Quantum clocks are more accurate than classical ones

# Quantum clocks are more accurate than classical ones

Mischa P. Woods Institute for Theoretical Physics, ETH Zurich, Switzerland Department of Computer Science, University College London, UK    Ralph Silva Institute for Theoretical Physics, ETH Zurich, Switzerland Département de Physique Appliquée, Université de Genève, Switzerland    Gilles Pütz Institute for Theoretical Physics, ETH Zurich, Switzerland    Sandra Stupar Institute for Theoretical Physics, ETH Zurich, Switzerland    Renato Renner Institute for Theoretical Physics, ETH Zurich, Switzerland
###### Abstract

A clock is, from an information-theoretic perspective, a system that emits time information. One may therefore ask whether the theory of information imposes any constraints on the maximum precision of clocks. We find that, indeed, the accuracy of the time information generated by a clock is fundamentally limited by the clock’s size or, more precisely, the dimension  of its quantum-mechanical state space. Furthermore, compared to a classical clock, whose evolution is restricted to stochastic jumps between perfectly distinguishable classical states, a genuine quantum clock can achieve a quadratically improved accuracy.

## I Introduction

M.W. and R.S. contributed equally to the results.

The definition of units of time is traditionally based on periodic processes, such as the moon cycle, the rotation of the earth, or the transition between two energy levels of an atom SI2006 (). However, not any system with a periodically evolving state can serve as a clock. A necessary requirement to a clock, apart from having a time-dependent internal state, is hence that it also emits information about this state to the outside — a principle formalised in RaLiRe15 (), and employed by the Quasi-Ideal Clock used for quantum control in WSO16 (), and the thermodynamic clock in thermoClockErker ().

This information-theoretic viewpoint points to an inherent limitation that any clock is subject to, and which can be seen as an instance of the “information-disturbance” tradeoff FuchsPeres96 (). According to quantum theory, the transfer of information about the clock’s internal state to the outside will in general disturb the former. As an illustration, one may think of a mechanical clock, where the time information is represented by the clock’s hands. Modelled within quantum mechanics, the state of the clock’s hands would correspond to a wave packet well localised in position and momentum space — at least as long as the clock is in normal operating mode. But if, in an attempt to measure time very precisely, we observed the position of the clock’s hands too closely, we would necessarily spread the wave packet in momentum space Schroedinger31 (). This, in turn, would disrupt the dynamics of the clock, thus making any future time measurement less accurate.

The example also provides intuition for what makes an accurate clock. Clearly, if the clock’s hands are massive, reading off their position will not cause much disturbance to them. Conversely, if the clock’s state was just represented by the orientation of, say, a single molecule, even a moderately precise time measurement could alter it significantly. One would therefore expect that the maximum achievable accuracy of a clock is size-dependent — small clocks cannot be as accurate as large ones.

The size of a clock can in principle be quantified in many different ways, e.g., by its mass SaleckerWigner58 (). We take here an approach that is motivated by information theory and consider the size of its state space. This is measured in terms of the number, , of perfectly distinguishable states the clock can be in, i.e., the dimension of the associated Hilbert space. In other words, a clock of size  is a clock that could in principle store bits of information in its internal state.

Continuing this information-theoretic description, it is useful to distinguish between two types of devices for measuring time: timepieces that output time information on request, like a stop watch, and clocks that output time information autonomously, like a chiming clock. They serve different purposes. Stop watches are used to measure a time interval between events triggered by external process (e.g., between the event that a train leaves the station at  and the event that it arrives at ) OptimalStopwatch (). Conversely, chiming clocks “generate” events themselves, which may then be used to trigger external events (e.g., that the train leaves the station at ).111A chiming clock may of course also be used to measure a time interval: simply count how many ticks generated by the clock fit into the interval. However, such a use appears to be rather wasteful in terms of the time information that needs to be generated.

This work is concerned with the second type of time-keeping. Hence, from now on (and with the exception of the review of earlier work at the beginning of Section II) we use the term clock for devices that output time information autonomously.222The word “clock” derives from the Medieval Latin “clocca”, which means “bell”. The hourly ringing of the bells may be regarded as an autonomous process. Specifically, we take a clock to be a device that generates a sequence of individual events, which we call ticks. For the purpose of this discussion, we assume that the ticks are the only information output by the clock.

Ideally, one would like to quantify the accuracy of clocks in an operational manner, and in such a way that no absolute time reference is necessary. Such an operational measure has been introduced in RaLiRe15 (). The idea is most easily phrased in terms of the Alternate Ticks Game. Suppose you have two copies of a clock. The task is to initialise them such that they tick in alternating order as long as possible; the number of ticks during which this order is maintained is then a measure for the clock’s accuracy (see RaLiRe15 () for a discussion).

