Bounding the Minimum Time of a Quantum Measurement

Measurements take a singular role in quantum theory. While they are often idealized as an instantaneous process, this is in conﬂict with all other physical processes in nature. In this Letter, we adopt a standpoint where the interaction with an environment is a crucial ingredient for the occurrence of a measurement. Within this framework, we derive lower bounds on the time needed for a measurement to occur. Our bound scales proportionally to the change in entropy of the measured system, and inversely proportional to the number of possible measurement outcomes and the interaction strength driving the measurement. We evaluate our bound in two examples where the environment is modelled by bosonic modes and the measurement apparatus is modelled by spins or bosons.


Motivation
Quantum measurements are one of the most controversial and enigmatic aspects of quantum theory.Although most agree on the end result of said measurements, the dynamics behind them has been a subject of debate since the initial 'quantum-boom' in the early twentieth century [1][2][3][4].The textbook description of a quantum measurement is the 'collapse of the wave function' [5,6], which was later codified by von Neumann in his mathematical formulation of quantum mechanics.In it, a quantum measurement is defined as a probabilistic, non-unitary, irreversible, and instantaneous process [7].This is in direct contention with another postulate of quantum mechanics: that all microscopic processes are unitary and reversible.This peculiar exemption for measurements (along with other concerns) is dubbed the 'measurement problem' [8][9][10][11][12].
One of the most successful attempts to address the measurement problem without changing the fundamental postulates of quantum mechanics is the decoherence mechanism introduced by Zeh [13].Decoherence can be viewed as the practically irreversible generation of correlations between a system of interest and a large environment.Although the global evolution of the system and environment is unitary, the system follows a non-unitary dynamics that rapidly damps coherence within the system, destroying quantum interference effects, and driving the state to a statistical mixture that, for all practical purposes, can be described by a classical probability distribution [9,9,11].To what extent decoherence can solve the measurement problem is still matter of debate [8,14,15].However, it did have a huge impact on current interpretations of quantum mechanics and in the development of many modern branches of quantum theory, motivating carefully controlled experiments designed to track the dynamics of a quantum measurement [16][17][18][19].
In this Letter, we derive a general bound on the minimum time it takes for a measurement to occur in decoherence-based interpretations.We stress that our results are not aimed at deciding whether a measurement has occurred or not.Instead, this Letter caters to those that follow the school of thought in which a measurement is facilitated by the interaction with an environment.Alternatively, whatever philosophy the reader may have, the results we show imply bounds on the decoherence timescales of the Figure 1: Our measurement scheme is composed of three parts: a system Q and a measurement apparatus A in contact with an environment E. During the pre-measurement stage, A interacts with the system Q, correlating to the potential outcomes of a measurement.At this stage, the process is still reversible.However, the posterior interaction with an environment E drives the system and apparatus to a state physically indistinguishable from a statistical mixture of measurement outcomes, in a process that is practically irreversible.
correlations of arbitrarily complex quantum systems, setting a saturable bound on the minimum timescales needed for a quantum system to effectively behave classically due to the interaction with an environment [10].

Measurement Model
We consider a setup where a quantum system Q interacts with a measurement apparatus A capable of recording the outcome of the measurement.Thereafter, an external environment E interacts with the measurement apparatus [20].Note that the assumption of a mesoscopic measurement device ensures that outcomes are observable at a macroscopic level.The apparatus consists solely of the elements that interact with the measured system, e.g. the free charges in an avalanche photodiodes or the mean photon number in a superconducting microwave resonator.All the remaining constituents that would conventionally be considered part of the device are treated as a thermal environment that conveys the information related to the quantum state in the classical world.In order to differentiate between a measurement and a noisy process, we assume that Q does not interact with E, see Fig. (1).
