A quantum clock must satisfy two basic constraints. The first is a bound on the time resolution of the clock given by the difference between its maximum and minimum energy eigenvalues. The second follows from Holevo's bound on how much classical information can be encoded in a quantum system. We show that asymptotically, as the dimension of the Hilbert space of the clock tends to infinity, both constraints can be satisfied simultaneously. The experimental realization of such an optimal quantum clock using trapped ions is discussed.A quantum clock must satisfy two basic constraints. The first is a bound on the time resolution of the clock given by the difference between its maximum and minimum energy eigenvalues. The second follows from Holevo's bound on how much classical information can be encoded in a quantum system. We show that asymptotically, as the dimension of the Hilbert space of the clock tends to infinity, both constraints can be satisfied simultaneously. The experimental realization of such an optimal quantum clock using trapped ions is discussed.

arXiv:quant-ph/9808042Physical Review LettersMar 8, 1999212 citations

Relativistic Limits on the Discretization and Temporal Resolution of a Quantum Clock

Tommaso Favalli

We provide a brief discussion regarding relativistic limits on the discretization and temporal resolution of time values in a quantum clock. Our clock is characterized by a time observable chosen to be the complement of a bounded and discrete Hamiltonian that can have an equally spaced or a generic spectrum. In the first case, the time observable can be described by a Hermitian operator, and we find a limit in the discretization for the time eigenvalues. Nevertheless, in both cases, the time observable can be described by a POVM, and, by increasing the number of time states, we show how the bound on the minimum time quantum can be reduced and identify the conditions under which the clock values can be treated as continuous. Finally, we find a limit for the temporal resolution of our time observable when the clock is used (together with light signals) in a relativistic framework for the measurement of spacetime distances.We provide a brief discussion regarding relativistic limits on the discretization and temporal resolution of time values in a quantum clock. Our clock is characterized by a time observable chosen to be the complement of a bounded and discrete Hamiltonian that can have an equally spaced or a generic spectrum. In the first case, the time observable can be described by a Hermitian operator, and we find a limit in the discretization for the time eigenvalues. Nevertheless, in both cases, the time observable can be described by a POVM, and, by increasing the number of time states, we show how the bound on the minimum time quantum can be reduced and identify the conditions under which the clock values can be treated as continuous. Finally, we find a limit for the temporal resolution of our time observable when the clock is used (together with light signals) in a relativistic framework for the measurement of spacetime distances.

arXiv:2504.05848EntropyOct 14, 20250 citations

Quantum Clocks are More Accurate Than Classical Ones

Mischa P. Woods, Ralph Silva, Gilles Pütz +2 more

A clock is, from an information-theoretic perspective, a system that emits information about time.One may therefore ask whether the theory of information imposes any constraints on the maximum precision of clocks.Here we show a quantum-over-classical advantage for clocks or, more precisely, the task of generating information about what time it is.The argument is based on information-theoretic considerations: we analyze how the accuracy of a clock scales with its size, measured in terms of the number of bits that could be stored in it.We find that a quantum clock can achieve a quadratically improved accuracy compared to a purely classical one of the same size.A clock is, from an information-theoretic perspective, a system that emits information about time.One may therefore ask whether the theory of information imposes any constraints on the maximum precision of clocks.Here we show a quantum-over-classical advantage for clocks or, more precisely, the task of generating information about what time it is.The argument is based on information-theoretic considerations: we analyze how the accuracy of a clock scales with its size, measured in terms of the number of bits that could be stored in it.We find that a quantum clock can achieve a quadratically improved accuracy compared to a purely classical one of the same size.

arXiv:1806.00491PRX QuantumFeb 3, 202240 citations

Faster than Hermitian Time Evolution

Carl M. Bender

For any pair of quantum states, an initial state |I and a final quantum state |F , in a Hilbert space, there are many Hamiltonians H under which |I evolves into |F .Let us impose the constraint that the difference between the largest and smallest eigenvalues of H, E max and E min , is held fixed.We can then determine the Hamiltonian H that satisfies this constraint and achieves the transformation from the initial state to the final state in the least possible time τ .For Hermitian Hamiltonians, τ has a nonzero lower bound.However, among non-Hermitian PT -symmetric Hamiltonians satisfying the same energy constraint, τ can be made arbitrarily small without violating the time-energy uncertainty principle.The minimum value of τ can be made arbitrarily small because for PT -symmetric Hamiltonians the path from the vector |I to the vector |F , as measured using the Hilbert-space metric appropriate for this theory, can be made arbitrarily short.The mechanism described here is similar to that in general relativity in which the distance between two space-time points can be made small if they are connected by a wormhole.This result may have applications in quantum computing.For any pair of quantum states, an initial state |I and a final quantum state |F , in a Hilbert space, there are many Hamiltonians H under which |I evolves into |F .Let us impose the constraint that the difference between the largest and smallest eigenvalues of H, E max and E min , is held fixed.We can then determine the Hamiltonian H that satisfies this constraint and achieves the transformation from the initial state to the final state in the least possible time τ .For Hermitian Hamiltonians, τ has a nonzero lower bound.However, among non-Hermitian PT -symmetric Hamiltonians satisfying the same energy constraint, τ can be made arbitrarily small without violating the time-energy uncertainty principle.The minimum value of τ can be made arbitrarily small because for PT -symmetric Hamiltonians the path from the vector |I to the vector |F , as measured using the Hilbert-space metric appropriate for this theory, can be made arbitrarily short.The mechanism described here is similar to that in general relativity in which the distance between two space-time points can be made small if they are connected by a wormhole.This result may have applications in quantum computing.