Unfortunately, the operational measure is rather difficult to analyse for general quantum clocks.333See however ATGRandomWalk (), that analyses for certain classical clocks. But, clearly, a clock has a high accuracy if the time elapsed between any two subsequent ticks does not fluctuate too much. We may thus alternatively quantify accuracy by a value , which, roughly, corresponds to the number of ticks that the clock can generate until it is off by a time interval that is as large as the time between two ticks444This is an exact correspondence for clocks whose ticks form an independent sequence, see Appendix E.2.1., as was introduced for thermodynamic clocks in thermoClockErker (). While, intuitively, is a good approximation for  and vice versa, the precise relation between these two quantities is still an open question (see also the discussion section III.1.4).

We use the term quantum clocks for a clock whose dynamics is not subject to any constraints other than those imposed by quantum theory. Their internal state can therefore be represented by a density operator in a -dimensional Hilbert space, and the transition from the clock’s state at a time to its state at time corresponds to a trace-preserving completely positive map. We also consider the special case of classical clocks. Their state space is, by definition, restricted to a fixed set consisting of perfectly distinguishable states and their probabilistic mixtures — the “classical” states. In this case, a state transition from time to is most generally represented by a stochastic map.

Our main results are bounds on the accuracy of clocks depending on the clock’s size . On the one hand, we prove that, for any fixed , there exist quantum clocks of large enough dimension whose accuracies scale as

 Rquantum≥Ω(d2−η). (1)

That is, quantum clocks can have an accuracy that grows essentially quadratically in the clock’s size for large . We prove this statement by construction, showing that the so-called Quasi-Ideal Clocks proposed in WSO16 () achieve this bound. On the other hand, we prove that the accuracy of any classical clock is upper bounded by

 Rclassical≤d, (2)

and show that a simple stochastic clock, previously studied in ATGRandomWalk () in the context of the Alternate Ticks Game, saturates this bound. Combining Eqs. (1) and (2), we conclude that for large size , quantum clocks outperform classical ones quadratically in terms of their accuracy .

This result allows us to add a new item to the list of tasks with a quantum-over-classical advantage — the task of time-keeping. Other items on this list come from the areas of computation and metrology. For example, Grover’s quantum algorithm exhibits a quadratic speed-up compared to any classical algorithm for database search Grover1996 (). Similarly, the number of basic measurement steps555A typical example is the measurement of a distance using laser interferometry, where the relevant resource is the energy of the beam or the number of photons that are involved in the measurement process. In this case one may regard the generation and detection of a single photon as a basic measurement step. required to determine an experimental parameter to a given precision drops like a square root if one uses quantum instead of classical methods GiLlMa11 (). However, despite the superficial similarities between these results, they are of a fundamentally different kind, for they refer to different resources.666In particular, we are not aware of a result that would allow us to obtain statements like Eqs. (1) and (2) from known bounds in quantum metrology; see Section III.1.2 for more details. In computation, the relevant resource is the number of computational steps, and in the case of metrology, it is the number of basic measurement steps. In contrast, for the task of time keeping considered here, the relevant resource is the size of the clock.

## Ii Modelling clocks

To motivate our framework for describing clocks, we first have a look at existing models that have been considered in the literature and discuss their limitations. (An extensive review on prior literature regarding clocks and the general issue of time in quantum mechanics can be found in TiQMVol1 (); TiQMVol2 ().)

Pauli regarded an “ideal clock” as a device that has an observable whose value is in one-to-one correspondence to the time parameter  in the quantum-mechanical equation of motion. The observable would need to satisfy . Furthermore, since neither nor the Hamiltonian of the system, , should depend on time explicitly, they would need to satisfy the commutation relation .777We set , so that . Pauli then argued that this implies that has as its spectrum the full real line Pashby14 (). Since such Hamiltonians are unphysical, he concluded that an observable with the desired properties, and hence an ideal clock, cannot exist pauli1 (); PauliGeneralPrinciples ().888We note that this conclusion has been challenged and it has been argued that the relation can be satisfied for Hamiltonians with semi-bounded spectrum if one considers operators with restricted domains of definition (see Pashby14 () for a discussion). Such restrictions however still correspond to unphysical assumptions, such as infinite potentials to keep a particle in a confined region. As such, these objects are referred to as Idealised Clocks.