Within the ideal measurement scheme devised by Von Neumann [7], a quantum measurement is partitioned into two distinct transformations: a 'pre-measurement' followed by a 'collapse'.Dur-ing the pre-measurement of an observable O = j o j Π O j , the apparatus A and system Q interact in such a way that the 'pointer states' |a j ⟩ of the apparatus, become correlated with the projectors Π O j [27].More precisely, if the system is initialized in |ψ Q ⟩, the state after the pre-measurement stage is Since |ψ QA ⟩ is a pure state that can be undone by applying local unitaries on the joint system of Q and A, the pre-measurement does not constitute a definite quantum measurement [11].Moreover, without an additional physical process, one cannot unambiguously identify the outcomes of a measurement.The secondary phase of the measurement, the 'collapse', resolves the ambiguity by transforming |ψ QA ⟩ into the statistical mixture This state is physically indistinguishable from a classical probability distribution where outcome state |a j ⟩⟨a j |, corresponding to the outcome o j , occurs with probability (2) is conventionally thought of as an instantaneous jump process [7], decoherence theory introduces an environment E which allows modelling this transition as a continuous transformation on the joint state of QAE.
A Hamiltonian which governs the transition dynamics of |ψ QA ⟩ → ρ QA M can be expressed as a sum of terms where the subscripts indicates which system(s) a specific Hamiltonian affects, for example H E acts exclusively on the environment.We assume that the Hamiltonian commutes with the pointer basis, [H, Π O j |a j ⟩⟨a j | Π O j ] = 0, which ensures that measurement outcomes in the pointer basis are stable under the evolution [9,11].Moreover, given a large environment, the process |ψ QA ⟩ → ρ QA M typically occurs rapidly 1 .Note that we have 1 One may be concerned by a 'fuzzy' definition of measurement that relies on the state of the system merely becoming close to ρ QA M .More definite, objective notions arise if quantum gravity implies fundamental uncertainties in measurements [54] excluded interaction terms between Q and E in Eq. (3), i.e.H QE and H QAE , this is to distinguish a measurement from a noisy process.Thus the unique term driving the measurement is H AE .Nevertheless, the results which follow in the next sections are easily extended to Hamiltonians which contain an interaction term between Q and E.
The exact amount of time it takes for the process |ψ QA ⟩ → ρ QA M to occur could be computed by solving the appropriate master equation.However, doing so often relies on a series of assumptions.Further, this definition of measurement time can bring about ambiguity due to the fact that typical decoherence inducing Hamiltonians cause the state of QA to become arbitrarily close to ρ QA M , but not necessarily equal.In what follows, we derive a lower bound on the minimum timescales needed for this 'collapse' stage of the measurement process to occur, while circumventing solving the dynamics of a Master equation, and ambiguity of deciding 'when' a measurement has concluded.We accomplish this by applying techniques from quantum limits [21][22][23][24][25], from which we obtain a bound which is solely dependent on the relative entropy between |ψ QA ⟩ and ρ QA , the number of measurement outcomes, and the interaction Hamiltonian between the apparatus and the environment H AE .In a similar spirit, the authors of Ref. [26] derived bounds on the timescale of the pre-measurement stage using quantum speed limit techniques.However, Ref. [26] focuses on unitary dynamics, whereas we take an open-system approach, resulting in different bounds on the timescales of measurements (see [21] and [22][23][24] respectively).

Bounding the Timescale of a Measurement
Our bound is based on a simple premise: if a car travels at a maximum speed of v max , then the time needed to travel a distance d must be greater than d/v max .In quantum theory, one can consider the speed of a state or of an operator instead of a car, and derive quantum speed limits [21][22][23][24][25].
We aim for a bound on the time it takes for the system and apparatus to reach the post-measurement state ρ QA M .For that purpose, we focus on the speed of the relative en- between the joint state of the system and apparatus at a given time, ρ QA (t), and the post-measurement state, ρ QA M .Here, S(t) := − Tr ρ QA (t) ln ρ QA (t) is the entropy of ρ QA (t).The relative entropy serves as a proxy of distance between two states, characterizing the probability to distinguish them [29].In a hypothesis testing scenario, the probability to confuse two states ρ 1 and ρ 2 after K measurements satisfies p error ∼ e −K•S(ρ 1 ∥ρ 2 ) [30].