arXiv:0712.3910Symmetry Integrability and Geometry Methods and ApplicationsDec 26, 20072 citations

Quantum clocks observe classical and quantum time dilation

Alexander R. H. Smith, Mehdi Ahmadi

Abstract At the intersection of quantum theory and relativity lies the possibility of a clock experiencing a superposition of proper times. We consider quantum clocks constructed from the internal degrees of relativistic particles that move through curved spacetime. The probability that one clock reads a given proper time conditioned on another clock reading a different proper time is derived. From this conditional probability distribution, it is shown that when the center-of-mass of these clocks move in localized momentum wave packets they observe classical time dilation. We then illustrate a quantum correction to the time dilation observed by a clock moving in a superposition of localized momentum wave packets that has the potential to be observed in experiment. The Helstrom-Holevo lower bound is used to derive a proper time-energy/mass uncertainty relation.Abstract At the intersection of quantum theory and relativity lies the possibility of a clock experiencing a superposition of proper times. We consider quantum clocks constructed from the internal degrees of relativistic particles that move through curved spacetime. The probability that one clock reads a given proper time conditioned on another clock reading a different proper time is derived. From this conditional probability distribution, it is shown that when the center-of-mass of these clocks move in localized momentum wave packets they observe classical time dilation. We then illustrate a quantum correction to the time dilation observed by a clock moving in a superposition of localized momentum wave packets that has the potential to be observed in experiment. The Helstrom-Holevo lower bound is used to derive a proper time-energy/mass uncertainty relation.

Jan 1, 202074 citations

Nonasymptotic bounds for quantum purity amplification

Thilo Scharnhorst, Jack Spilecki, John Wright

In quantum purity amplification, one is given $n$ copies of a noisy quantum state $ρ\in \mathbb{C}^{d \times d}$ and asked to prepare $k$ copies of its principal eigenstate $|v_d\rangle$ . Several prior works have derived information-theoretically optimal algorithms for this problem, but the bounds they prove are only shown in the asymptotic regime as the number of samples $n$ tends to infinity. In this paper, we establish the following nonasymptotic guarantee: if $ρ$ 's eigenvalues are sorted $p_1 \leq \cdots \leq p_d$ and $p_{d-1} < p_d$ , then \begin{equation*} n = O\Big(k + \frac{k}δ \cdot \frac{1-p_d}{(p_d-p_{d-1})^2}\Big) \end{equation*} copies suffice to output a state with fidelity at least $1-δ$ with $|v_d^{\otimes k}\rangle$ . Our bound holds for arbitrary spectra, and is independent of the dimension $d$ . In the case of depolarizing noise, our finite-sample guarantee matches the optimal asymptotic scaling. Our proof is based on the combinatorics of random Young diagrams.In quantum purity amplification, one is given $n$ copies of a noisy quantum state $ρ\in \mathbb{C}^{d \times d}$ and asked to prepare $k$ copies of its principal eigenstate $|v_d\rangle$. Several prior works have derived information-theoretically optimal algorithms for this problem, but the bounds they prove are only shown in the asymptotic regime as the number of samples $n$ tends to infinity. In this paper, we establish the following nonasymptotic guarantee: if $ρ$'s eigenvalues are sorted $p_1 \leq \cdots \leq p_d$ and $p_{d-1} < p_d$, then \begin{equation*} n = O\Big(k + \frac{k}δ \cdot \frac{1-p_d}{(p_d-p_{d-1})^2}\Big) \end{equation*} copies suffice to output a state with fidelity at least $1-δ$ with $|v_d^{\otimes k}\rangle$. Our bound holds for arbitrary spectra, and is independent of the dimension $d$. In the case of depolarizing noise, our finite-sample guarantee matches the optimal asymptotic scaling. Our proof is based on the combinatorics of random Young diagrams.

quant-pharXiv:2605.27262May 27, 20260 citations

In quantum purity amplification, one is given

n

copies of a noisy quantum state

ρ\in \mathbb{C}^{d \times d}

and asked to prepare

k

copies of its principal eigenstate

|v_d\rangle

. Several prior works have derived information-theoretically optimal algorithms for this problem, but the bounds they prove are only shown in the asymptotic regime as the number of samples

n

tends to infinity. In this paper, we establish the following nonasymptotic guarantee: if

ρ

's eigenvalues are sorted

p_1 \leq \cdots \leq p_d

and

p_{d-1} < p_d

, then \begin{equation*} n = O\Big(k + \frac{k}δ \cdot \frac{1-p_d}{(p_d-p_{d-1})^2}\Big) \end{equation*} copies suffice to output a state with fidelity at least

1-δ

with

|v_d^{\otimes k}\rangle

. Our bound holds for arbitrary spectra, and is independent of the dimension

d

. In the case of depolarizing noise, our finite-sample guarantee matches the optimal asymptotic scaling. Our proof is based on the combinatorics of random Young diagrams.