A Unitary dynamics

Generalized Geometric Quantum Speed Limits

Abstract

The attempt to gain a theoretical understanding of the concept of time in quantum mechanics has triggered significant progress towards the search for faster and more efficient quantum technologies. One of such advances consists in the interpretation of the time-energy uncertainty relations as lower bounds for the minimal evolution time between two distinguishable states of a quantum system, also known as quantum speed limits. We investigate how the non uniqueness of a bona fide measure of distinguishability defined on the quantum state space affects the quantum speed limits and can be exploited in order to derive improved bounds. Specifically, we establish an infinite family of quantum speed limits valid for unitary and nonunitary evolutions, based on an elegant information geometric formalism. Our work unifies and generalizes existing results on quantum speed limits, and provides instances of novel bounds which are tighter than any established one based on the conventional quantum Fisher information. We illustrate our findings with relevant examples, demonstrating the importance of choosing different information metrics for open system dynamics, as well as clarifying the roles of classical populations versus quantum coherences, in the determination and saturation of the speed limits. Our results can find applications in the optimization and control of quantum technologies such as quantum computation and metrology, and might provide new insights in fundamental investigations of quantum thermodynamics.

pacs:
03.65.Yz, 03.67.-a
12

I Introduction

Quantum mechanics relies on counterintuitive features which challenge our merely classical perception of Nature. One of the most fundamental quantum aspects lies in the impossibility of knowing simultaneously and with certainty two incompatible properties of a quantum system Heisenberg (1927); ?; ?. Contrarily to the well understood uncertainty relation between any two non-commuting observables, the time-energy uncertainty relation still represents a controversial issue Pauli (1926); ?; ?, although the last decades witnessed several attempts towards its explanation Aharonov and Bohm (1961); ?; ?. This effort led to the interpretation of the time-energy uncertainty relation as a so-called quantum speed limit (QSL), i.e. the ultimate bound imposed by quantum mechanics on the minimal evolution time between two distinguishable states of a system Mandelstam and Tamm (1945); Fleming (1973); Bhattacharyya (1983); Anandan and Aharonov (1990); Pati (1991); Uhlmann (1992); Vaidman (1992); Uffink (1993); Pati (1995); Anandan and Pati (1997); Pati (1999); Brody (2003); Margolus and Levitin (1998); Levitin and Toffoli (2009); Giovannetti et al. (2003); Jones and Kok (2010); Deffner and Lutz (2013a); Pfeifer (1993); Pfeifer and Fröhlich (1995); Hegerfeldt (2013, 2014); Luo (2004, 2005); Russell and Stepney (2014); Andersson and Heydari (2014); Pires et al. (2015); Mondal et al. (2016); Taddei et al. (2013); del Campo et al. (2013); Deffner and Lutz (2013b); Uzdin et al. (2014); Zhang et al. (2014); Liu et al. (2015); Xu et al. (2014); Wu et al. (2015); Mondal and Pati (2016); Hamma et al. (2009); Marvian and Lidar (2015). QSLs have been widely investigated within the quantum information setting, since their understanding offers a route to design faster and optimized information processing devices Caneva et al. (2009); ?, thus attracting constant interest in quantum optimal control, quantum metrology Giovannetti et al. (2011), quantum computation and communication Yung (2006). Interestingly, it has been recently recognized that QSLs play a fundamental role also in quantum thermodynamics Deffner and Lutz (2010).

In a seminal work, Mandelstam and Tamm (MT) Mandelstam and Tamm (1945) reported a QSL for a quantum system that evolves between two distinguishable pure states, and , via a unitary dynamics generated by a time independent Hamiltonian . The ensuing lower bound on the evolution time is given by , where is the variance of the energy of the system with respect to the initial state. Several years later, Anandan and Aharanov Anandan and Aharonov (1990) extended the MT bound to time dependent Hamiltonians by using a geometric approach which exploits the Fubini-Study metric defined on the space of quantum pure states. Specifically, they simply used the fact that the geodesic length between two distinguishable pure states according to the Fubini-Study metric, i.e. their Bures angle, is a lower bound to the length of any path connecting the same states. Over half a century after the MT result, Margolus and Levitin (ML) Margolus and Levitin (1998) provided a different QSL on the time evolution of a closed system whose Hamiltonian is time independent and evolving between two orthogonal pure states. This bound reads as , where is the mean energy. Although the ML bound is tight, it does not recover the MT one whatsoever. Therefore, the quantum speed limit for unitary dynamics, when restricting to orthogonal pure states, can be made tighter by combining these two independent results as  Levitin and Toffoli (2009).

