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Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control

TL;DR: In this paper, Paczynska et al. presented a visual representation of Einstein's gedanken experiment, Fig. 4, which was supported by the U.S National Science Foundation under Grant No. CHE-1648973.
Abstract: We are grateful to Marta Paczy´nska for creating the visual representation of Einstein’s gedankenexperiment, Fig. 1, and Lu (Lucy) Hou for providing the resources for Fig. 4. SD would like to thank Eric Lutz for many years of insightful discussions and supporting mentorship, and in particular for inciting our interest in quantum speed limits. This work was supported by the U.S. National Science Foundation under Grant No. CHE-1648973.

Summary (15 min read)

Jump to: [1. Introduction][Quantum speed limits 3][Quantum speed limits 4][Quantum speed limits 6][2.2. The uncertainty relation of Mandelstam and Tamm][Quantum speed limits 7][2.4. The unified bound is tight][Quantum speed limits 9][2.5. Generalisations to arbitrary angles and driven dynamics][Quantum speed limits 10][Quantum speed limits 11][3.1. Bremermann-Bekenstein Bound][Quantum speed limits 12][3.2. Quantum thermodynamics][Quantum speed limits 13][Quantum speed limits 14][3.3. Quantum computation][3.4. Quantum metrology][Quantum speed limits 16][4. Optimised quantum evolution and the minimal time approach][4.1. Optimal control theory][Quantum speed limits 18][Quantum speed limits 20][4.2. Parametric Hamiltonians and driven dynamics][Quantum speed limits 21][5. Maximal quantum speed from the geometric approach][Quantum speed limits 22][5.1. Defining the geometric quantum speed][Quantum speed limits 23][5.2. Differential geometry][5.3. Open systems and non-Markovian dynamics][Quantum speed limits 25][Quantum speed limits 27][Quantum speed limits 28][Quantum speed limits 29][5.4. Dynamics of multi-particle systems][Quantum speed limits 30][5.5. Quantum speed limits for other quantities][Quantum speed limits 31][Quantum speed limits 32][5.6. Universal, geometric quantum speed limits][Quantum speed limits 33][5.7. Applications: Shortcuts to adiabaticity][Quantum speed limits 34][6. Quantum speed limits in non-Schrödinger quantum mechanics][6.1. Relativistic systems and Dirac dynamics][Quantum speed limits 36][6.2. PT-symmetric quantum mechanics][Quantum speed limits 37][Quantum speed limits 38][6.3. Non-linear systems][Quantum speed limits 39][Quantum speed limits 40][7.3. Lieb-Robinson bound][Quantum speed limits 41][7.4. Maximal rate of quantum learning and Holevo’s information][8. Final remarks][Quantum speed limits 42][Acknowledgments][Quantum speed limits 43][Quantum speed limits 44][Quantum speed limits 45][Quantum speed limits 46][Quantum speed limits 47][Quantum speed limits 48][Quantum speed limits 49][Quantum speed limits 50] and [Quantum speed limits 51]

1. Introduction

  • The theory says a lot, but does not really bring us any closer to the secret of the old one.
  • At a preset time the shutter opens the hole for a short period and lets photons escape.

Quantum speed limits 3

  • About the intrinsic time scale of unitary quantum dynamics.
  • Hence, one should rather write ∆t & ~/∆E, where ∆t is interpreted as the time a quantum system needs to evolve from an initial to a final state.
  • More specifically Mandelstam and Tamm derived the first expression of the quantum speed limit time τQSL = π~/2∆H, where ∆H2 is the variance of the Hamiltonian, H, of the quantum system [10].
  • As an application of their bound, they also argued that τQSL naturally quantifies the life time of quantum states [10], which has found widespread prominence in the literature [11–16].
  • Nevertheless, the desire to formalise time as a proper quantum observable persisted [17, 18].

Quantum speed limits 4

  • This means Eq. (2) sets the fastest attainable time-scale over which a quantum system can evolve.
  • In particular, Eq. (2) sets the maximal rate with which quantum information can be communicated [25], the maximal rate with which quantum information can be processed [26], the maximal rate of quantum entropy production [27], the shortest time-scale for quantum optimal control algorithms to converge [28], the best precision in quantum metrology [29], and determines the spectral form factor [30].
  • The next major milestone in the development of the field was achieved only relatively recently with the generalisation of the quantum speed limit to open systems.
  • The authors will begin with a historical overview in Section 2, where they will also summarise the original derivations by Mandelstam-Tamm and Margolus-Levitin.
  • When writing this Topical Review, the authors strove for objectivity and completeness.

Quantum speed limits 6

  • To avoid any misconceptions the authors emphasize that other time-like variables such as arrival or tunneling times [38,39] can very well be expressed as operators.
  • “time” will always be understood as the quantity with which one commonly associates an uncertainty relation of the form (5).
  • From its first appearance of such a notion of time in Heisenberg’s paper in 1927 [2] it took almost twenty years before a mathematically sound and physically insightful treatment was proposed by Mandelstam and Tamm [10].

2.2. The uncertainty relation of Mandelstam and Tamm

  • Mandelstam and Tamm’s analysis [10] rests on the fact that for quantum systems evolving under Schrödinger dynamics the evolution of any observable A is given by the Liouville-von-Neumann equation ∂A ∂t = i ~ [H, A] , (8) where H is the Hamiltonian of the system.
  • It is interesting to note that the proper interpretation of the energy-time uncertainty principle as a bound on the minimal time of quantum evolution was formalised by Aharonov and Bohm [48].
  • They pointed out that Eq. (12) must not be interpreted as an uncertainty relation between the duration of a measurement and the energy transferred to the observed system.
  • Rather, the quantum speed limit time, τQSL, sets an intrinsic time scale of any quantum evolution [48,49].

Quantum speed limits 7

  • It was Uffink who realised [19] that in many situations ∆H gives a very unreasonable measure for the speed of a quantum evolution.
  • The lower bound in Eq. (12) can be arbitrarily small, since the variance of the Hamiltonian, ∆H, can diverge even if the average energy is finite [23].

2.4. The unified bound is tight

  • As a consequence it was simply assumed without much justification that the minimal time a quantum system needs to evolve between two orthogonal states is given by τQSL = max { π.
  • To this end, they proved the following theorem:.
  • This state |ψ〉 Eq. (18) is unique up to degeneracy of the first excited state and arbitrary phase factors.
  • That |ψ〉 is the only state to attain the Margolus-Levitin bound is easy to see from Eq. (15).
  • In complete analogy to above the authors again expand the initial state, |ψ0〉, and the time-evolved state, |ψt〉, in the energy eigenbasis, Eqs. (13) and (14), respectively.

