Entropic Energy-Time Uncertainty Relation

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As Published http://dx.doi.org/10.1103/PhysRevLett.122.100401

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Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW LETTERS 122, 100401 (2019)

Entropic Energy-Time Uncertainty Relation

Patrick J. Coles,1 Vishal Katariya,2 Seth Lloyd,3,4 Iman Marvian,5 and Mark M. Wilde2,6 1Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 2Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA 3Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 4Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 5Departments of Physics & Electrical and Computer Engineering, Duke University, Durham, North Carolina 27708, USA 6Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA (Received 11 June 2018; published 12 March 2019)

Energy-time uncertainty plays an important role in quantum foundations and technologies, and it was even discussed by the founders of quantum mechanics. However, standard approaches (e.g., Robertson’s uncertainty relation) do not apply to energy-time uncertainty because, in general, there is no Hermitian operator associated with time. Following previous approaches, we quantify time uncertainty by how well one can read off the time from a quantum clock. We then use entropy to quantify the information-theoretic distinguishability of the various time states of the clock. Our main result is an entropic energy-time uncertainty relation for general time-independent Hamiltonians, stated for both the discrete-time and continuous-time cases. Our uncertainty relation is strong, in the sense that it allows for a quantum memory to help reduce the uncertainty, and this formulation leads us to reinterpret it as a bound on the relative entropy of asymmetry. Because of the operational relevance of entropy, we anticipate that our uncertainty relation will have information-processing applications.

DOI: 10.1103/PhysRevLett.122.100401

Introduction.—The uncertainty principle is one of the special case of the harmonic oscillator, this pair corre- most iconic implications of quantum mechanics, stating that sponds to number and phase, and number-phase uncer- there are pairs of observables that cannot be simultaneously tainty is relevant to metrology [9], e.g., phase estimation in known. It was first proposed by Heisenberg [1] for the interferometry. The energy-time pair is arguably the most position qˆ and momentum pˆ observables and then rigorously general observable pair in the sense that it applies to all stated by Kennard [2] in the familiar form using standard physical systems (i.e., all systems have a Hamiltonian). deviations: ΔqˆΔpˆ ≥ ℏ=2. Robertson [3] later formulated a Despite the lack of a Hermitian observable associated similar relation for a different class of observables, namely, with time, relations with the feel of energy-time uncertainty for pairs of bounded Hermitian observables Xˆ and Zˆ (e.g., the relations have been formulated. Mandelstam and Tamm 1 [10] related the energy standard deviation ΔE to the time τ Pauli spin operators), as ΔXˆ ΔZˆ ≥ jh½X;ˆ Zˆ ij. Since then, 2 that it takes for a state to move to an orthogonal state: many alternative formulations have been proven for similar τΔE ≥ ðπℏ=2Þ. This relation can be thought of as a speed Hermitian operator pairs (e.g., Refs. [4,5]). limit—a bound on how fast a quantum state can move— Unfortunately, these relations do not apply to energy and and other similar speed limits have been formulated [11]. time since time does not, in general, correspond to a Alternatively, it can be thought of as bounding how well a ’ Hermitian operator. In particular, Pauli s theorem states quantum system acts as a clock, since the time resolution of that the semiboundedness of a Hamiltonian precludes the the clock is related to the time τ for the system to move to existence of a Hermitian time operator, or in other words, if an orthogonal state. there was such an operator, then the Hamiltonian would be In this work, we take the clock perspective on time unbounded from below and thus unphysical [6]. Hence, uncertainty: one’s uncertainty about time corresponds to formulating a general energy-time uncertainty relation is a how well one can “read off” the time from measuring a nontrivial task. We point to Ref. [7] for an overview on time quantum clock. A natural measure for this purpose is to in quantum mechanics. consider the information-theoretic distinguishability of the Nevertheless, the energy-time pair is of significant various time states. As such, we propose using entropy to importance both fundamentally and technologically. quantify time uncertainty, and our main result is an entropic Energy-time uncertainty was already discussed by the energy-time uncertainty relation. founders of quantum mechanics: Bohr, Heisenberg, Entropy has been widely employed in uncertainty rela- Schrödinger, and Pauli (see Ref. [8] for a review). In the tions for position-momentum [12] and finite-dimensional