All these results attracted a considerable interest in the subject. Giovannetti et al. Giovannetti et al. (2003) extended the ML QSL to the case of arbitrary mixed states and also showed that entanglement can speed up the dynamical evolution of a closed composite system. A plethora of other extensions and applications of QSLs for unitary processes has been investigated in Refs. Fleming (1973); Bhattacharyya (1983); Pati (1991); Uhlmann (1992); Vaidman (1992); Uffink (1993); Pati (1995); Anandan and Pati (1997); Pati (1999); Brody (2003); Jones and Kok (2010); Deffner and Lutz (2013a); Pfeifer (1993); Pfeifer and Fröhlich (1995); Hegerfeldt (2013, 2014); Luo (2004, 2005); Russell and Stepney (2014); Mondal et al. (2016); Andersson and Heydari (2014); Pires et al. (2015). For example, in Ref. Pires et al. (2015) some of us have recently shown that the rate of change of the distinguishability between the initial and the evolved state of a closed quantum system can provide a lower bound for an indicator of quantum coherence based on the Wigner-Yanase information between the evolved state and the Hamiltonian generating the evolution.

Since any information processing device is inevitably subject to environmental noise, QSLs have been also investigated in the nonunitary realm. Taddei et al. Taddei et al. (2013) and del Campo et aldel Campo et al. (2013) were the first to extend the MT bound to any physical process, being it unitary or not. Specifically, Ref. Taddei et al. (2013) exploits the quantum Fisher information metric on the whole quantum state space and represents a natural extension of the idea used in Ref. Anandan and Aharonov (1990), whereas Ref. del Campo et al. (2013) exploits the relative purity. Then, Deffner and Lutz Deffner and Lutz (2013b) extended the ML bound to open quantum systems by adopting again a geometric approach using the Bures angle. These authors have also introduced a new sort of bound, which is tighter than both the ML and MT ones, and shown that non-Markovianity can speed up the quantum evolution. Some other works have then provided a QSL for open system dynamics by using the relative purity, whose usefulness ranges from thermalization phenomena Uzdin et al. (2014) to the relativistic effects on the QSL Zhang et al. (2014). Further developments include the role of entanglement in QSL for open dynamics Liu et al. (2015); Xu et al. (2014); Wu et al. (2015), QSL in the one-dimensional perfect quantum state transfer Yung (2006), and the experimental realizability of measuring QSLs through interferometry devices Mondal and Pati (2016). Finally, a subtle connection was recently highlighted between QSLs and the maximum interaction speed in quantum spin systems Hamma et al. (2009), with implications for quantum error correction and the relaxation time of many-body systems Marvian and Lidar (2015).

Distinguishing between two states of a system being described by a probabilistic model stands as the paradigmatic task of information theory. Information geometry, in particular, applies methods of differential geometry in order to achieve this task Amari and Nagaoka (2000). The set of states of both classical and quantum systems is indeed a Riemannian manifold, that is the set of probability distributions over the system phase space and the set of density operators over the system Hilbert space, respectively. Therefore it seems natural to use any of the possible Riemannian metrics defined on such sets of states in order to distinguish any two of its points. However, it is also natural to assume that for a Riemannian metric to be bona fide in quantifying the distinguishability between two states, it must be contractive under the physical maps that represent the mathematical counterpart of noise, i.e. stochastic maps in the classical settings and completely positive trace preserving maps in quantum. Interestingly, Čencov’s theorem states that the Fisher information metric is the only Riemannian metric on the set of probability distributions that is contractive under stochastic maps Čencov (1982), thus leaving us with only one choice of bona fide Riemannian geometric measure of distinguishability within the classical setting. On the contrary, it turns out that the quantum Fisher information metric Wootters (1981); Braunstein and Caves (1994) is not the only contractive Riemannian metric on the set of density operators, but rather there exists an infinite family of such metrics Gibilisco and Isola (2003), as characterized by the Morozova, Čencov and Petz theorem Morozova and Čencov (1991); Petz and Hasegawa (1996); ?; ?; ?.