Quantum speed limits 9

  • This alternative proof, however, immediately allows us to determine when the inequality in Eq. (22) becomes an equality.
  • As main results Levitin and Toffoli [24] showed that the unified bound, Eq. (17), is tight and that it is attained only by states of the form Eq. (18).
  • From a few more rather technical considerations they further proved that no mixed state can have a larger speed, which means that Eq. (17) sets the ultimate speed limit for the evolution between orthogonal states and for time-independent Hamiltonians.

2.5. Generalisations to arbitrary angles and driven dynamics

  • The natural question arises whether the expression for the quantum speed limit, Eq. (19), can be further sharpened for arbitrary angles and for driven dynamics.
  • Here and in the following “driven” refers to dynamics under parametrically varying Hamiltonians.
  • Whereas for pure states addressing this question is rather straight forward [60], for general mixed quantum states the situation is technically significantly more challenging.
  • The difficulty arises from the fact that for pure states the angle is simply given by [61] (25) Equipped with the latter, Uhlmann was able to generalise the Mandelstam-Tamm bound to mixed states and driven dynamics [66].
  • Interestingly, without using its name Uhlmann already worked with the infinitesimal quantum Fisher information, on which the authors will expand shortly in Sec. 3.4.

Quantum speed limits 10

  • (27) Note that the superoperator R−1ρ is here defined as describing the infinitesimal Bures angle L, and hence differs by a factor 4 from the one used in Ref. [68], where R−1ρ is determined by the infinitesimal statistical distance.
  • Ht, ρt] , (28) since the expectation value of the energy 〈Ht〉 is a real number that can be included in the commutator.
  • Similar results were also found by Braunstein and Milburn [69] and summarised by Braunstein et al. in Ref. [70].
  • Similar to its original discovery, the generalisation of the Margolus-Levitin bound (16) proved to be a significantly harder.

Quantum speed limits 11

  • This result (34) is particularly remarkable since Giovannetti, Lloyd, and Maccone [71–73] numerically verified the analytical treatment of Pfeifer [60] and Uhlmann [66].
  • Their study also highlighted that the Margolus-Levitin bound does not generalise as intuitively as one might hope.
  • Finally, it is interesting to note that not all works were exclusively interested in lower bounds for the quantum speed limit time.
  • Further attempts at generalisations of the bounds to driven dynamics and arbitrary angles were undertaken by Jones and Kok [75], Zwierz [76], and Deffner and Lutz [67].
  • Therefore, the next two sections will focus on the physical significance and conceptual insights from the quantum speed limit for time-independent dynamics, Eqs. (17) and (34), before the authors return to a more detailed discussion of driven systems and the geometric approach in Section 5.

3.1. Bremermann-Bekenstein Bound

  • A natural playground for exploring the ramifications of quantum speed limits are information processing systems.
  • One of the earliest explicit considerations was proposed by Bremermann [77], who considered the physical limitations of any computational device.
  • In particular, he argued that such a device must obey the fundamental laws of physics namely special relativity, quantum mechanics, and thermodynamics.
  • Thus the rate with which information is processed has to be bounded simultaneously by the light barrier, the quantum barrier, and the thermodynamic barrier.
  • In an almost heuristic way, Bremermann invoked the quantum speed limit by first considering Shannon’s seminal work on classical channel capacities and the associated noise energy, coupled with a maximum speed of propagation given by the speed of light, and then imposing the energy-time uncertainty principle.

Quantum speed limits 12

  • Where R is the radius of a sphere enclosing the system and 〈H〉 is the mean energy.
  • If the system’s entropy is maximal then using all available internal states allows for up to S/(kB ln 2) bits of information to be stored, therefore their system (enclosed by the sphere) can store at most I = (2π 〈H〉R)/(~c ln 2) bits.
  • If the authors are now interested in learning about a system, information has to be exchanged between this system and an outside observer.
  • It is interesting that in these considerations the limits on transmission are imposed by the fundamental physical laws, such as the speed of light, rather than explicitly invoking Heisenberg’s energy-time uncertainty relation.
  • The Bremerman-Bekenstein bound [78, 79] is an important result in cosmology, since it gives an upper bound on how much can be learned about non-accessible objects in the Universe, such as black holes.

3.2. Quantum thermodynamics

  • Recent years have seen a surge of interest in exploring the thermodynamics of quantum systems [80].
  • When dealing with quantum thermodynamics the notion of quantum speed limit times become fundamentally important, as is evidenced by two simple considerations.
  • Firstly, irreversibility is a core aspect of thermodynamics and indeed understanding the emergence of this irreversibility will allow us to understand the arrow of time.
  • In the following the authors examine these two situations more closely.
  • Consider a closed quantum system with Hamiltonian H0 initially in thermal equilibrium at inverse temperature β.

Quantum speed limits 13

  • A total elapsed time τ , typically the system will be forced out-of-equilibrium and therefore lead to some degree of irreversible entropy production, 〈Σ〉 ≡ β (〈W 〉 −∆F ) , (38) where 〈W 〉 is the total work done on the system during time τ and ∆F is the free energy difference.
  • The last term on the right-hand side is equal to ∆F , while the first two are (1/β) times the quantum Kullback-Leibler divergence S(ρτ ||ρeqτ ), or quantum relative entropy [85], between the actual density operator of the system ρτ at time τ and the corresponding equilibrium density operator ρeqτ .
  • A tighter bound can be derived by considering the geometric distance between these states [27] 〈Σ〉 ≥ 8 π2 L2(ρτ , ρeqτ ). (42) Already the use of the Bures metric hints that some relation with the quantum speed limit might exist.
  • It is worth noting that if initial and final states are orthogonal and in the limit of high temperatures, Eq. (44) simplifies to the Bremermann-Bekenstein bound [25].

Quantum speed limits 14

  • 2.2. Efficiency and power of quantum machines.
  • The study of thermal quantum engines has grown substantially in recent years and the quantum Otto cycle receiving particular focus, see for example Ref. [87].
  • To circumvent this issue the use of “shortcuts to adiabaticity” has been proposed to ensure the compression/expansion strokes are performed in a finite time, τ [81, 88], see Ref. [89] for a review of these techniques.
  • Interestingly, a Margolus-Levitin-type quantum speed limit on the time required to achieve the transformations can be defined [94] τ ≥ τQSL = ~L(ρi, ρf ) 〈HSA〉 (48) Finally, also the speed and efficiency of incoherent engines [95] and the effect of finite-sized clocks [96] has been studied.