0031-9007=19=122(10)=100401(6) 100401-1 © 2019 American Physical Society PHYSICAL REVIEW LETTERS 122, 100401 (2019) observables [13,14]—see Ref. [15] for a recent detailed secret key whose security is guaranteed by our relation. review of entropic uncertainty relations. The key benefits We provide more details regarding applications in the of entropy as an uncertainty measure are its clear opera- Supplemental Material (Appendix A) [22]. tional meaning and its relevance to information-processing Uncertainty relations can be understood in the frame- applications. Indeed, entropic uncertainty relations form work of a guessing game involving two players, Alice and the cornerstone of security proofs for quantum key dis- Bob [15,19], and Fig. 1 shows this game for the energy- tribution and other quantum cryptographic tasks [15]. They time pair. Bob prepares system A in an arbitrary state ρA furthermore allow one to recast the uncertainty principle in and sends it to Alice. Alice then flips a coin. If she gets terms of a guessing game, as we do below for energy heads, she performs an energy measurement, and Bob then and time. must guess the outcome (possibly with the help of a An entropic uncertainty relation for energy and time was memory system R that is initially correlated to A). If she previously given in Ref. [16] by constructing an almost- gets tails, she applies a time evolution e−iHt in which t is periodic time observable and using a so-called almost- randomly chosen from some predefined set, and then sends periodic entropy for time. This approach was extended in A back to Bob, who then tries to guess which time t Alice Ref. [17], where the Holevo information bound was used to applied. All of our uncertainty relations can be understood derive an entropic energy-time uncertainty relation. in terms of this guessing game and can be viewed as However, as indicated in Ref. [16], an almost-periodic constraints on Bob’s probability of winning this game (i.e., time observable serves as a poor quantum clock for guessing both the energy and time correctly). There are aperiodic systems. In Ref. [18], the entanglement between other variations of this energy-time uncertainty guessing a system and a clock was used to derive an entropic energy- game that are possible, one of which is discussed in the time uncertainty relation for a Hamiltonian with a uni- Supplemental Material (Appendix B) [22]. formly spaced spectrum. In what follows, we give some necessary preliminaries In this Letter, we derive entropic energy-time uncertainty before stating our main result for the R´enyi entropy family relations for general, time-independent Hamiltonians. We in the discrete-time case, and then we extend to the first derive a relation for discrete and arbitrarily spaced time, continuous-time case for the von Neumann entropy. and then we extend this relation to infinitesimally closely Finally, we apply our relation to an illustrative example spaced (i.e., continuous) time. Our results apply to systems of a spin-1=2 particle. with either finite- or infinite-dimensional Hamiltonians. Preliminaries.—We begin by considering a finite-dimen- A novel aspect of our energy-time uncertainty relation is sional Hamiltonian H that acts on a quantum system A, and þ that it allows the observer to reduce their uncertainty suppose that it has NE ∈ Z real energy eigenvalues taken through access to a quantum memory system, as was the from aP set E ⊂ R. We thus write the Hamiltonian as H εΠε Πε case in prior uncertainty relations [19]. The two main A ¼ ε∈E A, where A denotes the projector onto benefits of allowing for quantum memory are that (1) it the subspace spanned by energy eigenstates with eigen- ε ε0 ε dramatically tightens the relation when the clock is in a value ε. The projectors obey Π Π ¼ Π δε;ε0, where δε;ε0 ¼ 0 mixed state, and (2) it makes the relation more relevant to 1 if ε ¼ ε and δε;ε0 ¼ 0 otherwise. cryptographic applications in which the eavesdropper may We now recall how to encode the classical state of a hold the memory system (e.g., see Ref. [19]). Furthermore, clock into a quantum system. Inspired by the Feynman- by allowing for quantum memory, we can reinterpret our Kitaev history state formalism [33–35], as well as the uncertainty relation as a bound on the relative entropy of (1) asymmetry [20], and we discuss below the implications of this reinterpretation.

The fact that our uncertainty relation is stated using (2) (3) Bob estimates operationally relevant entropies implies that it should be time t useful for information processing applications. For example, if one can distinguish between the time states well, then Energy it is possible to extract randomness by performing an measurement Bob guesses Alicemeasurement Bob energy measurement. True random bits are critical to the outcome execution of secure protocols and numerical computations. In this case, the randomness of energy measurement FIG. 1. Guessing game for energy-time uncertainty. (1) Bob prepares a quantum clock in the state ρA and sends it to Alice. outcomes is certified by our bound. Entropic uncertainty (2) Alice flips a coin and (3) either measures the clock’s energy or relations also find use in proving the security of quantum randomly sets the clock’s time (i.e., applies a time evolution e−iHt key distribution (QKD) protocols [21]. If one party is able with t randomly chosen from a predefined set). Bob’s goal is to, to prepare states in both the phase and number bases of depending on Alice’s coin flip, guess the clock’s energy or guess t photons, and if another party is able to perform measure- by reading the clock. Our uncertainty relations constrain Bob’s ments in these two bases, then both parties can distill a ability to win this game.