Substituting relative entropy in place of distance in the aforementioned car analogy, we arrive at a precursor for the bound on the measurement time and since the post-measurement state is diagonal in the pointer basis, so is the operator ln ρ QA M .Therefore, where the final equality makes use of the fact that Π O j |a j ⟩⟨a j | Π O j commutes with the total Hamiltonian.Hence, it is enough to focus on the change in entropy S(t) of combined state of Q and A to compute the bound expressed in Eq. (4).However, determining the maximum of dS(t) dt often requires solving complex dynamics.In order to circumvent this challenging task, we make use of the inequalities proven in Ref. [25].Here, is the incoherent portion of the quantum Fisher information of ρ QA (t) with eigenvalues {λ j }, is the varentropy of state ρ QA (t), and is the variance of the interaction Hamiltonian H AE 2 .The quantities I d , ∆S and ∆H AE are time-dependent, but their dependence is not made explicit in Eq. ( 7), nor subsequent equations, for the sake of clarity.The final inequality we employ is a bound on ∆S with respect to the number of non-zero eigenvalues of ρ QA (t) [32], which is dictated by the number of distinct outcomes A = { |a j ⟩} that the apparatus can measure, A proof can be found in the Appendix.Note that f A = 1 for a measurement with a binary output (such as the spin of a qubit), and Combining Eqs.(4), ( 7) and (11), we obtain bounds on the minimum timescale of a measurement The tighter bound in Eq. ( 12) requires the information-theoretic quantity ∆S √ I d , typically hard to calculate, whereas the looser bound only requires knowledge of the uncertainty in the interaction Hamiltonian.
The right-most bound in Eq. ( 12) can be further simplified in the strong-coupling regime [33], where H AE dominates the evolution and is timeindependent, since in this situation ∆H AE is (approximately) time-independent.Then, we obtain the main result of this Letter, Henceforth, we assume the strong-coupling regime and a time-independent Hamiltonian H AE 2 While alternative bounds on the entropy rate have been derived [55][56][57], the main advantage of Eq. ( 7) is that it involves standard deviations instead of operator norms, which typically results in tighter bounds [25].so that the right hand side of the bound solely involves time-independent quantities.In this way, our result provides a simple-to-evaluate expression for the minimum time needed for a measurement to occur.
Note how the quantities that define the lower bound are naturally linked to the problem at hand.The further away the initial state is to the post-measurement state, as quantified by δS, the larger the lower bound on τ is.In contrast, a stronger coupling to the environment, as quantified by ∆H int , leads to potentially faster measurement processes.

Measurement times in the spinboson model
Let us consider the spin-boson, often used as a toy model to study environmental decoherence [11,12,35].In it, spins representing the apparatus interact with a bosonic environment represented by a collection of harmonic oscillators in a state The subscript k spans through the environmental modes, ω k is the energy of the kth mode, β = 1/(k B T ) is the inverse temperature, and Z k are normalization constants.The environment is assumed to be large such that ρ QAE (t) = ρ QA (t) ⊗ ρ E .We assume Q to be a two-level system in an initial state |ψ Q ⟩ = x |0⟩ + y |1⟩, and that the state of QA after the pre-measurement is Here, we assume that the apparatus is composed of N distinct spin systems (N ≈ 1 corresponds to microscopic apparatus, and N ≫ 1 corresponds to a macroscopic apparatus).If we are interested in measuring the state of the system Q in the computational basis { |0⟩ , |1⟩}, the postmeasurement state will be ) Finally, the apparatus and environment interaction is modeled via the Hamiltonian [20,35] where the local Hamiltonian σ (i) Z = |↓⟩⟨↓| − |↑⟩⟨↑| acts as such exclusively on the ith spin and identity on all the others, the ladder operators a k and a † k act on the kth environmental modes degrees of freedom and g k is a positive coupling constant.