3.3. Quantum computation

  • It is interesting to note that in their original paper Margolus and Levitin make explicit reference to interpreting their result in the context of the number of gate operations a computing machine can achieve per second [23].
  • Lloyd re-examined this question by first assuming that the authors have a given amount of energy with which to perform a computation, which they denote 〈H〉 for consistency of notation.
  • As noted by Lloyd, the rate at which a computer can process a computation is limited by the energy available.
  • The authors will revisit the relation between employing shortcuts to adiabaticity and the quantum speed limit in Sec. 5.7.
  • For realistic bounds additional assumptions about the information density and information transmission speed are necessary, see also Sec. 4.1.1 on quantum communication.

3.4. Quantum metrology

  • Quantum metrology deals with the use of techniques to achieve the best possible precision in estimating an unknown parameter, or parameters, of a given system.
  • Imagine the authors wish to determine some unknown parameter, µ of a given quantum system, %(µ).
  • The authors choose a measurement strategy for their estimate µ̂ and repeat this M times.
  • Then through some smart data-processing the authors can arrive at an estimate for µ.
  • Under the assumption that their strategy is unbiased, i.e. 〈 µ̂ 〉 = µ, the uncertainty in their estimate, is related to the variance of µ̂ and is lower bounded by the so-called quantum.

Quantum speed limits 16

  • 102], and hence the relation to the Mandelstam-Tamm bound (12) becomes immediately apparent.
  • While from the outset the relation between quantum speed limits and quantum metrology may not be immediately obvious, considering Eq. (52) is a type of uncertainty bound it seems wholly plausible that a strict relationship might exist.
  • In particular, consider if the parameter the authors wish to estimate is the elapsed time from a given evolution governed by a time-independent Hamiltonian, H. Then Eq. (52) bounds the time uncertainty by the QFI.
  • The authors will return to discuss some of these ideas in more detail in Sec. 5 and Sec. 7.1.

4. Optimised quantum evolution and the minimal time approach

  • While the authors have stressed previously that the quantum speed limit time, τQSL, is associated with the intrinsic properties of the system, it can of course correspond to a physically elapsed time.
  • Indeed for a given time-independent Hamiltonian the quantum speed limit time implies that there exists a driven dynamic achieving this maximal speed.
  • The relation and importance of this first became evident in the seminal work of Caneva et al. [28] in the context of optimal control.
  • In the following the authors explore the remarkable emergence of the quantum speed limit time as a fundamental limit in determining effective means to control quantum systems when they fix the physically allowed passage of time to be finite.

4.1. Optimal control theory

  • It is a well established fact that quantum systems are inherently fragile.
  • Put simply, this method involves choosing an initial ‘guess pulse’ for the functional form of the tuneable Hamiltonian parameter and then iteratively solving a Lagrange multiplier problem such that the fidelity, F = | 〈ψτ |ψT 〉 | → 1.
  • In a remarkable work, Caneva et al. [28] showed explicitly the connection between the quantum speed limit and optimal control.
  • The Krotov algorithm was then performed taking various values for the elapsed time, τ .
  • Remarkably they found that there was a minimum value of τ given by Battacharyya’s bound [51] (i.e. the Mandelstam-Tamm bound for arbitrary angles), below which the algorithm failed to converge, while for values above this they consistently found F → 1.

Quantum speed limits 18

  • From Battacharyya’s bound [51] it is easily shown τQSL = arccos | 〈ψ0|ψT 〉 | ∆H ≈ 1.56881. (58) Therefore, Caneva et al. showed that using the Krotov algorithm the minimal elapsed time one can consider corresponds exactly to τQSL.
  • This approach to quantum communication, first proposed by Bose [112], has lead to a wide ranging field of study.
  • The question of how fast this information can be propagated along the chain then becomes one of both fundamental and practical interest.
  • By employing the Krotov algorithm to optimise the profile of the magnetic field, Bn(t), they showed that there was a cutoff time τQSL, below which the algorithm failed to converge, in close analogy to the discussions from Sec. 4.1.
  • Therefore, the Quantum speed limits 19 (a) optimised evolution performs a controlled propagation of the excitation wave, and can be understood as a cascade of effective swaps, each of which has a duration shorter.

Quantum speed limits 20

  • The total quantum speed limit time is then TQSL = γ(N − 1)τQSL. (61) with γ < 1 a dimensionless constant that quantifies the effective swap duration in terms of the orthogonal swap.
  • The results show the interesting features that can emerge when dealing with many-body systems and puts into evidence the role that interactions and entanglement play in dictating the quantum speed limit.
  • In many systems these quantum phase transitions (QPTs) happen when the spectral gap between the ground and first excited state closes.
  • The adiabatic limit ensures that the system remains in its ground state at all times.
  • For an action s < π, which means τ < τQSL i.e. the driving time is less than the the quantum speed limit time, defects are produced [115].

4.2. Parametric Hamiltonians and driven dynamics

  • The emergence of the quantum speed limit time as a fundamental limitation for optimal control methods outlined previously is indeed remarkable, however, it should be noted the somewhat special circumstances considered: namely the application of a particular algorithm to design the control pulses, and the form of the initial and final states.
  • Returning to the Landau-Zener example Eq. (57), the minimal time approach.

Quantum speed limits 21

  • The crucial difference in their approach is to revisit the notion of minimal times while also recalling that by engineering special control pulses for the Hamiltonian their system was no longer truly time-independent.
  • This is particularly important considering the quantum speed limit time that emerged as the fundamental bound from Caneva et al. [28], from Eq. (58), actually assumes that the system’s Hamiltonian is time-independent.
  • Through a careful re-examination of the problem Hegerfeldt showed that the optimal control problem for a two level system is analytically treatable for arbitrary initial and final states.
  • This indicates that care must be taken when applying the quantum speed limit for time-independent Hamiltonians verbatim to certain control problems.
  • 3. Further reading on the minimal time approach.

5. Maximal quantum speed from the geometric approach

  • In the preceding section the authors discussed the so-called minimal time approach [122].
  • Within this paradigm one is interested in characterising the time optimal dynamics, or more generally the optimal generator of the quantum dynamics that drives the quantum system from a particular initial state to a particular final state, in the shortest time allowed under the laws of quantum mechanics.
  • In that the authors are no longer interested in determining the shortest evolution time, but rather the maximal speed.
  • This approach has become known as the geometric approach.