100401-2 PHYSICAL REVIEW LETTERS 122, 100401 (2019) quantum time proposal of Ref. [36], we introduce a register SαðAjBÞρ ¼ −infDαðρABkIA ⊗ σBÞ; ð3Þ σ T for storing the time, which can be interpreted as a B background reference clock. A measurement on the time where the optimization is with respect to all density register is treated in this framework as a time measurement. operators σB on system B. The quantity SαðAjBÞρ is in Let T ¼ft1; …;tKg denote a set of times, for integer K ≥ 2 t ∈ R k ∈ 1; …;K turn defined from the sandwiched Rényi relative entropy of , such that k for all f g, and ξ ζ t ≤ t ≤ … ≤ t T a density operator and a positive semi-definite operator , 1 2 K. We suppose that the register has a α ∈ 0; 1 ∪ 1; ∞ t K which is defined for ð Þ ð Þ as [37,38] complete, discrete, and orthonormal basis fj kigk¼1. The time values need not be evenly spaced, which means that 1 T D ξ ζ ζð1−αÞ=2αξζð1−αÞ=2α α : the basis for register can include any combination of αð k Þ¼α − 1 log2Tr½ð Þ ð4Þ jT j¼K distinct and orthonormal kets. Now consider a clock system A that may initially be If α > 1 and the support of ξ is not contained in the support correlated to a memory system R, together in a joint state of ζ, then it is defined to be equal to þ∞. The quantity ρ ρ ρ E AR with A ¼ TrRð ARÞ. Let random variable capture the DαðξkζÞ is defined for α ∈ f1; ∞g in the limit. outcomes of an energy measurement on the system A. The Entropic energy-time uncertainty relation.—Let us now outcomes can be stored in a classical register, which we state our uncertainty relation for energy and time. For a E also denote without ambiguity by in what follows. To pure state ρA ¼jψihψjA uncorrelated with a reference quantify energy uncertainty, we employ the Rényi condi- system R, it is as follows: tional entropy SαðEjRÞ (defined below) of the classical- quantum state SαðTjAÞκ þ SβðfpðεÞgÞ ≥ log2jT j; ð5Þ X ε ωER ≡ jεihεjE ⊗ TrAfΠAρARg; ð1Þ holding for all α ∈ ½1=2; ∞, with β satisfying 1=αþ ε∈E ε 1=β ¼ 2, where pðεÞ¼hψjΠAjψi. The above inequality ψ 0 [Eq. (5)] is saturated, e.g., when j i is an energy eigenstate. where the kets fjεigε∈E are orthonormal, obeying hε jεi¼ Such states also maximize the time uncertainty, SαðTjAÞκ ¼ δε0;ε, and thus serve as classical labels for the energies of the log2 jT j, since they are stationary states. Hamiltonian. To quantify the time uncertainty, we employ The concavity of entropy and concavity of conditional the Rényi conditional entropy SαðTjAÞ of the following entropy [39] then directly imply that the same inequality in classical-quantum state: Eq. (5) holds for a mixed state uncorrelated with a reference R ρ 1 XK system .However,if A is a maximally mixed state, the κ ≡ t t ⊗ e−iHtk ρ eiHtk : inequality in Eq. (5) yields a trivial bound on the total TA T j kih kjT A ð2Þ j j k¼1 uncertainty. This is because the inequality does not capture the inherent uncertainty of the initial state. ℏ 1 κ In the above and henceforth, we set ¼ . The state TA One of our main results remedies this deficiency, A can be interpreted as the joint state of system (the local capturing the inherent uncertainty mentioned above and T quantum clock) and the background reference clock ,at holding nontrivially for mixed states: an unknown time tk ∈ T chosen according to the uniform distribution. Equivalently, this state can be understood as a SαðTjAÞκ þ SβðEjRÞω ≥ log2jT j: ð6Þ time-decohered version of the Feynman-Kitaev history state [33–35], the latter of which has the entire history The entropic energy-time uncertainty relation in Eq. (6) of the state ρAðtÞ encoded and entangled with a time holds for all α ∈ ½1=2; ∞, where β satisfies 1=αþ register in superposition. The classical-quantum states in 1=β ¼ 2, with the proof given in the Supplemental Eqs. (1) and (2) are in one-to-one correspondence with the Material [22] (Appendix C). The quantity SαðTjAÞκ rep- following labeled ensembles, respectively: resents the uncertainty about the time tk from the perspec- A κ ε tive of someone holding the system of the state TA in fpðεÞ; jεihεjE ⊗ TrAfΠAρARg=pðεÞgε∈E; Eq. (2). The quantity SβðEjRÞω, which is determined by the 1= T ; t t ⊗ e−iHtk ρ eiHtk ; ρ H f j j j kih kjT A gtk∈T state AR and the Hamiltonian A, represents the uncertainty about the outcome of an energy measurement from ε where pðεÞ¼TrfΠAρARg. the perspective of someone who possesses the R system of — p ω ρ Rényi entropies. For a probability distribution f jg, the state ER in Eq. (1). In the case that AR is pure, then the α ∈ 0; 1 ∪ 1; ∞ the Rényi entropies are definedP for ð Þ ð Þ by quantity SβðEjRÞω is determined by the reduced state ρA S p 1= 1 − α pα α ∈ 0; 1; ∞ αðf jgÞ ¼ ½ ð Þlog2 j j , and for f g and the Hamiltonian HA. According to Eq. (6), a good in the limit. This entropy family is generalized to quantum quantum clock state ρA, for which SαðTjAÞκ ≈ 0, neces- states via the sandwiched Rényi conditional entropy [37], sarily has a large uncertainty in the energy measurement, in defined for a bipartite state ρAB with α ∈ ð0; ∞ as the sense that SβðEjRÞω ≳ log2 jT j. Conversely, a state with