This raises the question whether one can at least approximate an Idealised Clock. Salecker and Wigner SaleckerWigner58 () and Peres Peres80 () considered finite-dimensional constructions. Specifically, they showed that for any dimension  and for any fixed time interval  there exists a clock, which we will refer to as the SWP Clock, whose Hamiltonian satisfies

 ∀k∈{0,…,d−1}:eiHΔ|ϑk⟩=|ϑk+1(modd)⟩

where is the SWP basis — an orthonormal basis of the clock’s Hilbert space. Hence, if the clock was initialised to state and if one did read the clock at a time by applying a projective measurement with respect to the SWP basis, the outcome would be precise time information . However, in between these particular points in time, the amplitudes of the basis states are in general all non-zero Gross2012 (). Hence, if the clock was measured, say, at , the outcome would be uncertain999At intermediate time intervals, the variance of the state w.r.t. the basis states is as much as .. In addition, such a measurement would disturb the clock’s state, effectively resetting it to a random time. This problem was resolved in recent work by Woods, Silva, and Oppenheim, who introduced a clock, called the Quasi-Ideal Clock WSO16 (), which is able to approximate the dynamical behaviour of Pauli’s Idealised Clock while maintaining a finite dimension.

The constructions from SaleckerWigner58 (); Peres80 () do however not include a mechanism to output time information autonomously. Hence, to use the terminology introduced earlier, they are stop watches rather than chiming clocks. To extract time information from them, one would have to apply measurements from the outside. But then the outcome depends on when and how these measurements are performed. Thus, in order to reasonably talk about their accuracy — in terms of operationally motivated quantities such as and — we need a more complete description. The definition of quantum clocks, as outlined in the following section, accounts for this.

### ii.1 Quantum Clocks

The modelling of clocks that we use here follows the operational approach introduced in RaLiRe15 () with some adjustments. A -dimensional quantum clock consists of a (generally open) quantum system  whose evolution is assumed to be time-independent.101010Since any realistic clock is subject to noise, i.e., environmental perturbations that may vary over time, this assumption is not usually met perfectly. But since our aim is to explore information-theoretic constraints on a clock’s accuracy, we work here under the assumption that the clock is shielded from time-dependent perturbations. In fact, perturbations with a known time-dependence would need to be counted as an additional resource from which the clock could extract time information. The transition of a clock’s state at some time to its state at a later time can hence most generally be described by a trace-preserving completely positive map

 MΔC→C:ρC,t↦ρC,t+Δ,

which depends on  but not on . Note that these maps form a family parameterised by . For the particular choice it is the identity map,

 M(0)C→C=IC. (3)

Furthermore, the maps are mutually commutative under composition, i.e.,

 MΔ+Δ′C→C=MΔ′C→C∘MΔC→C=MΔC→C∘MΔ′C→C, (4)

for any . In other words, the evolution of is determined by a one-parameter family of maps, , and thus Markovian.

Assuming that the energy that drives the clock’s evolution is finite, we may additionally assume that the clock’s state changes at a finite speed. This means that the function is continuous. But, using Eqs. (3) and (4), this is in turn equivalent to the requirement that

 limΔ→0MΔC→C=IC, (5)

which may be regarded as a strengthening of Eq. (3).

Since we assumed that the clock’s evolution is time-independent, its description in terms of the entire family , for , is highly redundant. Indeed, using Eq. (4) we may write111111We use here the notation .

 MΔC→C=limδ→0(MδC→C)⌊Δδ⌋ (6)

It thus suffices to specify the evolution map for arbitrarily small time parameters, which we will in the following denote by . (The evolution is thus governed by the Lindblad equation, a fact that we will exploit in Section II.2.)

The maps describe the evolution of the state on . But, as argued above, we are generally interested in the information that the clock transmits to the outside. This can be included in our description by virtue of extensions of the maps . That is, we consider maps whose target space is a composite system, consisting of and an additional system , such that

 MδC→C=\textuptrT∘MδC→CT. (7)

We call  the tick register, alluding to the idea that the basic elements of information emitted by a clock are its “ticks” RaLiRe15 ().

After these general remarks, we are now ready to state the technical definition.

###### Definition 1 (RaLiRe15 ()).

A (quantum) clock is a pair , consisting of a density operator on a -dimensional Hilbert space  together with a family of trace preserving and completely positive maps from to , where is an arbitrary system, such that the following limits exist and take on the value

 limΔ→0limδ→0(\textuptrT∘MδC→CT)⌊Δδ⌋=IC. (8)

Using Eq. (6), it is easy to see that any family of maps whose reduction to  satisfies Eqs. (4) and (5) also satisfies Eq. (8). The converse is however not necessarily true. Nevertheless, given a family of maps as in Def. 1, one may always define a family of maps

 (9)

which meet both Eqs. (4) and (5). In this sense, specifying a map that satisfies Eq. (8) is indeed sufficient to define the continuous and time-independent evolution of a clock.

The definition does not yet impose any constraints on the tick register, . Since we want to compare different clocks, it will however be convenient to assume that contains two designated orthogonal states, and , which we interpret as “tick” and “no tick”, respectively. The idea is that ticks are the most basic units of time information that a clock can emit. Roughly speaking, a tick indicates that a certain time interval has passed since the last tick.