Quantum speed limits 22

  • In the geometric approach one is interested to find an estimate for the maximal quantum speed under a given quantum dynamics, which the authors will write as a quantum master equation, ρ̇t = L(ρt) . (65) Here, L(ρt) is an arbitrary, linear or non-linear, Liouvillian super-operator.
  • At first glance, these two approaches appear to be equivalent, since they give the same results for time-independent generators [117, 122].
  • Above in Sec. 2 the authors already mentioned that for mixed quantum states defining an angle is rather involved.
  • Finding such a measure of distinguishability, however, is necessary in order to be able to define the quantum speed, which should be given by the derivative of some distance.
  • For isolated systems it quickly became clear that the Bures angle (25) would do the job [66,67], whereas for open systems the situation has been less obvious.

5.1. Defining the geometric quantum speed

  • In its standard interpretation [43] quantum mechanics is a probabilistic theory, in which the state of a physical system is described by a wave function ψ(x).
  • The modulus squared of ψ(x) is the probability to find the quantum system at position x.
  • Hence to be fully consistent, a proper measure of distinguishability of two wave functions ψ1(x) and ψ2(x) should be equivalent to the distance between |ψ1〉 and |ψ2〉.
  • Čencov’s theorem states that the Fisher-Rao metric is (up to normalisation) the unique metric whose geodesic distance is a monotonic function [136].
  • Hence it is the only metric on the probability simplex that exhibits invariant properties under probabilistically natural mappings [137].

Quantum speed limits 23

  • It is worth emphasising that within the geometric approach τQSL no longer describes the actual evolution time of the quantum system.
  • In a certain sense mixed states are the quantum analogues of marginal probability distributions.
  • The latter property is very desirable since coarse graining means that discarded information cannot increase the distinguishability of quantum states.
  • Finally, the Bures angle (71) is the Fisher-Rao distance (67) maximised over all possible purifications.

5.2. Differential geometry

  • An approach inspired by differential geometry on density operator space was proposed by Taddei et al. [31].
  • Then the infinitesimal Bures angle δL can be expanded in powers of δt, δL = 1− FQ(t) 4 δt2 +O(δt3) , (73) where FQ(t) is again the quantum Fisher information, i.e., the quantum generalisation of the Fisher-Rao metric.
  • Similar results were obtained by Andersson and Heydari with more mathematical rigor, who put forward a geometric construction [138] to study the relations between Hamiltonian dynamics and Riemannian structures [139], and to characterise timeoptimal Hamiltonians [140].
  • The authors conclude this section with an observation: Therefore, Taddei et al. [31] only obtained a Mandelstam-Tamm type bound (32), whereas a Margolus-Levitin type bound (16) is beyond the scope of this approach.

5.3. Open systems and non-Markovian dynamics

  • Generally, the Bures angle L (70) is a mathematically rather involved quantity, since it is defined in terms of square roots of operators.
  • Note that for general quantum dynamics (65) the timedependent state ρτ will be mixed, even if the initial state is pure.
  • (78) From the latter both a Mandelstam-Tamm type as well as a Margolus-Levitin type bound are easily derived.

Quantum speed limits 25

  • For a Hermitian operator, they are given by the absolute value of the eigenvalues of A, and are positive real numbers.
  • If A1 and A2 are simple functions of density operators acting on the same Hilbert space, Eq. (69) remains true for arbitrary dimensions [142].
  • Next, the authors also derive a Madelstam-Tamm type bound.
  • (88) That Eq. (88) is a bound of the Mandelstam-Tamm type is obvious, since the HilbertSchmidt norm reduces for Hamiltonian dynamics to the variance of the energy [33].
  • The timedependent decay rate, γt, and the time-dependent Lamb shift, λt, are fully determined by the spectral density, J(ω), of the cavity mode.

Quantum speed limits 27

  • Deffner and Lutz [33] then found that τQSL is a monotonically decreasing function of γ0/ω0, which suggests that the non-Markovian backflow of information from the environment into the system can accelerate its dynamics.
  • In the lower panel, the authors show the quantum speed limit vQSL (81) for different values of γ0.
  • This observation was made more precise by Xu et al. [146].
  • Without loss of generality they chose the qubit to be initially prepared in its excited state, ρ0 = |1〉 〈1|.
  • Non-Markovian dynamics are characterised by an information backflow from the environment, and the convergence of ρ(t) towards the stationary state is accompanied by oscillations, cf non-monotonic behaviour of vQSL in Fig. 3 (b).

Quantum speed limits 28

  • (100) Equation (100) clearly demonstrates that the more non-Markovian the dynamics, the faster a quantum system can evolve.
  • Similar conclusions where found by Zhang et al. for Ohmic spectral densities [150] and nonequilibrium environments [151].
  • It was also pointed out that the expression for τQSL, and hence its behaviour sensitively depends on the choice of the initial state [152], which was confirmed by Zhu and Xu [153].
  • Moreover, quantum speed-ups can also be achieved by a judicious choice of the external driving protocol [154].
  • Finally, Liu et al. [155] showed that in addition to non-Markovianity, the preparation of the initial state, and the choice of the driving.

Quantum speed limits 29

  • Protocol, the characteristics of the energy spectrum govern the maximal speed of quantum evolution.
  • Finally, it is worth noting that the quantum speed limit time also sets the time-scale over which decoherence is effective [30,156,157].
  • All this theoretical work posed the natural question whether environment assisted speed-ups could be observed in an experiment.
  • To accomplish this feat, however, Cimmarusti et al. [34] had to deviate from the conventional view on cavity QED systems.
  • Moreover, the rate of evolution of the cavity field enhances – speeds up – with increasing the coupling.

5.4. Dynamics of multi-particle systems

  • Non-Markovianity can arise from collaborative excitations and correlations in the environment [160] (see Ref. [161] for a recent review of non-Markovian dynamics).
  • Thus from studying the effect of correlations in its environment on the speed of a quantum system it is only natural to also analyse the influence of collaborative effects in the system itself.
  • Moreover, Song et al. [167] found that these cooperative effects can be enhanced by a judicious choice of the external driving.
  • Generically, quantum phase transitions are accompanied by a critical slowing down close to the critical point [171, 172].
  • The LMG model was originally introduced to describe the tunneling of bosons between degenerate levels in nuclei [174], but it recently attracted attention as a testbed for shortcuts to adiabaticity [175] and for quantum thermodynamics [176].

Quantum speed limits 30

  • Note that HS can be understood as a probe system, which interacts through HSB with the LMG-model HB .
  • The question is now, if the quantum speed limit for the probe system only, ρS(t) = trB {ρtot(t)}, exhibits features of the phase transition in the environment.
  • Conversely, when the bath is characterised by the symmetric phase, where the spectral gap becomes vanishingly small with system size, τQSL witnesses a significant drop.
  • Therefore the authors can conclude that the dynamics of the probe is now significantly slower than τQSL.
  • Interestingly the authors see τQSL → 0 in the vicinity of the critical point, thus reflecting the characteristic critical slowing down mentioned previously.