It holds that where J(ω) is the spectral density of the coupling constants [11].Then, Eq. (13) implies that the measurement time satisfies with δS = −|x| 2 ln |x| 2 − |y| 2 ln |y| 2 ≤ ln 2. Decreasing the coupling constants g k or the 'size' N of the measurement apparatus leads to necessarily larger measurement times.The bound is illustrated in Fig. (2) with a microscopic apparatus (N = 1) and a single environmental mode, in which the predominant order of magnitude is in the range of nanoseconds up to zeptoseconds.Current technologies might be able to witness a carefully engineered measurement, which in principle could be used as a test-bed to benchmark against objective collapse theories [19,36,37] or fundamentally non-unitary dynamics [38][39][40].
Even though the bound Eq. ( 13) is extremely general, Eq. ( 19) correctly captures the timescales on the spin-boson model for a small apparatus.By solving the master equation in the Born-Markov approximation, one finds that the off-diagonal terms of the state acquire an exponentially decaying factor r(t) = e −Γ , with [12] With this, we can find the time it takes for the relative entropy to fall to ε ≪ 1.In an experiment, ε would be set, for instance, as a function of the precision with which the state can be estimated from tomography.Assuming |x| = |y| = 1/ √ 2 for simplicity, we get which simplifies to ε ≈ e −2Γ by using that ln 1 ± e −Γ ≈ ±e −Γ .Assuming that the measurement timescale is much shorter than the dynamical timescales imposed by the environmental modes, t ≪ ℏ/ω k , we obtain the analytic solution This expression is remarkably similar to the lower bound in Eq. (19), with the exception of the scaling in N .Notably, the Born-Markov approximation allows one to maximize ∆S √ I d and thus make use of the tighter bound in Eq. (12).By doing so, one recovers the √ N scaling A derivation is included in the Appendix.

Measurement times in the bosonboson model
Here, we consider a bosonic apparatus interacting with a bosonic environment.This was the the-oretical framework used in [16].The apparatus is represented by a cat state with pointer states |α⟩ and |−α⟩, where |α| 2 is the mean number of excitations.At t = 0 the state of QA is while the post-measurement state is We label the creation and annihilation operators of the environment and the apparatus by {a k , a † k } and {b, b † }, respectively.
In this example, when |α| ≫ 1, we find that Then, the measurement time satisfies We have explicitly left the varentropy dependence in the denominator, as opposed to using the inequality in Eq. (11), because the pointer basis only commutes with the Hamiltonian approximately.Moreover, |α⟩ and |−α⟩ are not exactly orthogonal.In any case, upon solving the master equation and using the Born-Markov approximation, one finds these corrections to be negligible [20,41], so that under a good approximation it holds that ∆S ≲ f A 3 .Then, the bound is very similar to the spinboson example, Eq. (19), so one can refer to Fig. (2) for insight on the dependencies with g for a single environmental mode.The most noteworthy difference is the scaling of 1/|α| = 1/ √ M , 3 Alternatively, one could have chosen an alternative H BB AE to avoid the commutativity issue, e.g.[41], however said Hamiltonian is representative of coupling the Fock states to the environmental modes, which is unrealistic and thus not typically used.
where M is the average number of excitations, which differs with the scaling of 1/N in the spinboson model 4 This model can be generalized to include more measurement outcomes, A > 2, by assigning additional phases to a coherent state for the additional measurement outcomes.For example if A = 4 one could use |α⟩, |iα⟩, |−α⟩ and |−iα⟩ [45].

Discussion
There is an uncomfortable dissonance between continuous physical processes in nature and the often-assumed instantaneous jump of a state during a quantum measurement.One proposed resolution relies on the fact that the interaction with an environment drives a measurement apparatus to a state that is, in practice, indistinguishable from a statistical mixture of definite measurement outcomes [11].
Within this framework, we derived a general bound on the minimum timescale of a quantum measurement.The bound is based on the principle that a measurement involves a change in entropy δS in the system and measurement apparatus, and that this change takes a finite amount of time [25].Crucially, the bound on the minimum measurement time, which depends on the energy variance of the apparatus-environment coupling, can be calculated without needing to solve for the exact complex dynamics induced by the environment.When the dynamics is simple enough, e.g., as in the spin-boson model with the Born-Markov assumption, the tighter bound Eq. ( 23) can be obtained by exploiting the information-theoretic bound in Eq. (12).