100401-3 PHYSICAL REVIEW LETTERS 122, 100401 (2019) a small uncertainty in the energy measurement, i.e., XK X 1 −iHt iHt ε ε S E R ≈ 0 e k ρAe k Π ρAΠ : 9 βð j Þω , is necessarily a poor quantum clock state, T ¼ ð Þ j j k¼1 ε i.e., SαðTjAÞκ ≈ log2 jT j. Note that the uncertainties in Eq. (6) are entropic and One way to satisfy Eq. (9) is if ½ρA;H¼0, and hence hence do not quantify the uncertainties of time and energy ρ in their units, but rather the amount of information (in bits) the relation is tight for states A that are diagonal in the energy eigenbasis.P Another way to satisfy Eq. (9) is if that we do not know about the respective quantities. For 0 1= T K eiðε−ε Þtk δ ε ε0 example, if a system can equally likely take on one of two ð j jÞ k¼1 ¼ ε;ε0 for all combinations of , . T energies E1 and E2, then the entropic uncertainty in energy If the j j times are equally spaced, this implies that 0 iðε−ε ÞtK 0 constitutes only one bit, and it does not depend on the e ¼ 1 and ðε − ε ÞtK ¼ 2π. This can be understood magnitudes of E1 or E2. Each entropy in Eq. (6) is as an exact inverse relationship between the conjugate analogous to a guessing probability, which quantifies variables, which is a signature of a saturated uncertainty how well one can guess the time t given the state ρAðtÞ, relation. or the energy given the ability to measure a memory system Equation (8) can be generalized to nonuniform proba- R. In fact, SαðAjBÞ converges to the negative logarithm of bilities for the various times. As shown in the Supplemental the guessing probability as α → ∞ [37,40]. Material (Appendix E) [22], the right-hand side of Eq. (8) Considering the special case of jT j¼2, one finds a gets replaced by the entropy SðTÞκ of the time distribution simple, yet interesting corollary of Eq. (6): under the for this generalization. — Hamiltonian HA, a quantum state ρA can evolve to a Relative entropy of asymmetry formulation. As shown perfectly distinguishable state, only if SβðEjRÞω ≥ 1 for in the Supplemental Material (Appendix F) [22],an β ∈ ½1=2; ∞. In other words, for SβðEjRÞω < 1, the alternative way of stating our main result in Eq. (6) is orthogonalization time τ in the Mandelstam-Tamm bound by employing the sandwiched R´enyi relative entropy of is infinite, which cannot be seen using Mandelstam-Tamm asymmetry [41], which generalizes an asymmetry measure or other standard quantum speed limits. put forward in Ref. [20]: By means of a quantum memory, one can also reduce the time uncertainty instead of only reducing the energy