5.5. Quantum speed limits for other quantities

  • Conventionally, geometric quantum speed limits are derived as upper bounds on the rate of change of a geometric measure of distinguishability.
  • In the preceding subsections the authors presented the case for the Bures angle, which is the generalised geometric angle between density operators.
  • For many applications one is not necessarily interested in how exactly the quantum states evolve, but one rather needs a quick estimate of, for instance, the typical flow of entropy [177] or the rate of decoherence [178].
  • Thus, a plethora of other speed limits have been discussed in the literature.
  • A particularly insightful treatment was proposed by Uzdin and Kosloff [179].

Quantum speed limits 31

  • (106) Equation (106) can be interpreted as a Mandelstam-Tamm type bound for the rate with which the purity, P(t), decays in open systems.
  • Note that the denominator in Eq. (106) only depends on the Lindblad operators, and thus the resulting quantum speed limit time, τPQSL, is independent of the time-dependent quantum states ρt.
  • A Margolus-Levitin type bound follows from re-writing the density operator in Liouville space, i.e., ρt is re-shaped as a vector in |ρt〉 ∈ C1×N 2 , and the authors have |ρ̇t〉 = H |ρt〉, where the Hamiltonian superoperator is given by [180].

Quantum speed limits 32

  • (111) Similarly to the case of speed limits based on the Bures angle (89) Uzdin and Kosloff [179] again find that the Margolus-Levitin type bound (110) is tighter, but that the Mandelstam-Tamm type bound (106) is much easier to compute.
  • This is illustrated by Uzdin and Kosloff [179] among other examples for dephasing channels and erasure of classical and quantum correlations.
  • The quantum speed limit based on the dynamics of the purity (111) serves as an important and illustrative example of various studies.
  • Dehdashti et al. [181] used the quantum speed limit based on the relative purity [32] to study decoherence in the spin-deformed boson model, whereas Jing et al. [182] quantified the generation of quantumness, also known as Other examples include.
  • Finally, Sun et al. [187] focused on computable bounds.

5.6. Universal, geometric quantum speed limits

  • Comparing the expressions for the quantum speed limit time, τQSL, based on the Bures angle (89) and based on the purity Eq. (111), as well as the expressions for the relative purity [32] and the Winger-Yanase information [135] the authors notice that the choice of the measure of distinguishability only determines the numerator.
  • The actual speed limit is set in all of these cases by a time-averaged Schatten-p-norm of the generator of the dynamics.
  • This observation can be made more precise, if one starts the derivation of the speed limit directly with such a norm [188].
  • For p = 2 the authors have the Hilbert-Schmidt distance, for p = 1 the trace distance, and for p =∞ the operator norm.

Quantum speed limits 33

  • Computing most of the Schatten-p-norms is rather involved, since the singular values of the generator of the dynamics have to be determined.
  • An equivalent, and much more tractable quantum speed limit can be derived, if one considers the quantum state in its.
  • In conclusion, it was shown in Ref. [188] that, independent of the choice of measure, the quantum speed is universally characterised by the Schatten-p-norm of the generator of the dynamics.
  • Moreover, this Schatten-p-norm can be computed from the mathematically tractable Wasserstein-p-norm of the corresponding Wigner function.

5.7. Applications: Shortcuts to adiabaticity

  • The authors conclude this section on the geometric approach by highlighting an important application that was recognised only very recently.
  • In the area of quantum control so-called shortcuts to adiabaticity have become a prominent and active topic [89].
  • A shortcut to adiabaticity is a fast process with the same final state that would result from infinitely slow driving.
  • Among all of these techniques transitionless quantum driving is unique.
  • In its original formulation [90–92] one considers a time-dependent Hamiltonian H0(t) and constructs an additional counterdiabatic field, H1(t), such that the joint Hamiltonian.

Quantum speed limits 34

  • At first glance, it seems that such an energetically free shortcut to adiabaticity could be implemented for any arbitrarily fast process of arbitrarily short duration τ .
  • That this is not the case and how the quantum speed limit enters the picture was formalised by Campbell and Deffner [93].
  • In Ref. [88] a family of functionals has been proposed to quantify the cost associated with implementing H1(t).
  • It is easy to see that for a single 2-level spin, ∂tC is proportional to the average power input [88], i.e., H1(t) reduces to an orthogonal, magnetic field.
  • More generally, C can be interpreted as the additional action arising from the counterdiabatic driving.

6. Quantum speed limits in non-Schrödinger quantum mechanics

  • In the above sections the authors have been aiming to provide an overview over the various notions of a quantum speed limit in standard quantum mechanics.
  • So far, the authors have always assumed that the dynamics of an isolated systems is described by the vonNeumann equation (8), and that open systems can be described by a quantum master equation (65).
  • And discuss generalisations and applications of the quantum speed limit for dynamics that are beyond the applicability of the Schrödinger equation.the authors.

6.1. Relativistic systems and Dirac dynamics

  • Since its inception [218] the Dirac equation (126) has proven to be one of the most versatile results of theoretical physics.
  • Originally it was designed as a relativistic wave equation to describe massive spin-1/2 particles, such as electrons and quarks [216,217].
  • It is only natural to ask whether and how the quantum speed limit generalises to relativistic Dirac dynamics.
  • To keep things as simple as possible Villamizar and Duzzioni [227] then considered systems in the x-y plane withB(x) =.
  • B ẑ, and an initial state that is is a homogeneous superposition of two radial eigenstates in different Landau energy levels.

Quantum speed limits 36

  • Note that in the non-relativistic limit τSQSL and τDQSL become equivalent.
  • Villamizar and Duzzioni ’s [227] work has been followed by several further analyses for which the authors refer the reader to the literature [228–231].
  • Finally, it is interesting to note that in passing Villamizar and Duzzioni [227] resolved the longstanding debate of Einstein and Bohr (see Sec. 1) by bringing the energy-time uncertainty principle in full agreement with special relativity.

6.2. PT-symmetric quantum mechanics

  • Another area of research that has attracted considerable attention are non-Hermitian [232–236] and PT -symmetric quantum systems [178, 237–240].
  • Since its originial development as a rather mathematical theory [237, 241], PT -symmetric quantum mechanics has found experimental realisation as systems with balanced loss and gain [241–247].