To know if the clock has ticked after the application of the map , one has to measure the tick register in the “tick” basis . In general, this represents an additional map on the clock and register, as Def. 1 allows for the tick register to be coherent in the tick basis, and even entangled with the clock system . However, in this work we are only concerned with the probability distribution of ticks (as we characterise the performance of the clock from this alone), and so we incorporate the additional measurement into the map itself. This is equivalent to requiring the map to restrict the state of the clock and tick register to be block-diagonal states in the basis .121212An ideal measurement of the register in the basis will remove any coherence between and .

Furthermore we consider the behaviour of the tick register in the limit . In principle, the probability of a tick in this limit need not be zero. However, such a clock would correspond to one that has a probabilitiy of ticking on every application of the map independently of the state of the clock system, and thus does not provide any time information.131313More precisely, we could express such a clock via the convex combination of two maps, one that does have a zero tick probability for , and one that does not. The second one would provide no time information, and thus only worsen the performance of the clock. Note that the times when the clock does not tick are equally necessary for synchronization with other clocks (see Fig. 1).

Following the above considerations, we continue with clocks whose maps provide states on that are diagonal in the tick basis , and also satisfy the limit

 \textuptrC[limδ→0MδC→CT(ρC)]=|0⟩⟨0|T. (10)

One may feel inclined to think of a clock whose “ticks” convey additional information, such as the number of previous ticks produced by the clock. For example, often a church bell will produce different chimes to specify the passing of different hours. To treat this within our model, one may think of a (classical) counter, which merely counts the number of tick registers in the state . This way, if a tick occurs, one can read the counter and discern the time. Clearly the counter does not need any additional timing devices to function. Importantly, since such a counter only interacts with the tick registers and not the clock, it does not directly affect the evolution of the clock system .141414The state of the counter is independent of given the tick registers.

This concludes our discussion of the generic model of clocks. Real life clocks may also be subject to additional constraints, such as unavoidable de-coherence or power constraints Erker (),thermoClockErker ().

### ii.2 Representation in Terms of Generators

As explained above, the specification of the individual maps of the family is redundant. The following lemma, which is basically a variant of the Lindblad representation theorem Lindblad (), asserts that the family can equivalently be specified in terms of generators.

###### Lemma 1.

Let be a clock with a classical tick register, having as a basis the states . Then there exists a Hermitian operator as well as two families of orthogonal operators and on such that

 MδC→CT(ρC)=ρC⊗|0⟩⟨0|T−δ(i[H,ρ]+m∑j=112{L†jLj+J†jJj,ρ}−LjρL†j)⊗|0⟩⟨0|T+δm∑j=1JjρJ†j⊗|1⟩⟨1|T+FδC→CT(ρC), (11)

for , and where . Conversely, given any Hermitian operator and orthogonal families of operators and on , Eq. (11) defines a clock with a classical tick register.

In the case of classical clocks with basis , is the zero operator and the operators and can all be chosen to be proportional to operators of the form .

The proof of this Lemma, which is provided in Appendix A.1, follows the description in Section 3.5.2 of PreskillLectureNotes (). We call the initial state of the clock. Furthermore, the operators are called tick generators. While Eq. (11) does not define a dynamical semigroup , one can do so quite simply via the map , where the Lindbladian on the clock and register is

 LCT(⋅) =−i[~H,(⋅)]+m∑j=1∑a∈0,1~Lja(⋅)~L†ja−12{~L†ja~Lja,(⋅)}+m∑j=1~Jj(⋅)~J†j−12{~J†j~Jj,(⋅)}, (12)

where the extended operators are , , , and . The clock map is then defined via

 MδC→CT(ρC)=eδLCT(ρC⊗|0⟩⟨0|T). (13)

The tick generators determine when the clock ticks depending on its state, and they also define the clock’s state after a tick. Clocks for which this state coincides with the initial state are of special interest, for they have a particularly appealing mathematical structure and are optimal in terms of their accuracy in the case of classical clocks.151515We expect the same is true for quantum clocks, although a proof has not yet been constructed.

###### Definition 2.

A reset clock is a quantum clock whose tick generators induce a mapping to the clock’s initial state161616More generally, the tick generators induce a mapping to some fixed state, but there is very little loss of generality setting the initial state to be the same, since only the first tick of the clock is affected, every subsequent tick behaves as if the initial state is the fixed state., i.e.,

 m∑j=1JjρJ†j∝ρ0C∀ρ∈S(HC). (14)

### ii.3 The continuous limit of clocks

One may also use the Lindbladian generators to describe the evolution of the clock system as continuous, parametrised by a real variable . From Lemma 1, the following differential equation governs the evolution of the clock,

 ddtρC(t) =limδ→0\textuptrT[MδC→CT(ρC(t))]−ρC(t)δ=−i[H,ρC(t)]+m∑j=1LjρC(t)L†j+JjρC(t)J†j−12{L†jLj+J†jJj,ρC(t)}. (15)

For the tick register, one may take the same limit to find the probability density of a tick being recorded, via the probability that the register is in the state ,

 P(t) =limδ→0\textuptrCT[|1⟩⟨1|TMδC→CT(ρC(t))]δ=\textuptrC[m∑j=1JjρC(t)J†j]. (16)

This limit and the sequence of ticks is discussed in more detail in Section II.6.