Figure 1: (Color online). Illustration of geometric quantum speed limits. The dashed blue curve is the path in the quantum state space representing a generic evolution between an initial state and a final state , parameterized by time . Given a metric on the quantum state space, the length of this path is denoted by . The solid red curve denotes the geodesic connecting to , whose length is given by . Quantum speed limits originate from the fact that the geodesic amounts to the path of shortest length among all physical evolutions between the given initial and final states: . Such inequality can be interpreted as follows. For any given physical evolution from to , and according to any valid metric, the maximum distance between the initial and the final state is the length of the path followed by the system. The ensuing minimal time necessary for this distance to reach a chosen value is the time at which the path length reaches this value. This interpretation provides a neat criterion for the saturation of the lower bound on the evolution time, that is when the dynamical evolution coincides with a geodesic of the considered metric. Here we establish a general family of geometric quantum speed limits with respect to an infinite hierarchy of contractive Riemannian metrics on the space of quantum states, unifying and extending previous results under an information geometry framework.

In this paper, we construct a new fundamental family of geometric QSLs (see Fig. 1) which is in one to one correspondence with the family of contractive Riemannian metrics characterized by the Morozova, Čencov and Petz theorem. We demonstrate how such non uniqueness of a bona fide measure of distinguishability defined on the quantum state space affects the QSLs and can be exploited in order to look for tighter bounds. Our approach is versatile enough to provide a unified picture, encompassing both unitary and nonunitary dynamics, and is easy to handle, requiring solely the spectral decomposition of the evolved state. This family of bounds is naturally tailored to the general case of initial mixed states and clearly separates the contribution of the populations of the evolved state and the coherences of its time variation, thus clarifying their individual role in driving the evolution.

We formulate in rigorous terms the problem of identifying the tightest bound within our family for any given dynamics. While such a problem is unfeasibly hard to address in general, we establish concrete steps towards its solution in practical scenarios. Specifically, we show explicit instances of QSLs which make use of some particular contractive Riemannian metric such as the Wigner-Yanase skew information and can be provably tighter than the corresponding QSLs obtained with the conventional quantum Fisher information. These instances are relevant in metrological settings. Overall this work provides one of the most comprehensive and powerful approaches to QSLs, with potential impact on the characterization and control of quantum technologies.

The paper is organized as follows. In Sec. II we review the relation between statistical distinguishability and the contractive Riemannian metrics on the quantum state space characterized by the Morozova, Čencov and Petz theorem. Section III provides a new generalized geometric derivation of a family of QSLs which is in one to one correspondence with the family of such metrics. In Sec. IV we illustrate and compare the obtained bounds for both unitary and nonunitary evolutions. Finally, in Sec. V we present our conclusions.

Ii Geometric measures of distinguishability

According to the standard formulation of quantum mechanics, any quantum system is associated with a Hilbert space and its states are represented by the Riemannian manifold of density operators over , i.e. the set of positive semi-definite and trace one operators over the carrier Hilbert space. A Riemannian metric over is said to be contractive if the corresponding geodesic distance contracts under physical maps, which means satisfy the inequality for any completely positive trace preserving map and any . The Morozova, Čencov and Petz theorem provides us with a characterization of such metrics in the finite-dimensional case, by constructing a one to one correspondence between them and the Morozova-Čencov (MC) functions, a function which is () operator monotone: for any semi-positive definite operators and such that , then ; () self-inversive: it fulfils the functional equation ; and () normalized: . Specifically, the Morozova, Čencov and Petz theorem states that every contractive Riemannian metric assigns, up to a constant factor, the following squared infinitesimal length between two neighboring density operators and  Petz (1996)

(1)

with

(2)

where and are any two traceless hermitian operators, and

(3)

is a symmetric function, , which fulfills , with being a MC function, and finally are two linear superoperators defined on the set of linear operators over as follows: and . We stress again that each contractive Riemannian metric is arbitrary up to a constant factor. In accordance with Ref. Bengtsson and Życzkowski (2006), we have chosen the factor in order to make the entire family of contractive Riemannian metrics collapse to the classical Fisher information metric when and commute.