Quantum speed limits 37

  • Since T also changes the sign of the imaginary unit i, canonical commutation relations such as [x, p] = i~ are invariant under PT .
  • The major difference between Hermitian and PT -symmetric quantum mechanics is the definition of the inner product [238,248].
  • This observation will become crucial below, since this means that for the geometric approach the angle Eq. (67) will have to be re-defined.
  • Thus, one would naively expect that both the MandelstamTamm (12) as well as the Margolus-Levitin bound (16) for time-dependent generators to remain valid.
  • To analyse this claim Bender et al. [249] considered a simple case study.

Quantum speed limits 38

  • Bender et al. [249] also note that such |ψ0〉 and |ψT 〉 are no longer orthogonal, since under PT -symmetric Hamiltonians the inner product, and hence the geometry of Hilbert space has to be modified (137).
  • Thus, Uzdin et al. [250] revisited the geometric approach to the quantum speed limit.
  • In their analysis Uzdin et al. [250] then use Eq. (143) to identify time-optimal Hamiltonians, for which the fastest evolution between specific initial and final states can be achieved.
  • Identifying time-optimal Hamiltonians is sometimes also dubbed the quantum brachistochrone problem, which has been further elaborated in the literature for Hermitian [251–257] as well as non-Hermitian systems [258–264].

6.3. Non-linear systems

  • Finally, the notion of quantum speed limits has also been generalised to non-linear systems.
  • Non-linear Schrödinger equations arise from effective descriptions of manyparticle systems, such as Bose-Einstein condensates (BEC) [265] or in non-linear optics [266].
  • The second model is more involved and describes oscillations between atomic and molecular BECs.
  • It had been claimed by Dou et al. [268] that nonlinear interactions strongly affect the minimal time a quantum system needs to evolve between distinct states.

Quantum speed limits 39

  • Claim was refuted by Chen et al. [267], and they showed explicitly that the maximal quantum speed of nonlinear systems for unconstrained controls is independent of the strength of the non-linearity, and that non-linear systems at maximal speed evolve as fast as equivalent linear systems.
  • In particular, they found for unconstrained control protocols Γ(t), τmin = |θi − θf | 2ω0 , (146) where θi and θf are initial and final azimuth on the Bloch sphere.
  • Note that τmin is independent of κ and that τmin is similar to the result for linear qubits with more than one control parameter (64).

Quantum speed limits 40

  • Of course, in the situation where the parameter to estimate is time and H corresponds to the system Hamiltonian, then the analysis reduces to the “standard” quantum speed limit.
  • The remarkable insight of Ref. [269] is that for any pair of conjugate variables this reasoning can be used to establish tight bounds on the precision with which they can be estimated.

7.3. Lieb-Robinson bound

  • The authors saw in Sec. 4.1.1 that the quantum speed limit fundamentally bounds the rate at which a quantum state can be communicated along a quantum channel.
  • Indeed, while the emergence of τQSL is remarkable, the fact that information cannot propagate arbitrarily fast when the interaction range is limited is quite intuitive.
  • In fact, this notion is formalised in the form of Lieb-Robinson bounds [114].
  • In essence, these bounds set constraints on the speed with which local effects can propagate through.

Quantum speed limits 41

  • A quantum system by rigorously showing that correlations between the expectation values of local observables are exponentially suppressed outside an effective “spacetime cone”.
  • The authors refer to Ref. [277] for an introduction to the more formal aspects of Lieb-Robinson bounds and some of their more far reaching implications.
  • If the authors consider only nearest-neighbour interactions, then the speed with which correlations can travel along a one-dimensional chain is bounded linearly by the so-called Lieb-Robinson velocity, vLR.
  • This behaviour was recently verified experimentally in Ref. [278], while the analysis for long range interactions, where such bounds can be violated is found in Ref. [118].
  • To the best of their knowledge, it is currently an open question regarding a more stringent relation between vLR and the speed arising from the quantum speed limit, vQSL.

7.4. Maximal rate of quantum learning and Holevo’s information

  • As a final example the authors return to the Bremerman-Bekenstein bound (36).
  • In its original formulation it gives an upper bound on the rate with which information can be extracted from a quantum system.
  • Generally this is not the case [279] due to the quantum back action of measurements [280].
  • Imagine that the authors have an observable A = ∑ n anΠn, where an are the measurement outcomes, and Πn are the projectors into the eigenspaces corresponding to an.
  • (153) In Ref. [281] this bound was studied for the harmonic oscillator and for the PöschlTeller potential by means of time-dependent perturbation theory.

8. Final remarks

  • Throughout this Topical Review the authors have striven to focus on the practical applications of the quantum speed limit.
  • In particular, with the rapid development of new quantum technologies and the recent surge of interest in exploring the thermodynamic working principles of quantum systems it is important to understand the limits on controlling such systems.
  • Arguably the most basic question one can ask is: how fast can the authors achieve.

Quantum speed limits 42

  • To which one then naturally would attempt to reduce the time or, seemingly equivalently, speed up the process.
  • In most set-ups for quantum computation one works hard to isolate the system against noise and decoherence from the environment.
  • While of fundamental interest, arguably it is the practical ramifications of the energy-time uncertainty principle that has lead to the renewed interest.
  • As the authors have tried to evidence in this Topical Review, the seeming ubiquity of the quantum speed limit in controlling quantum systems indicates more is to be learned.
  • Over the last century, however, quantum theory has undoubtedly become the cornerstone of physics and its “peculiarities” are what make the Universe work.

Acknowledgments

  • The authors are grateful to Marta Paczyńska for creating the visual representation of Einstein’s gedankenexperiment, Fig. 1, and Lu (Lucy) Hou for providing the resources for Fig.
  • SD would like to thank Eric Lutz for many years of insightful discussions and supporting mentorship, and in particular for inciting their interest in quantum speed limits.
  • This work was supported by the U.S. National Science Foundation under Grant No. CHE-1648973.

Quantum speed limits 43

  • The uncertainty principle for energy and time.
  • J. D. Bekenstein. Energy Cost of Information Transfer.
  • Dynamical time and time indeterminacy, also known as Part I.

Quantum speed limits 44

  • Generalized time-energy uncertainty relations and bounds on lifetimes of resonances.
  • M. Andrecut and M. K. Ali. Maximum speed of quantum evolution.
  • An extension of Kakutani’s theorem on infinite product measures to the tensor product of semifinite w*-algebras.