While we focused on the time needed for an environment to drive the system and apparatus to the post-measurement state ρ QA M , an expression like Eq. (13) also bounds the time needed for a correlated quantum system to decohere.
Bounds on the timescale of a quantum measurement have also been considered recently in Refs.[26,46,47].However, the bounds in the literature assume particular models of open quantum dynamics or require solving the associated master equation for the density matrix of the system of interest.In contrast, we derive bounds that rely on few assumptions on the measurement model.We showcased our bound in two simple models.Remarkably, under the right regime, the bounds on the measurement times are within experimental reach, as illustrated in Fig. (2).It would be fascinating to devise an experiment, perhaps similar to [16], to probe these fundamental bounds.This may require exploring more realistic measurement models.
Note that Eq. ( 13) is reminiscent of the Bremmerman-Bekenstein bound, which provides an upper limit on the rate at which information can be transferred given an energy constrain [48].The original derivation of this bound is based on heuristic arguments, but more rigorous versions of it have been explored [25,49].Our bound provides a formal version that applies to the dynamics during a quantum measurement, providing new insights into the physics of information processing.Finally, it would be interesting to study the relationship between our bound on the timescale of a measurement and the energetic constraints and resource costs of performing a measurement [50][51][52][53].
[43] Alternatively, one could have chosen an alternative H int to avoid the commutativity issue, e.g.
, however said Hamiltonian is representative of coupling the Fock states to the environmental modes, which is unrealistic and thus not typically used.
[44] The scaling of 1/|α| in our bounds seemingly disagrees with the one found in Refs.[16,42], where they found a decoherence time that scales as 1/|α| 2 .The difference is due to the different choice of interaction Hamiltonian [42].
The above expression is subjected to two maximizations; the first yields that for any r ∈ [0, 1], the optimal choice is n = A − 1 and m = 1, whereas the second yields an optimal value r = r A satisfying Using the above in tandem with the inequalities we arrive at the inequality B Calculations for the spin-boson model In the main text, we focus on the setting of a single qubit measurement.For the sake of generality, we perform the calculation for an m qubit quantum state.Recall that the environment is modeled as where the subscript k indicates a property or operator unique to the kth mode.In this generalization, each of the m correlate to a measurement apparatus composed of N spins; the overall interaction Hamiltonian is where h i,j is the Hamiltonian which has the form σ Z = |↓⟩⟨↓| − |↑⟩⟨↑| when acting on the jth spin associated with the ith qubit and identity on all other spins, and g k is the positive coupling constant of the kth environmental mode.Notice that we can write and To compute the variance of H SB AE , i.e we make use of the fact that This greatly simplifies the expression to Typically, when assuming a continuum of modes, one makes the substitution k |g k | 2 → ∞ 0 J(ω)dω, where J(ω) is the spectral density of the coupling constants, hence Finally, recall that all of the spins assigned to a qubit are correlated with each other, thus for any j 1 and j Using this innate symmetry with the spins of the apparatus, one obtains the expression where is a time-independent pre-factor dependent on the initialization of the quantum state; note that χ = 1 for a single qubit measurement (m = 1).Therefore, the timescale of a measurement can be bounded via In the single-qubit case, a tighter bound can be obtained by employing the Born-Markov approximation and maximizing ∆S √ I d .As stated in the main text, the off-diagonal terms of ρ QA (t) acquire a decay term e −Γ , with and therefore (B17) where we use the approximation in Eq. (B13) and note that Γ/t 2 is thus time-independent.By combining the above two equations, one obtains (∆S)

C Calculations for the boson-boson model
The boson-boson example calculations are very similar to the spin-boson calculations.For a multiqubit example, we could consider m qubits, each coupled to a measurement apparatus with outputs |α⟩ and |−α⟩.This generalization leads to a similar pre-factor of χ, Eq. (B11).Thus, for compactness, we only consider a single qubit in this set of calculations.Using the only non-vanishing terms in the variance are From b |α⟩ = α |α⟩, it follows that where we assume that |α| 2 ≫ 1, hence 1 + |α| 2 ≈ |α| 2 .