Furthermore, consider the case of a clock in which one focuses on a single tick, and tracks the state of the clock only up to the first tick. In this case one can remove the “tick” channel from the Lindbladian of the clock in Eq. (15), as it represents the state of the clock after a tick (see Lemma 1). Thus the description of the entire family of tick generators is redundant. Labelling the state of the clock for just a single tick as , its dynamics are given by (taking Eq. (15) with the tick channel removed)

 ddtρ(1)C(t)=−i[H,ρ(1)C(t)]−{V,ρ(1)C(t)}+m∑j=1Ljρ(1)C(t)L†j−12{L†jLj,ρ(1)C(t)}, (17)

where

 V=12m∑j=1J†jJj (18)

is an arbitrary positive operator representing the ticking dynamics of the clock. In this case, the probabability density of the first tick being recorded is, from Eq. 16,

 P(1)(t) =\textuptrC[2Vρ(1)C(t)]. (19)

This proves handy in the case of reset clocks. As we discuss later, the ticks of a reset clock are a sequence of independent and identically distributed random variables, and thus the first tick suffices to characterise such a clock.

### ii.4 Example

When describing a clock, one may want to distinguish between the intrinsic evolution of the state of the “clockwork” and the mechanism that transfers information about this state to the outside. A rather generic way to do this is to describe the evolution of the clock by a Hamiltonian on the system , and the transfer of information to the outside by a continuous measurement of the system’s state with respect to a fixed basis , which we will refer to as the time basis. In order to ensure that the measurement does not disturb the clock’s state too much, the coupling between clockwork and measurement mechanism must be weak. We quantify it in the following by assigning a coupling parameter to each of the elements of the time basis and consider a reset clock (Def. 2). We could then define a quantum clock with initial state and maps

 MδC→CT:ρC↦e−iδ^HC^Mδ0ρC^Mδ0†eiδ^HC⊗|0⟩⟨0|+d−1∑j=0e−iδ^HC^Mδ1,jρC^Mδ1,j†eiδ^HC⊗|1⟩⟨1| (20)

where and with

For sufficiently small , the quantities together with form a positive-operator valued measure (POVM), since

 ^Mδ0†^Mδ0+d−1∑i=0^Mδi,1†^Mδi,1=∣∣\mathbbm1C−2δ^VC∣∣+2δ^VC.

As such, one can interpret Eq. (20), in the following light. The initial state of the clock is measured via the POVMs, followed by allowing the clock to freely evolve according to its internal Hamiltonian for an infinitesimal time step and repeating the process. In accordance with Eq. (20) one would then associate the outcome “no-tick” with the POVM element and the “tick” outcome with the elements . Since the POVM defines a measurement with classical outcome, one may regard the tick as a classical value, i.e., the tick register could be assumed to be classical in this case.

Furthmore, by expanding in , Eq. (20) can be written in the form

 MδC→CT:ρC↦ρC⊗|0⟩⟨0|−δ(i[^HC,ρ]+m∑j=112{J†jJj,ρ})⊗|0⟩⟨0|+δm∑j=1JjρJ†j⊗|1⟩⟨1|+O(δ2), (21)

where we have defined . With the further identifications , and with the set of zero operators, we see that Eq. (21) is in the form prescribed by Lemma (1). This ensures that the map is indeed a clock, according to our definition 8. Consequently, it is easily verified that the operators satisfy Eq. (14) and the clock is thus a reset clock. It also follows from Section II.3 that in the continuous limit of clocks, the probability of not getting a “tick” in the time interval followed by a tick in the interval time is

 δtP(1)(t)=δt\textuptrC[2Vρ(1)C(t)]=δt\textuptrC[2^VCρ(1)C(t)], (22)

where , with

 |¯ψt⟩=e−it^HC−t^VC|ψ0⟩. (23)

### ii.5 Classical Clocks as a Special Case

Roughly speaking, a classical clock may be regarded as a clock that satisfies Def. 1, but whose state space is restricted to classical states. These are defined relative to a fixed orthonormal basis, , of the Hilbert space of the clock as

 C{|ck⟩}={ρC=d−1∑i=0pi|ci⟩⟨ci|:pi≥0,d−1∑i=0pi=1}.
###### Definition 3.

