (cid:126) Y t + ω X t (cid:104) x + y + i (cid:0) x − y (cid:1) (cid:16) ω ˙ X t X t + ˙ Y t Y t (cid:17)(cid:105)(cid:19) . (25)Here, X t and Y t are the solutions of the force free harmonic oscillator, ¨ X t + ω t X t = 0,with the boundary conditions X = 0, ˙ X = 1, Y = 1, and ˙ Y = 0.It is then a simple exercise to numerically obtain the quantum speed limits, v QSL (15) and v W QSL (22). Without loss of generality, we computed v QSL (15) from thetrace norm (12), and v W QSL (22) from the Wasserstein-1-norm (19). This is sufficient,since for pure states both the Schatten- p -norms as well as the Wasserstein- p -normsare monotonic in p [45].Specifically, v QSL can be obtained from a numerical singular value decompositionof ρ ( x, y, t ) (25), which is a computationally rather expensive task. On the other hand, v W QSL is obtained directly from the corresponding Wigner function (16). In Fig. 1 weplot the numerical results for a linear quench, ω t = ω − ( ω − ω ) t/τ , for severalvalues of the quench time τ . For the ease of comparison, we further normalized thequantum speed limits v QSL (15) and v W QSL (22) by their maximal value during the timeinterval t ∈ [0 , τ ]. We observe perfect agreement of v QSL (15) and v W QSL (22). Hence,we conclude that our initial expectation is indeed verified, namely we find that v QSL and v W QSL are fully equivalent. In particular for the present case v QSL and v W QSL onlydiffer by a factors that is determined by their maximal value during the quench time τ . It is worth emphasizing again that the quantum speed limit is significantly easier tocompute in Winger phase space, since the computationally expensive task of havingto determine the singular values can be fully avoided.
6. Quantum speed in the semiclassical limit
We conclude the analysis by briefly studying the high-temperature, semi-classical limit.Within the geometric approach to quantum speed it often proves useful to define thequantum speed limit time, τ QSL , which is given by the inverse of the time-averagedquantum speed limit, v QSL , [27]. Note that τ QSL is not a physical time, but rather acharaterisitic of the internal dynamics [7, 9].We can thus define the quantum speed limit time in Wigner space as τ W QSL ≡ D ( W t , W )1 /τ (cid:82) τ dt v W QSL . (26)For unitary dynamics it is easy to see that τ W QSL is proportional to (cid:126) , and hence, τ W QSL ,can be understood as an expression of the Heisenberg indeterminacy principle forenergy and time. For open systems, however, the interpretation is less obvious [27].Since the quantum speed limit v W QSL is fully determined by the metric properties ofthe generator of the dynamics (22), it is not ad hoc obvious how τ W QSL behaves in thesemiclassical, high-temperature limit k B T (cid:29) (cid:126) γ , where T is the temperature and γ the damping coefficient. eometric quantum speed limits: a case for Wigner phase space ν ���� � ��� ��� ��� ��� ��� ��� � / τ ��������������� ν ���� � ��� ��� ��� ��� ��� ��� � / τ ��������������� ν ���� � ��� ��� ��� ��� ��� ��� � / τ ��������������� ν ���� � ��� ��� ��� ��� ��� ��� � / τ ��������������� Figure 1.
Quantum speed limits in density operator space v QSL (15) (blue,dashed line) and Wigner phase space v W QSL (22) (red, solid line) for p = 1, theharmonic oscillator (23), and normalized by their maximal value during the timeinterval t ∈ [0 , τ ]. Parameters are ω = 1, ω = 2, M = 1, (cid:126) = 1 and: upperleft panel: τ = 0 .
1; upper right panel: τ = 1; lower left panel: τ = 5; lower rightpanel: τ = 10. To address this question we turn again to the harmonic oscillator. Since we arenow interested merely the behavior of τ W QSL in the limit high-temperatures, we nowconsider an un-driven system with potential, V ( x ) = 1 / M ω x . In this case theexact master equation in Wigner space can be written as [53, 54] ∂ t W ( x, P, t ) = (cid:20) − PM ∂ x + V (cid:48) ( x ) ∂ P + ∂ P ( γP + D P P ∂ P ) + D xP ∂ xP (cid:21) W ( x, P, t )(27)where D P P = M γ/β + M βγ (cid:126) ( ω − γ ) /
12 and D xP = βγ (cid:126) /
12. In Fig. 2 we plot theresulting quantum speed limit, v W QSL (22), together quantum speed limit time, τ W QSL (26), again for p = 1. As initial state we chose a narrow Gaussian W ( x, P ) = 12 π σ x σ P exp (cid:18) − ( x − µ x ) σ x (cid:19) exp (cid:18) − ( P − µ P ) σ P (cid:19) (28)As expected, the quantum speed limit time τ W QSL vanishes in the classical, high-temperature limit. This observation confirms that the quantum speed limit is a purelyquantum phenomenon also in open systems.