SαðTjAÞκ þ inf DαðρkσÞ ≥ log2jT j; ð10Þ σ∶½H;σ¼0 uncertainty. This can be accomplished by considering the memory system R to be a bipartite system R1R2. One can and holds for all α ∈ ð0; ∞. The inequality in Eq. (10) then write the uncertainty relation in Eq. (6) as follows: delineates a trade-off, given the Hamiltonian H, between how well a state ρA can serve as a quantum clock and the SαðTjAR1Þκ þ SβðEjR2Þω ≥ log2jT j; ð7Þ asymmetry of ρA with respect to time translations. with full details given in the Supplemental Material [22] Moreover, this connection is exact for pure states. α → 1 D ρ σ (Appendix D). This shows that the tightening of Eq. (5) to In the limit , the quantity infσ∶½H;σ¼0 αð k Þ give Eq. (6) using quantum memory can reduce the reduces to the relative entropy of asymmetry [20] uncertainties in both energy and time. We note that this rewriting is achieved only by relabeling systems, and is thus lim inf DαðρkσÞ¼ inf DðρkσÞ a consequence of our earlier result in Eq. (6). α→1 σ∶½H;σ¼0 σ∶½H;σ¼0 α β 1 An important special case of Eq. (6) is ¼ ¼ where ≡ ΓHðρÞ¼SðΔðρÞÞ − SðρÞ; ð11Þ both entropies are the von Neumann conditional entropy. This results in the following entropic uncertainty relation: where the quantum relative entropy is definedP as DðρkσÞ ≡ ε ε Tr½ρ½log2 ρ − log2 σ [42] and ΔðρÞ¼ ε∈EΠ ρΠ (in the SðTjAÞκ þ SðEjRÞω ≥ log2 jT j; ð8Þ context of asymmetry, the function SðΔðρÞÞ − SðρÞ was where the von Neumann conditional entropy of a bipartite first studied in Ref. [43]). Then the entropic uncertainty S T A Γ ρ ≥ T state τCD can be written as SðCjDÞτ ¼ −Tr½τCD log2 τCDþ relation in Eq. (10) reduces to ð j Þκ þ Hð Þ log2j j. — Tr½τD log2 τD. In fact, we show in Appendix E of the Extension to continuous time. We now extend Supplemental Material [22] that the following equality the uncertainty relation in Eq. (6) so that it is applicable holds for the von Neumann case when ρAR is pure: to continuous, as opposed to discrete, time, and to Hamiltonians with countable spectrum. From Eq. (6) and X ε ε Ref. [44], we derive an inequality applicable to the von SðTjAÞκ þ SðEjRÞω ¼ log2jT jþD κAk Π ρAΠ : ε Neumann entropies. Full details are available in the Supplemental Material (Appendix G) [22]. Consider time As discussed in the Supplemental Material (Appendix E) to be continuous in theP interval ½0;TF. Given a state ρA and H εΠε E [22], when ρAR is pure,P equality in Eq. (8) is achieved a Hamiltonian A ¼ ε∈E A, with countably infinite, ε ε [equivalently, DðκAk εΠ ρAΠ Þ¼0] if and only if we then have that