A clock is called classical if there exists a basis (called the classical basis) such that

 ρ0∈C{|ck⟩}andMδ(C{|ck⟩})⊆C{|ck⟩}

for any .

With this definition at hand, we find that the clock generators take on a particularly simple form, namely:

###### Corollary 1.

Let be a classical clock with basis and suppose that the tick register has basis . Then there exist -matrices and such that

 MδC→CT(ρC)=ρC⊗|0⟩⟨0|+δ∑m,n⟨cm|ρC|cm⟩|cn⟩⟨cn|C⊗(Nmn|0⟩⟨0|T+Tmn|1⟩⟨1|T)+O(δ2).

with171717Eq. (24) will be relaxed in the appendix by replacing the “” sign with “”. By doing so, we prove that our results for classical clocks hold under more general circumstances. The example of the maximally accurate classical clock in Section C.1 satisfies Eq. (24).

 d∑n=1Nnm+d∑n=1Tnm=0. (24)

for any and

 Nnm {≤0for n=m≥0for n≠m Tnm ≥0.

for any .

See Appendix A.2 for a proof of this corollary.

In the case of quantum clocks, we defined an alternative description in terms of the Lindbladian generators rather than the maps (Section II.3). We can do the same for classical clocks, by taking the limit . However, in this case, since the state is always diagonal w.r.t. a fixed orthonormal basis, we only require the dynamics of the diagonal elements, which are seen to be

 (25)

and the probability density of a tick being recorded becomes

 P(t) (26)

A simpler description of the above is to label by , the vector of diagonal elements of . W.r.t. , the dynamics and tick density take on the simple form of a vector differential equation,

 ddtvC(t) =(N+T)vC(t), (27) P(t) =||TvC(t)||1, (28)

where is the vector -norm, which here is the same as the sum of the elements, as both and are element-wise non-negative.

### ii.6 Accuracy of Clocks

The ticks of a clock may be regarded as the most basic time information it produces, in the sense that any other time information generated by the clock could be obtained simply by a counting of ticks. We will therefore use the regularity of the ticks output by a clock as a measure for its accuracy. As described in the introduction, this could be quantified operationally, and without reference to an external time parameter, by the Alternate Ticks Game RaLiRe15 (). However, since an analytical study of this measure appears to be rather involved, we consider a closely related measure of accuracy, which we term . We will now introduce definitions that allow us to express this quantity formally in terms of the clock maps.

A clock (Def. 1) after applications of the map gives rise to a probability distribution of ticks occurring; the 1st of which at time , the 2nd at time , etc. Here we will be interested in the closure of this distribution, corresponding to the limit which gives rise to a probability density of ticks, such as in Eq. 16. Formally, we have the following definition.

###### Definition 4.

The tick pattern of a clock is a sequence of random variables over whose marginal probability density , for any , satisfies

 ∫t10⋯∫tk0dt′1⋯dt′kPT1⋯Tk(t′1,…,t′k)=limδ→0⌊t1δ⌋∑τ1=0⌊t2δ⌋∑τ2=⌈t1δ⌉⋯⌊tkδ⌋∑τk=⌈tk−1δ⌉Pδtick({τ1,…,τk})

where, for any and for any subset of ,

 Pδtick(θ)=limN→∞\textuptr[(MδC→CT)N(ρ0)(\mathbbm1C⊗π1∈θ⊗⋯⊗πN∈θ)],

with if and otherwise.

The value of the random variable , for any , can thus be interpreted as the time of the tick. Accordingly, is the probability that the first tick occurs in the interval , the second in the interval , and so on. Such probability densities are known as delay functions. In particular, can only be non-zero for .

The expected time distance between the and the tick as well as the variance are then given by

 μj =⟨Tj⟩=∫∞0dtjPTj(tj)tj σ2j =⟨T2j⟩=∫∞0dtjPTj(tj)t2j,

for any . is the delay function associated with the tick. Based on these quantities, we can now define the clock accuracies . Note that this is different from the clock accuracy , which will be defined below for the particular case of reset clocks.

###### Definition 5.

The clock accuracies of a clock is a sequence of real numbers, where the element is the accuracy of the tick,

 Rj=μ2jσ2j.

We will use this definition to define a partial ordering of clocks. For any two clocks , and , with clock accuracies and respectively, we will say that is more accurate than iff every tick of is more accurate than the corresponding tick of , i.e., iff . It is this definition that we refer to when we later prove that quantum clocks are more accurate than classical ones.