In order to make Eq. (1) more explicit, we can write the density operator in its spectral decomposition, , with and , and get Bengtsson and Życzkowski (2006)

(4)

where and we note that the summation is constrained to the requirement . Equation (4) is crucial since it clearly identifies two separate contributions to any contractive Riemannian metric. The first term, which is common to all the family, depends only on the populations of and can be seen as the classical Fisher information metric at the probability distribution . The second term, which is responsible for the non uniqueness of a contractive Riemmanian metric on the quantum state space, is instead only due to the coherences of with respect to the eigenbasis of and is a purely quantum contribution expressing the non-commutativity between the operators and . Finally, for all the contractive Riemannian metrics that can be naturally extended to the boundary of pure states, such that , the Fubini-Study metric appears always to be such extension up to a constant factor, so that the non uniqueness of a contractive Riemannian geometry can be only witnessed when considering quantum mixed states. This is the reason for which only the mixed states will be relevant in our analysis, whose aim is exactly to investigate the freedom in the choice of several inequivalent bona fide measures of indistinguishability in order to get tighter QSLs.

As pointed out by Kubo and Ando Kubo and Ando (1980), among the MC functions there exists a minimal one, , and a maximal one, , such that a generic MC function must satisfy . Interestingly, the maximal MC function is the one corresponding to the celebrated quantum Fisher information metric, whereas the Wigner-Yanase information metric corresponds to an intermediate MC function, , as illustrated in Fig. 2.

Figure 2: (Color online). The family of Morozova-Čencov (MC) functions is upper bounded by a maximal one, , corresponding to the quantum Fisher information metric (red solid line), and lower bounded by a minimal one, (purple dotted line). Any MC function satisfies , i.e., its graph falls into the shaded area, as shown in the particular case of the MC function corresponding to the Wigner-Yanase information metric (blue dashed line). All the MC functions collapse to at . This reflects the fact that the corresponding Riemannian metrics are regular on the maximally mixed state.

Each of these metrics plays a fundamental role in quantum information theory since the corresponding geodesic length , being by construction contractive under quantum stochastic maps, represents a bona fide measure of distinguishability over the quantum state space. However, finding such geodesic distance is unfortunately a very hard task in general. In fact, analytic expressions are known only for the geodesic distance related to the quantum Fisher information metric Uhlmann (1993); ?,

(5)

where is the Uhlmann fidelity, and for the one related to the Wigner-Yanase information metric Gibilisco and Isola (2003),

(6)

where is known as quantum affinity.

Iii Generalized Geometric Quantum speed limits

We are now ready to present our main result, that is, a family of geometric QSLs which hold for any physical process and are in one to one correspondence with the contractive Riemannian metrics defined on the set of quantum states. The most general dynamical evolution of an initial state can be written in the Kraus decomposition as , where are operators satisfying and depending on a set of parameters which are encoded into the input state , in such a way that depends analytically on each parameter (). In the unitary case, the evolution is given in particular by , where is a multiparameter family of unitary operators, fulfilling . In this case, the observables , with , are the generators driving the dynamics.

Consider a dynamical evolution in which the set of parameters is changed analytically from the initial configuration to the final one . Geometrically, this evolution draws a path in the quantum state space connecting and whose length is given by the line integral and depends on the chosen metric (see Fig. 1). Since is an arbitrary path between and , its length need not be the shortest one, which is instead given by the geodesic length between and . Therefore the latter represents a lower bound for the length of the path drawn by the above dynamical evolution. This observation will play a crucial role in the imminent derivation of our family of QSLs, in analogy with Refs. Anandan and Aharonov (1990) and Taddei et al. (2013).

Since the density operator evolves analytically with respect to the parameters , we can write

(7)

Let be the spectral decomposition of , with and . We note that both the eigenvalues and eigenstates of may depend on the set of parameters , i.e. and , so that

(8)

and thus

(9)

where we used the identity . Combining Eq. (7) and Eq. (9), we get

(10)

where we define . By using Eq. (10), in the case of we get

(11)

whereas in the case of , by using the fact that is hermitian, we obtain

(12)

Finally, by substituting Eqs. (11) and (12) into Eq. (4), the squared infinitesimal length between and according to any contractive Riemannian metric becomes