7. Concluding remarks
What is the ultimate limit on how fast a quantum system can evolve? At the vergeof the age of quantum computing this century old question is more topical than ever. eometric quantum speed limits: a case for Wigner phase space �� �� �� �� �� �� �� � � � / ℏγ ��������������� τ ���� � � � / ℏγ = �� � � � / ℏγ = �� � � � / ℏγ = ���� ��� ��� ��� ��� � / τ ������� ν ���� Figure 2.
Quantum speed limit time τ W QSL (26) (left panel) and quantumspeed limit v W QSL (22) for an open harmonic oscillator (27) with the initial state ofEq. (28) for p = 1. Parameters are γ = 2, (cid:126) = 1, M = 1, ω = 1, τ = 2, µ x = 2, σ x = 0 . µ P = 0, and σ P = 0 . In the present work we highlighted that the maximal quantum speed can be fullycharacterized by the Schatten- p -norms of the generator of quantum dynamics. Wefurther showed that equivalent expressions can be found in Wigner phase space, wherethe computationally expensive operator norm is replaced by the absolute value of a realvalued function. The utility of the novel approach to quantum speed was illustrated bycomparing the quantum speed limits in density operator space and Wigner phase spacefor the parametric harmonic oscillator. As a consistency check we finally verified thatthe bound is a pure quantum phenomenon, and that the dynamics of open classicalsystems is not restricted by the quantum bound. Therefore, we imagine that ourresults could prove useful for practical consequences and applications of the quantumspeed limit, since for any situation the new bound is significantly easier to computethan the operator norm of high-dimensional density matrices. Acknowledgments
The author would like to thank Steve Campbell for insightful discussions. This workwas supported by the U.S. National Science Foundation under Grant No. CHE-1648973.
Appendix A. Quantum speed from Schatten distance
This appendix is dedicated to a proof of Eq. (14). Consider the Schatten- p -distance (cid:96) p ( ρ t , ρ ) = (cid:107) ρ t − ρ (cid:107) p ≡ (tr {| ρ t − ρ | p } ) /p , (A.1)where p is an arbitrary, positive, real number, p ∈ [0 , ∞ ). Then the geometric speedcan be written as˙ (cid:96) p ( ρ t , ρ ) = (tr {| ρ t − ρ | p } ) p − tr (cid:26)(cid:104) ( ρ t − ρ ) (cid:105) p − ( ρ t − ρ ) ˙ ρ t (cid:27) . (A.2)The latter expression looks rather involved, but it can be simplified by using,˙ (cid:96) p ( ρ t , ρ ) ≤ | ˙ (cid:96) p ( ρ t , ρ ) | and employing the triangle inequality for operators, | tr { O } | ≤ eometric quantum speed limits: a case for Wigner phase space {| O |} , to read˙ (cid:96) p ( ρ t , ρ ) ≤ (tr {| ρ t − ρ | p } ) p − tr (cid:110) | ρ t − ρ | p − | ˙ ρ t | (cid:111) . (A.3)Equation (A.3) can be further simplified with the help of H¨older’s inequality [55]tr {| OB |} ≤ (tr {| O | q } ) /q (tr {| B | q } ) /q (A.4)which is true for all 1 /q + 1 /q = 1. Now choosing B = ˙ ρ t and q = p/ ( p − q = p , we finally obtain the desired result (14)˙ (cid:96) p ( ρ t , ρ ) ≤ (cid:107) ˙ ρ t (cid:107) p . (A.5) Appendix B. Quantum speed from Wasserstein distance
Finally, we proof the expression for the quantum speed limit in Wigner phase space(22). To this end, we start with the Wasserstein- p -distance [49], which is given by D p ( W t , W ) = (cid:107) W t − W (cid:107) p ≡ (cid:18)(cid:90) d Γ | W (Γ , t ) − W (Γ) | p (cid:19) /p . (B.1)Accordingly we have˙ D p ( W t , W ) = (cid:18)(cid:90) d Γ | W (Γ , t ) − W (Γ) | p (cid:19) p − × (cid:90) d Γ | W (Γ , t ) − W (Γ) | p − W (Γ , t ) − W (Γ) | W (Γ , t ) − W (Γ) | ˙ W (Γ , t ) , (B.2)which can be simplified again with the help of the triangle inequality to read˙ D p ( W t , W ) ≤ (cid:12)(cid:12)(cid:12) ˙ D p ( W t , W ) (cid:12)(cid:12)(cid:12) ≤ (cid:18)(cid:90) d Γ | W (Γ , t ) − W (Γ) | p (cid:19) p − (cid:90) d Γ | W (Γ , t ) − W (Γ) | p − (cid:12)(cid:12)(cid:12) ˙ W (Γ , t ) (cid:12)(cid:12)(cid:12) . (B.3)In complete analogy to the derivation in density operator space, we now consider againH¨older’s inequality [55] (cid:90) dx | f ( x ) g ( x ) | ≤ (cid:18)(cid:90) dx | f ( x ) | q (cid:19) /q (cid:18)(cid:90) dx | g ( x ) | q (cid:19) /q , (B.4)which holds for all 1 /q + 1 /q = 1. Once again choosing q = p/ ( p −
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