The characterisation of clocks provided by definition 5 has a number of nice properties. Firstly, it is scale invariant, meaning that the values are invariant under the mapping to , for constants . In other words, it is a measure of the accuracy of each tick, and is not affected by whether these ticks took place over a short or long time scale. Physically, this means that for every clock with accuracies , and mean tick times , there is another clock with the same accuracy, but with the ticks occurring on average at times . The new clock is constructed from the old clock by mapping to , which is equivalent to rescaling the generators , and the Hamiltonian , introduced in Lemma 1, by constant factors.

Furthermore, we can now appreciate the simplicity of reset clocks (Def. 2). Since every time such a clock produces a tick, it is reset to its initial state, the ticks represent a sequence of independent events, which are identically distributed. It thus follows that the delay function of the tick, , is the convolution of the delay function associated with the 1 tick , with itself times. This in turn, gives rise to a simple relationship between the accuracies of reset clocks, (see appendix B.1.1, and Remark 6 for a detailed argument)

 Rj=jR1, (29)

and takes on a particularly satisfactory interpretation. Namely, the accuracy of the 1 tick , is the number of ticks the clock generates (on average), before the next tick has a standard deviation equal to the mean time between ticks, . As such, roughly speaking, the clock’s useful lifetime is , beyond which one can no-longer distinguish between subsequent ticks. To compare two reset clocks, it follows that one only needs to compare their values. Given the special significance of , we will sometimes simply refer to it as .

A similar interpretation is also possible for the value of later ticks. For the purpose of illustration, suppose that the mean time between ticks, is one second. Then corresponds to the number of minutes (on average) that the clock can generate until the tick corresponding to the next minute has a standard deviation which is equal to one minute. As such, while according to Eq. (29), is 60 times larger than , this is not to say that “the 60 tick is more accurate than the 1 tick.”

## Iii Fundamental limitations for classical and quantum clocks

In this section, we will state our findings and explain their relevance and connection to related fields. There are two main theorems. The following one, which is about limitations on classical clocks, and Theorem 2, which shows how quantum clocks can outperform classical clocks.

###### Theorem 1 (Upper bound for classical clocks).

For every -dimensional classical clock, the clock accuracies satisfy

 Rj≤jd (30)

for all . Furthermore, for every dimension , there exists a reset clock whose accuracies saturate the bound Eq. (30),

 Rj=jd. (31)
###### Proof.

See Section E for a proof of the above inequalities and Section C.1 for an explicit construction of an optimal -dimensional reset clock which saturates bound Eq. (30). This clock is further discussed in Fig. 2 a). ∎

While the proof is a bit involved, there is an intuitive explanation to why reset clocks are optimal. If the clock is reset to its initial state after the 1st tick, then one can simply choose the initial state and dynamical map which has the highest possible accuracy for the 1st tick. Intuitively, the only way a non-reset clock could have a superior accuracy for later ticks than this one, would be for one to adjust the mean time of the following tick in the sequence to be longer or shorted than the previous one to make up for any lost or gained time due to the previous tick being “early” or “late”. However, determining whether the clock ticked too early or late would require an additional clock, which is not available within the model.

The optimal reset clocks which saturate the bound in Eq. (30), provide insight into our results. For these classical clock examples, the clock starts at one end of a length nearest neighbour chain, with the tick generator’s support region located at the other end. The clock dynamics produce a classical continuous biased random walk along this chain, see Fig a) 2 for details. The error in telling the time is a consequence of the state dispersing as it travels along the chain. Indeed, for these clocks, the standard result from random walk theory which predicts that the standard deviation in a state is proportional to the square root of the distance travelled, approximately holds. This dispersive behaviour achieves in the optimal case.

Before making a comparison with the quantum clocks described in this manuscript, it is illustrative to compare this result with a recent clock in the literature, thermoClockErker (). Here a quantum clock is powered by two thermal baths, at different temperatures. This temperature difference drives a classical random walk of an atomic particle up a -dimensional ladder, which spontaneously decays back to the initial state when reaching the top of the ladder, emitting a “tick” in the decay process. As such, it is a reset clock whose accuracy depends on the entropy generated by the clock. Interestingly, in the limit of weak coupling and vanishing frequency of ticks, it is found thermoClockErker () that clock dynamics becomes classical, represented by a biased random walk up the ladder. The accuracy of the clock is then

 R≈d(p↑−p↓p↑+p↓)≈d(eΔS/d−1eΔS/d+1) (32)

where , are the probabilities of moving up/down the ladder, induced by the thermal baths, and is the entropy generated by the clock for each instance that it ticks. As such, as far as the criterion of dimensionality is concerned, this classical thermodynamic clock is always less accurate than the optimal classical clock in Theorem 1 by a constant factor, and only approaches optimal accuracy in the limit of infinite entropy generation .

On the other hand, we can also compare the classical clock in Fig. 2 a) with the behaviour of the Idealised Clock of Pauli, introduced in Section II. Here the clock Hamiltonian is the generator of translations, and it is not disturbed by continuous measurements, thus leading to no dispersion, and a clock accuracy of , see Fig. 2 b). Of course, as previously discussed, this high accuracy is unfortunately an artefact of requiring infinite energy.