(13)

where

(14)

with

(15)

and

(16)

referring to, respectively, the contribution of the populations of and of the coherences of to the contractive Riemannian metric tensor . Herein we restrict to the case where the parameters are time-dependent, , for , and choose the parametrization such that and , where is the evolution time. Now, being the geodesic distance between the initial and final state, and , a lower bound to the length of the path followed by the evolved state when going from to , we have

(17)

where

(18)

Equation (17) represents the anticipated infinite family of generalized geometric QSLs and is the central result of this paper. Any possible contractive Riemannian metric on the quantum state space, and so any possible bona fide geometric quantifier of distinguishability between quantum states, gives rise to a different QSL. More precisely, we have that both the geodesic distance appearing in the left hand side and the quantity being in the right hand side of Eq. (17) depend on the chosen contractive Riemannian metric, specified by a MC function . In particular, by restricting to the celebrated quantum Fisher information metric, we recover the QSL introduced in Ref. Taddei et al. (2013).

It is intuitively clear that the contractive Riemannian metric whose geodesic is most tailored to the given dynamical evolution is the one that gives rise to the tightest lower bound to the evolution time as expressed in Eq. (17). In order to determine how much a certain geometric QSL is saturated, i.e. its tightness, we will consider the relative difference

(19)

that quantifies how much the dynamical evolution differs from a geodesic with respect to the considered metric .

By minimizing the quantity over all contractive Riemannian metrics, i.e., over all MC functions , one has a criterion to identify in principle the tightest geometric QSL, of the form given in Eq. (17), for any given dynamics . Formally, labelling by the optimal metric for the dynamics , the tightest possible geometric QSL is therefore defined by

(20)

where the minimization is over all MC functions .

Finding this minimum is, however, a formidable problem, which is made all the more difficult by the fact that the quantum Fisher information metric and the Wigner-Yanase information metric are the only contractive Riemannian metrics whose geodesic lengths are analytically known for general dynamics (as previously remarked).

Nevertheless, in this paper we will move the first steps forward towards addressing such general problem, by restricting the optimization in Eq. (20) primarily over these two paradigmatic and physically significant examples of contractive Riemannian metrics, namely the quantum Fisher information and the Wigner-Yanase skew information. Quite remarkably, this restriction will be enough to reveal how the choice of the quantum Fisher information metric, though ubiquitous in the existing literature, is only a special case which does not always provide the tightest lower bound. On the contrary, we will show how the Wigner-Yanase skew information metric can systematically produce tighter bounds in a number of situations of practical relevance for quantum information and quantum technologies, in particular in open system evolutions.

Iv Examples

In this section we will apply our general formalism to present and analyze QSLs based primarily on the quantum Fisher information and the Wigner-Yanase skew information in a selection of unitary and nonunitary physical processes. This will serve the purpose to illustrate how the choice of a particular bona fide geometric measure of distinguishability on the quantum state space affects the QSLs, therefore providing a guidance to exploit the freedom in this choice to obtain the tightest bounds in practical scenarios.

iv.1 Unitary dynamics

We start by restricting ourselves to a closed quantum system, so that our initial state undergoes a unitary evolution . Since the eigenvalues of a unitarily evolving state are constant, , we have that , and thus , along the curve drawn by the evolved state . In other words, the coherences of drive the evolution of a closed quantum system. Moreover, one can easily see that , where , , and are the generators of the dynamics, so that

(21)

In the following subsections we will focus on, respectively, the quantum Fisher information metric and the Wigner-Yanase information metric.

Quantum Fisher information metric

The quantum Fisher information metric corresponds to the MC function , so that and Eq. (21) becomes

(22)

Moreover, by using the following straightforward inequality

(23)

we get

(24)

where is the symmetrized covariance of and with respect to the evolved state, which reduces to the squared variance of the operator when , i.e. . By substituting the inequality (IV.1.1) into Eq. (17) we get the new bound

(25)

Although the QSL in Eq. (IV.1.1) applies to the very general -parameter case, let us restrict for simplicity to the one-parameter case where . Consequently, we have that and that the symmetrized covariance just reduces to the variance of the observable generating the dynamics of the system. Therefore, Eq. (IV.1.1) turns into the simpler bound

(26)

where is the mean variance of the generator . The following QSL is thus obtained

(27)

It is worth emphasizing that the bound in Eq. (27) applies to arbitrary initial and final mixed states and generic time-dependent generators of the dynamics. Moreover, we can immediately see that it exactly coincides with the one reported in Ref. Uhlmann (1992) and reduces to a MT-like bound, when further restricting to the case of a time-independent generator of the dynamics.