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We stated an entropic energy-time uncertainty relation for the R´enyi entropies for discrete time sets. This relation was strengthened for mixed states by allowing the observer to possess a quantum memory, a feature that also allowed us to reinterpret our relation as a bound on the relative entropy of asymmetry. For the von Neumann entropy, we extended our uncertainty relation to continuous time sets. Our relation is saturated for all states ρA diagonal in the energy eigenbasis. Expressed in terms of entropies, which are operationally important in information theory, our result should have uses in various tasks. Entropic uncertainty relations have been 1=2 FIG. 2. Our uncertainty relations applied to a spin- particle used previously to certify randomness and prove security of in a magnetic field. For Hamiltonian H ¼ κσz and jT j¼2 (in the quantum cryptography protocols, and we believe our result discrete-time case) or TF ¼ 2 (in the continuous-time case), the plot shows the variation in the uncertainties with θ, the angle will be an important tool used to develop such protocols the state makes with the z axis of the Bloch sphere. The quantity further. D ρ σ S T A infσ∶½H;σ¼0 ð Ak Þ is the energy uncertainty, while ð j Þ and s T A P. J. C. acknowledges support from the Los Alamos ð j Þ are respectively the time uncertainties for discrete and National Laboratory ASC Beyond Moore’s Law project. continuous time. The black dotted line shows log2 jT j¼log2 TF, i.e., our lower bounds on the total uncertainty from Eqs. (6) V. K. acknowledges support from the Department of and (12). Physics and Astronomy at Louisiana State University. S. L. was supported by IARPA under the QEO program, NSF, and ARO under the Blue Sky Initiative. M. M. W. inf DðρAkσÞþsðTjAÞ ≥ log2TF: ð12Þ acknowledges support from the US National Science σ∶½H;σ¼0 Foundation through Grant No. 1714215. M. M. W. is grateful to S. L. for hosting him for a research visit to For a continuously parametrized ensemble of states fpðxÞ; ρx University of Oxford, during which some of this research Bgx∈X , the differential conditionalR quantum entropy was conducted. s X B s X B − dx D p x ρx ρ ð j Þ is definedR as ð j Þ¼ X ð ð Þ Bk avgÞ, ρ dxp x ρx where avg ¼ X ð Þ B [44]. For our case, this means Z TF [1] W. Heisenberg, Über den anschaulichen inhalt der quanten- sðTjAÞ¼− dt DðρðtÞ=TFkρ¯Þ; theoretischen kinematik und mechanik, Z. Phys. 43, 172 Z0 1 TF (1927). ρ¯ dt e−iHtρ eiHt: [2] E. H. Kennard, Zur quantenmechanik einfacher bewegung- ¼ T A F 0 stypen, Z. Phys. 44, 326 (1927). [3] H. P. Robertson, The uncertainty principle, Phys. Rev. 34, Example: Spin in a magnetic field.—Consider a spin-1=2 163 (1929). particle in a magnetic field B ¼ Bzˆ. This is described by [4] P. Busch, P. Lahti, and R. F. Werner, Heisenberg uncertainty ˆ for qubit measurements, Phys. Rev. A 89, 012129 (2014). the Hamiltonian H ¼ κσz, where κ is a constant propor- [5] L. Maccone and A. K. Pati, Stronger Uncertainty Relations tional to B, and σz is the z-Pauli operator. Consider a pure for all Incompatible Observables, Phys. Rev. Lett. 113, state ρA ¼jψð0Þihψð0Þj that makes an angle θ with the z ψ 0 θ=2 0 260401 (2014). axis of the Bloch sphere, given by j ð Þi ¼ cosð Þj iþ [6] W. Pauli and N. Straumann, Die allgemeinen Prinzipien der θ=2 1 t sinð Þj i. After a time , this state evolves to Wellenmechanik (Springer, Berlin, Heidelberg, 1990). −iHt jψðtÞi ¼ e jψð0Þi. Figure 2 plots the variation of the [7] J. Butterfield, On time in quantum physics, in A Companion uncertainty (time, energy, and total uncertainty) with θ for to the Philosophy of Time, edited by H. Dyke and A. Bardon both our discrete- and continuous-time relations. For (John Wiley & Sons, Ltd., Hoboken, 2013), pp. 220–241. θ ¼ π=2, the energy uncertainty is maximal (one bit) while [8] V. V. Dodonov and A. V. Dodonov, Energy–time and the time uncertainty is minimal (although still nonzero in frequency–time uncertainty relations: Exact inequalities, this example). At the other extreme, for θ ¼ 0 or π, the Phys. Scr. 90, 074049 (2015). [9] V. Giovannetti, S. Lloyd, and L. Maccone, Quantum- energy uncertainty is zero while the time uncertainty is ’ enhanced measurements: Beating the standard quantum maximal (one bit), meaning that the clock s time states limit, Science 306, 1330 (2004). cannot be distinguished. One can see in Fig. 2 that our [10] L. Mandelstam and Ig. Tamm, The Uncertainty Relation uncertainty relation is tight in this extreme case. between Energy and Time in Non-Relativistic Quantum Discussion.—We gave a conceptually clear and opera- Mechanics, In Selected Papers (Springer, Berlin, tional formulation of the energy-time uncertainty principle. Heidelberg, 1991), pp. 115–123.

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