Quantum speed limits 45

  • The speed limit of quantum unitary evolution.
  • The role of quantum information in thermodynamics - a topical review.
  • Peter Talkner, Eric Lutz, and Peter Hanggi.
  • Trade-off between speed and cost in shortcuts to adiabaticity.
  • Fast quantum computation at arbitrarily low energy.

Quantum speed limits 46

  • Kind of entanglement that speeds up quantum evolution.
  • Nonlinear quantum metrology of many-body open systems.
  • Communication at the quantum speed limit along a spin chain.
  • Driving at the quantum speed limit: Optimal control of a two-level system.

Quantum speed limits 47

  • An introduction to quantum entanglement, also known as Geometry of quantum states.
  • A note on von Neumann’s trace inequalitv.
  • Quantum speed limit for arbitrary initial states.
  • Quantum dynamical speedup in a nonequilibrium environment.

Quantum speed limits 48

  • [161] Heinz-Peter Breuer, Elsi-Mari Laine, Jyrki Piilo, and Bassano Vacchini.
  • Criticality revealed through quench dynamics in the Lipkin-Meshkov-Glick model.
  • Sh. Dehdashti, M. Bagheri Harouni, B. Mirza, and H. Chen.
  • Quantum coherence sets the quantum speed limit for mixed states.

Quantum speed limits 49

  • Optimal frictionless atom cooling in harmonic traps: Shortcut to adiabaticity.
  • Fast-forward of adiabatic dynamics in quantum mechanics.
  • Generating shortcuts to adiabaticity in quantum and classical dynamics.
  • Optimal trajectories for efficient atomic transport without final excitation.
  • Suppression of work fluctuations by optimal control:.

Quantum speed limits 50

  • Resonantly enhanced pair production in a simple diatomic model.
  • Enhanced Schwinger pair production in many-centre systems.
  • Quantum speed limit for a relativistic electron in the noncommutative phase space.
  • Physical realization of PT -symmetric potential scattering in a planar slab waveguide.

Quantum speed limits 51

  • The quantum brachistochrone problem for non-hermitian hamiltonians.
  • Time-optimal quantum control of nonlinear two-level systems.
  • Experimental verification of landauer’s principle linking information and thermodynamics.

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Topical Review
Quantum speed limits: from Heisenberg’s
uncertainty principle to optimal quantum control
Sebastian Deffner
1
and Steve Campbell
2
1
Department of Physics, University of Maryland Baltimore County, Baltimore,
MD 21250, USA
2
Istituto Nazionale di Fisica Nucleare, Sezione di Milano & Dipartimento di
Fisica, Universit`a degli Studi di Milano, via Celoria 16, 20133 Milan, Italy
E-mail:
1
deffner@umbc.edu and
2
steve.campbell@mi.infn.it
Abstract. One of the most widely known building blocks of modern physics is
Heisenberg’s indeterminacy principle. Among the different statements of this fun-
damental property of the full quantum mechanical nature of physical reality, the
uncertainty relation for energy and time has a special place. Its interpretation
and its consequences have inspired continued research efforts for almost a cen-
tury. In its modern formulation, the uncertainty relation is understood as setting
a fundamental bound on how fast any quantum system can evolve. In this Topical
Review we describe important milestones, such as the Mandelstam-Tamm and the
Margolus-Levitin bounds on the quantum speed limit, and summarise recent ap-
plications in a variety of current research fields including quantum information
theory, quantum computing, and quantum thermodynamics amongst several oth-
ers. To bring order and to provide an access point into the many different notions
and concepts, we have grouped the various approaches into the minimal time ap-
proach and the geometric approach, where the former relies on quantum control
theory, and the latter arises from measuring the distinguishability of quantum
states. Due to the volume of the literature, this Topical Review can only present
a snapshot of the current state-of-the-art and can never be fully comprehensive.
Therefore, we highlight but a few works hoping that our selection can serve as a
representative starting point for the interested reader.
Keywords: Quantum speed limits, Heisenberg uncertainly principle, optimal
control theory, shortcuts to adiabaticity, quantum information theory, quantum
thermodynamics.
arXiv:1705.08023v2 [quant-ph] 16 Oct 2017

Quantum speed limits 2
1. Introduction
It is a historic fact that Einstein never seemed quite comfortable with the probabilistic
interpretation of quantum theory. In a letter to Born he once remarked [1]:
Quantum mechanics is certainly imposing. But an inner voice tells me that
it is not yet the real thing. The theory says a lot, but does not really bring
us any closer to the secret of the old one. I, at any rate, am convinced that
He does not throw dice.
Nevertheless, quantum mechanics is built on the very notion of indeterminacy, which
is rooted in Heisenberg’s uncertainty principles, and which can be expressed in terms
of the famous inequalities [2, 3],
p x & ~ and E t & ~ . (1)
Although physically insightful, these relations, Eq. (1), were originally motivated only
by plausibility arguments and by “observing” the commutation relations of canonical
variables in first quantization§.
While the uncertainty relation for position and momentum was quickly put on
solid grounds by Bohr [4] and Robertson [5], the proper formulation and interpretation
for the uncertainty relation of time and energy proved to be a significantly harder task.
In its modern interpretation the uncertainty relation for position and momentum
expresses the fact that the position and the momentum of a quantum particle cannot
be measured simultaneously with infinite precision [6]. However, if the uncertainty
principle is a statement about simultaneous events, the interpretation of an uncertainty
in time is far from obvious [7].
Thus, only three years after its inception Einstein challenged the validity of the
energy-time uncertainty relation with the following gedankenexperiment as depicted
in Fig. 1 [8]: Imagine a box containing photons, which has a hole in one of its walls.
This hole can be opened and closed by a shutter controlled by a clock inside the box.
At a preset time the shutter opens the hole for a short period and lets photons escape.
Since the clock can be classical, the duration can be determined with infinite precision.
From special relativity we know that energy and mass are equivalent, E = mc
2
. Hence,
by measuring the mass of the box in the gravitational field, the change in energy due
to the loss of photons can also be determined with infinite precision. As a consequence,
special relativity seems to negate the existence of an uncertainty principle for energy
and time!
Bohr’s counterargument in essence states that in order to measure time, position
and momentum of the hands of the clock have to be determined. In addition,
accounting for time dilation due to motion in the gravitational field, the uncertainty
relation for position and momentum implies an uncertainty relation for energy
and time. Although insightful, Bohr’s interpretation cannot be considered entirely
satisfactory [79], since it merely circumvents the problem of explaining the existence
of the uncertainty principle in non-relativistic quantum mechanics.
A major breakthrough was achieved by Mandelstam and Tamm [10], who realised
that E t & ~ is not a statement about simultaneous measurements, but rather
English translation of the German original: Die Quantenmechanik ist sehr achtung-gebietend. Aber
eine innere Stimme sagt mir, daß das doch nicht der wahre Jakob ist. Die Theorie liefert viel, aber
dem Geheimnis des Alten bringt sie uns kaum aher. Jedenfalls bin ich berzeugt, daß der nicht
urfelt.
§ Without any mathematical justification Heisenberg explicitly writes [2], Et tE = i~.