Wigner-Yanase information metric

The Wigner-Yanase information metric corresponds to the MC function , so that and Eq. (21) becomes

(28)

where reduces to the skew information between the evolved state and when ,  Gibilisco and Isola (2007). By putting Eq. (IV.1.2) into the bound in Eq. (17), we get

(29)

For simplicity, let us again analyze the one-parameter case, where , and reduces to the skew information between the evolved state and the observable generating the dynamics of the system. Therefore, the bound in Eq. (IV.1.2) turns into

(30)

where we define as the mean skew information between the evolved state and the generator of the evolution. The QSL thus becomes

(31)

As reported by Luo Luo (2006), the skew information is upper bounded by the variance of the observable , , so that and

(32)

The latter QSL strongly resembles the bound expressed in Eq. (27) and emerging from the quantum Fisher information metric, with the difference lying in the fact that we are now adopting the Hellinger angle instead of the Bures angle and a factor appears in the denominator. However, when the initial and final states commute, we have that the corresponding fidelity and affinity coincide, , and so the Bures angle is equal to the Hellinger angle, which implies that in this case the bound emerging from the Wigner-Yanase metric is less tight than the one corresponding to the quantum Fisher information by a factor of .

The above result could be intuitively expected due to the strict hierarchy respected by the MC functions corresponding to the two adopted metrics. To put such an intuition on rigorous grounds, in Appendix A we prove that the geometric QSL corresponding to the quantum Fisher information metric, as expressed directly by Eq. (17), is indeed tighter than the one corresponding to the Wigner-Yanase information metric, when considering any single-qubit unitary dynamics. However, we leave it as an open question to assess whether this is still the case when considering higher dimensional quantum systems, or other contractive Riemannian metrics in place of the Wigner-Yanase one.

Quite surprisingly, we will show instead in the next section that, for the realistic and more general case of nonunitary dynamics, the hierarchy of the MC functions does not automatically translate anymore into a hierarchy of tightness for the corresponding QSLs, not even in the case of a single qubit. This will reveal original consequences of our analysis in practically relevant scenarios.

iv.2 Nonunitary dynamics

We will now consider two paradigmatic examples of nonunitary physical processes acting on a single qubit: dephasing and amplitude damping.

Parallel and transversal dephasing channels

Figure 3: (Color online). Evolution path lengths for parallel dephasing processes by considering the contractive Riemannian metrics corresponding to the following MC functions: (red solid line), (blue dashed line) and (purple dotted line) for (a) , , (b) , , (c) , , and (d) , , . The insets in each panel show the relative difference , Eq. (19), for parallel dephasing processes by considering the quantum Fisher information metric (red solid line) and the Wigner-Yanase information metric (blue dashed line); such a relative difference can be regarded as an indicator of the tightness of the bounds (the smaller , the tighter the bounds).

Let us denote by the Bloch sphere representation of an arbitrary single qubit state , where is the Bloch vector, with , and , while denotes the identity matrix and is the vector of Pauli matrices.

Let us now consider a noisy evolution of this state governed by a master equation of Lindblad form

(33)

where describes the unitary evolution governed by a Hamiltonian while is the Liouvillian that describes the noise. We further consider as Hamiltonian , where is the unitary frequency, and as Liouvillian

(34)

where is the decoherence rate and with .

We can identify two main modalities of dephasing noise. When , the dephasing happens in the same basis as the one specifying the Hamiltonian of our system, a case that can be referred to as ‘parallel dephasing’. When instead , the dephasing occurs in a basis orthogonal to the one of the Hamiltonian, leading to the situation typically referred to as ‘transversal dephasing’ Chaves et al. (2013); Brask et al. (2015). We will explore these two cases separately.