The important question is whether one can do better than the classical clock, and achieve higher accuracy with a quantum clock. We will now show that this is indeed possible. To start with, the Quasi-Ideal Clock introduced in Section II is formed by taking a complex Gaussian amplitude superposition of the SWP clocks, namely

 |Ψ\textupQuasi-Ideal ⟩:=∑\mathclapk∈Sd(k0)Ae−πσ02(k−k0)2ei2πn0(k−k0)/d|ϑk⟩, (33)

where is a set of consecutive integers centred about , determines the mean energy of the clock state and its width in the basis. Its clock Hamiltonian is the 1st levels of a quantum harmonic oscillator with level spacing ; . The dynamics of the clock according to the total Hamiltonian , where with, the complementary basis to , formed by taking the discrete Fourier Transform. As such, and are diagonal in complementary bases to each other. This setup was introduced in WSO16 () with the aim of studying unitary control of other quantum systems. In the protocol, the clock underwent unitary dynamics without being measured — here we will use its construct to see how well quantum clocks can measure time. Indeed, it readily fits into our protocol developed here for measuring time by using the potential to implement continuous measurements rather than unitary operations. For this, we will here consider the special case in which the coefficients are pure imaginary, rather than real. It then follows that one can use the Quasi-Ideal Clock setup to perform continuous measurements on a reset clock as described in Section II.4.

Due to constructive/destructive quantum interference in the complex Gaussian amplitude tails in Eq. 33, the Quasi-Ideal Clock is able to mimic approximately the dynamical behaviour of the Idealised Clock, while maintaining finite energy and dimension WSO16 (), see Fig.2 c). The following theorem shows that the Quasi-Ideal Clock can exceed the classically permitted bound.

###### Theorem 2.

Consider Quasi-Ideal ClocksWSO16 () with and and the setup described in Section II.4. For all fixed constants , and bounded by , there exists such that the Quasi-Ideal Clock’s accuracy satisfies

 R1≥d2−ησ02+o(d2−ησ02), (34)

in the large limit, where we have used little-o notation. Furthermore, since it is a reset clock; the accuracies of later ticks satisfy

 Rj=jR1, (35)

for all .

###### Proof.

See Section F. The main difficulty of the proof is to come up with a potential which satisfies all the necessary properties — if its derivatives are too large, the clock dynamics are too disturbed by the continuous measurements, yet if they are not large enough, the measurements will not capture enough time information from the clock. ∎

In Section F we give an explicit construction of and a slightly more general version of the theorem. When the complex Gaussian amplitudes in the superposition are highly peaked in the time basis, namely , the clock will have an accuracy of effectively order while for Quasi-Ideal Clocks whose width in the time and energy basis are the same, namely when ; is only of order , saturating the bound of the optimal classical clock.

### iii.1 Discussion of the Quantum Bound: relationship to related fields and open problems

In this section, we discuss the relationship of the quantum bound to other concepts and fields which have been associated with time and clocks in quantum mechanics in the past. We finalise the section with a discussion of some open problems.

#### iii.1.1 Time-energy uncertainty relation

The time–energy uncertainty relation,

 ΔtΔE≥12, (36)

has been a controversial concept ever since its conception during the early days of quantum theory, with Bohr, Heisenberg, Pauli and Schrödinger giving it different interpretations and meanings. It still remains a controversial issue to this day, and multiple interpretations have been found Busch2008 (); PhysRevA.66.052107 (). Often, at the heart of the controversy, is that in quantum mechanics, as explained in the introduction Section I, time is usually associated with a parameter, rather than an operator.

Since in the present context, we do have operators for time, a lot of this controversy can be circumvented. Indeed, Peres introduced a time operator, , where is the period of the clock Hamiltonian Peres80 (). In WSO16 () it was shown that the standard deviation of the initial clock state Eq. (33) saturates a time-energy uncertainty relation; ,181818The saturation of the bound by the Quasi-Ideal Clock, is up to an additive correction term which decays faster than any polynomial in . where the standard deviations are calculated using the operators and respectively. One may be inclined to believe that one can increase the accuracy of the clock, by decreasing as a consequence of a larger . While indeed decreasing does have this effect, it would be naive to believe this paints the full picture.

To study this effect as the clock moves around the clock-face, let be the standard deviation of the clock coefficients de-phased in the energy basis, . We will define the standard deviation in the time basis similarly, but here one has to be careful since the time basis has circular boundary conditions, meaning . Consequently, the state will “jump” from the state to as it completes on period of its motion. We are not interested in the jumps due to the boundary effects, and therefore will denote