Quantum speed limits 3
Figure 1. Sketch of the gedankenexperiment envisaged by Einstein.
about the intrinsic time scale of unitary quantum dynamics. Hence, one should rather
write t & ~/E, where t is interpreted as the time a quantum system needs to
evolve from an initial to a final state. More specifically Mandelstam and Tamm derived
the first expression of the quantum speed limit time τ
QSL
= π~/2∆H, where H
2
is
the variance of the Hamiltonian, H, of the quantum system [10]. As an application of
their bound, they also argued that τ
QSL
naturally quantifies the life time of quantum
states [10], which has found widespread prominence in the literature [1116].
Nevertheless, the desire to formalise time as a proper quantum observable
persisted [17, 18]. To make matters worse, it was further argued that the variance
of an operator is not an adequate measure of quantum uncertainty [19,20], which only
highlighted that the uncertainty relation for time and energy needed to be put on even
firmer grounds.
With the advent of quantum computing [21, 22] Mandelstam and Tamm’s
interpretation of the quantum speed limit as intrinsic time-scale experienced renewed
prominence. Their interpretation was further solidified by Margolus and Levitin [23],
who derived an alternative expression for τ
QSL
in terms of the (positive) expectation

Quantum speed limits 4
value of the Hamiltonian, τ
QSL
= π~/2 hHi. Eventually, it was also shown that the
combined bound,
τ
QSL
= max
π
2
~
H
,
π
2
~
hHi
(2)
is tight [24]. This means Eq. (2) sets the fastest attainable time-scale over which
a quantum system can evolve. In particular, Eq. (2) sets the maximal rate with
which quantum information can be communicated [25], the maximal rate with which
quantum information can be processed [26], the maximal rate of quantum entropy
production [27], the shortest time-scale for quantum optimal control algorithms to
converge [28], the best precision in quantum metrology [29], and determines the
spectral form factor [30].
The next major milestone in the development of the field was achieved only
relatively recently with the generalisation of the quantum speed limit to open systems.
In 2013 three letters proposed, in quick succession, three independent approaches
of how to quantify the maximal quantum speed of systems interacting with their
environments. Taddei et al. [31] found an expression in terms of the quantum Fisher
information, del Campo et al. [32] bounded the rate of change of the relative purity, and
Deffner and Lutz [33] derived geometric generalizations of both, the Mandelstamm-
Tamm bound as well as the Margolus-Levitin bound. These three contributions [3133]
effectively opened a new field of modern research, since for the first time it became
obvious that the Heisenberg uncertainty principle for time and energy is not only
of fundamental importance, but actually of quite practical relevance. For instance,
it became clear that quantum processes in systems interacting with non-Markovian
environments can evolve faster than in systems coupled to memory-less, Markovian
baths which has been verified in a cavity QED experiment [34].
The purpose of this Topical Review is to take a step back and bring order into the
plethora of novel ideas, concepts, and applications. Thus, in contrast to earlier reviews
on quantum speed limits [3542], we will focus less on mathematical and technical
details, but rather emphasise the interplay between the various concepts, tradeoffs
between speed and physical resources, and consequences in real-life applications. We
will begin with a historical overview in Section 2, where we will also summarise
the original derivations by Mandelstam-Tamm and Margolus-Levitin. Section 3 is
dedicated to quantum systems with time-independent generators, whereas Section 4
focuses on optimal control theory. In Section 5 we will discuss the geometric approach,
which so far has been the most fruitful approach in the description of open quantum
systems, and which has led to the most interesting insights. The review is rounded
off with Section 6, in which we briefly summarise generalisations to relativistic and
non-linear quantum dynamics, and Section 7 which outlines the relation of quantum
speed limits to other fundamental bounds.
When writing this Topical Review, we strove for objectivity and completeness.
However, the sheer volume of publications demanded to select but a few works to
be discussed in detail. Our selection was motivated by accessibility, pedagogical
value, and conceptual milestones. Therefore, this review can never be a fully complete
discussion of the literature, but rather only serve as a starting point for further study
and research on quantum speed limits.

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Frequently Asked Questions (10)
Q1. What are the contributions mentioned in the paper "Quantum speed limits: from heisenberg’s uncertainty principle to optimal quantum control" ?

In this Topical Review the authors describe important milestones, such as the Mandelstam-Tamm and the Margolus-Levitin bounds on the quantum speed limit, and summarise recent applications in a variety of current research fields – including quantum information theory, quantum computing, and quantum thermodynamics amongst several others. Due to the volume of the literature, this Topical Review can only present a snapshot of the current state-of-the-art and can never be fully comprehensive. Therefore, the authors highlight but a few works hoping that their selection can serve as a representative starting point for the interested reader. 

An interesting caveat associated with using quantum systems as the working substance is the expansion/compression strokes should be performed adiabatically, which according to the quantum adiabatic theorem requires them to be performed (infinitely) slowly, and thus render the considered engine useless as its output power would be zero. 

In many systems these quantum phase transitions (QPTs) happen when the spectral gap between the ground and first excited state closes. 

In particular, with the rapid development of new quantum technologies and the recent surge of interest in exploring the thermodynamic working principles of quantum systems it is important to understand the limits on controlling such systems. 

Then using the tuneable parameters of the system’s Hamiltonian the authors seek to maximise the final fidelity of the evolved state with |ψT 〉. 

For s ≥ π, which corresponds to durations τ ≥ τQSL, the Krotov algorithm always converges and this is defined as a region in which adiabatic dynamics can be effectively achieved. 

In their analysis Uzdin et al. [250] then use Eq. (143) to identify time-optimal Hamiltonians, for which the fastest evolution between specific initial and final states can be achieved. 

An upper bound on the rate, İ, with which the information is communicated is given by the total information stored in the system divided by the minimal time it would take to erase all this information. 

To circumvent this issue the use of “shortcuts to adiabaticity” has been proposed to ensure the compression/expansion strokes are performed in a finite time, τ [81, 88], see Ref. [89] for a review of these techniques. 

In its modern interpretation the uncertainty relation for position and momentum expresses the fact that the position and the momentum of a quantum particle cannot be measured simultaneously with infinite precision [6].