Parallel dephasing. The parallel dephasing noise lets an initial state evolve as , where

(35)

are the Kraus operators, and with  Nielsen and Chuang (2000). Notice that the Kraus operators satisfy not only but also , as such a channel is unital. The effect of parallel dephasing is exactly the same as the one of phase flip and consists in shrinking the Bloch sphere onto the -axis of states diagonal in the computational basis, which are instead left invariant. Moreover, describes the rotation frequency around the -axis. One can easily see that the evolved state has the following spectral decomposition

(36)

where and

(37)

with and a normalization constant. By putting the above equations into Eqs. (15) and (16) one obtains, respectively,

(38)

and

(39)

The contractive Riemannian metric can be interpreted as the speed of evolution of . Equation (38), which corresponds to the contribution to common to all the MC family, is identically zero for all the initial states such that is either or , that are all the incoherent states lying on the -axis of the Bloch sphere (with density matrices diagonal in the computational basis), which are indeed left unaffected by the parallel dephasing dynamics. Although is a function of the initial purity and of time, it does not depend on the initial azimuthal angle since the eigenvalues of the evolved state do not depend on . Equation (39), which instead describes the truly quantum contribution to the speed of evolution and depends on the specific choice of the MC function , is identically zero for all the incoherent initial states such that is either or . Notice that in the case , for initial states lying in the equatorial -plane, is nonzero only when the frequency is also nonzero. Interestingly, does not depend on the initial azimuthal angle as well, even though the eigenstates of the evolved state do depend on . In summary, the speed of evolution is obviously zero for initial states belonging to the -axis, being them invariant under parallel dephasing; it is furthermore symmetric with respect to the initial azimuthal angle , and it arises only from the populations of the evolving state when starting from the equatorial -plane with zero frequency .

Figure 4: (Color online) Contour plot of the difference between the tightness parameter corresponding to the quantum Fisher information metric and the one corresponding to the Wigner-Yanase information metric, for the parallel dephasing process as a function of and , for (a) , , and (b) , . The QSL constructed with the Wigner-Yanase skew information is tighter than (respectively looser than) the one constructed with the quantum Fisher information when (resp. ), as in panel b (resp. a).

In Fig. 3 we compare the evolution path lengths appearing in the right hand side of Eq. (17) and corresponding to the three paradigmatic examples of contractive Riemannian metrics: the quantum Fisher information metric, the Wigner-Yanase information metric, and the metric corresponding to the minimal MC function. We consider the only initial parameters that play a role in all the above analysis, i.e. the initial purity and polar angle , and the dynamical parameter , while full details on the computation of all the quantities appearing in Eq. (17) are deferred to Appendix B. First, it can be seen that by fixing the initial purity (respectively, polar angle ), the speed of evolution increases as we increase the initial polar angle (respectively, purity ). In other words, the farther the initial state is from the -axis (the larger is its quantum coherence), the faster the corresponding evolution can be. Second, Fig. 3(d) in particular unveils the signature of the populations of the evolved state into the speed of evolution. Indeed, according to Eq. (39), the purely quantum contribution to the metric is equal to zero for and (). Thus, the speed of evolution is described solely by the term given in Eq. (38) and arising only from the populations of the evolved state. In this case, the speed of evolution remains invariant for any contractive Riemannian metric, since is common to all of them. However, it is still susceptible to changes depending on the purity and time.

Let us now investigate how the QSLs in Eq. (17) behave by considering the quantum Fisher information metric and the Wigner-Yanase information metric, whose geodesic lengths are known analytically. In the insets of Fig. 3 we compare the tightness parameter , as defined in Eq. (19), when considering these two metrics, for a parallel dephasing dynamical evolution. We can see that for the dynamics does not saturate the bound for any of the two metrics, although the quantum Fisher information metric provides in general a slightly tighter QSL. On the other hand, when and , we have that the QSL is saturated for both metrics, whereas for and , it is instead the Wigner-Yanase information metric that provides us with a slightly tighter lower bound.

More generally, it is sufficient to compare the difference between the tightness indicators for the two metrics in the whole parameter space of the parallel dephasing model, to identify in which regime each of the two corresponding bounds is the tightest. This analysis is reported in Fig. 4, showing that the Wigner-Yanase information metric does lead in general to a tighter QSL when the frequency is sufficiently small. This is in stark contrast with the case of unitary evolutions, discussed in the previous section, and constitutes a first demonstration of the usefulness of our generalized approach to speed limits in quantum dynamics.

Transversal dephasing. We now focus on the case of transversal dephasing noise, which lets an initial state evolve as , where is a hermitian matrix whose non-vanishing elements are given by , , , , and , with

(40)
(41)
(42)