Elementary derivation for passage times
Open accessAbstract
When a quantum system undergoes unitary evolution in accordance with a prescribed Hamiltonian, there is a class of states |ψ⟩ such that, after the passage of a certain time, |ψ⟩ is transformed into a state orthogonal to itself. The shortest time for which this can occur, for a given system, is called the passage time. We provide an elementary derivation of the passage time, and demonstrate that the known lower bound, due to Fleming, is typically attained, except for special cases in which the energy spectra have particularly simple structures. It is also shown, using a geodesic argument, that the passage times for these exceptional cases are necessarily larger than the Fleming bound. The analysis is extended to passage times for initially mixed states.
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TL;DR
This paper gives an elementary geometric derivation of the minimum time for a quantum state to evolve into an orthogonal state under a prescribed Hamiltonian, showing that the Fleming bound is usually saturated. It also examines exceptional spectra where the passage time is longer and extends the discussion to initially mixed states.
Key findings
- For a state that evolves along a geodesic in projective Hilbert space and is a superposition of two energy eigenstates, the minimum passage time is \(\tau_P=\pi\hbar/E\), where \(E=E_j-E_i\), equivalently \(\tau_P=\pi\hbar/(2\Delta H)\), the Fleming bound.
- The Fleming bound is typically attained, but for special spectra or states involving more than two energy eigenstates the passage time is larger. A three-level example with \(\alpha=\beta=\pi/4\) and \(\omega_{ki}/\omega_{ji}=(2m-1)/(2n-1)\) gives \(T=\pi\hbar/\omega_{ji}=3\pi\hbar/\omega_{ki}\) and \(T=\sqrt{6}\,\tau_P\).
- A spin-1 example with energies \(-1,0,+1\) and state \(\frac12|-\rangle+\frac1{\sqrt2}|0\rangle+\frac12|+\rangle\) becomes orthogonal after \(T=\pi\hbar\), which is \(\sqrt2\,\tau_P\).
- The Margolus-Levitin bound \(\pi\hbar/(2E)\) is argued not to be generally physically viable and never sharper than the derived bound unless the energy scale is shifted so all eigenvalues are nonnegative; it is attained only in a special case.
- For an initial mixture of two pure states of the two-level form, certainty of reaching an orthogonal state requires \(\tau_P=\pi\hbar\times \mathrm{LCM}(\omega_{ji}^{-1},\omega_{lk}^{-1})\), with a natural generalization to many such states.
Methods
- Theoretical derivation using projective Hilbert space, the Fubini-Study metric, Hermitian correspondence between points and hyperplanes, and geodesic motion on a projective line viewed as a two-sphere.
- The analysis uses the Anandan-Aharonov relation to connect the passage time to energy dispersion and to derive the Fleming bound.
- The paper treats two-level superpositions, a three-level superposition constraint, a spin-1 cyclic example, and mixed initial states represented by a density matrix.
Limitations
- The paper states that energy eigenvalues are assumed distinct for definiteness, though this is said not to be essential; it also notes that generic states will not evolve into orthogonal states, and that constraints become more severe for superpositions of more than three eigenstates.
- The mixed-state extension is limited: the paper states that it is not clear how to define orthogonality for impure density matrices meaningfully, or whether a passage time exists under such a definition, calling this an open problem.
- Evident from the text: the results are analytical and illustrated with special examples rather than experimental data, and the mixed-state treatment assumes initial pure states of a particular two-level form.
Model-written summary. Check it against the paper before relying on it.
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Claims the paper makes
0 of 5 cited sources checked · 0 supported · 0 flagged
- Findingnone given
There exist infinitely many special states for which unitary evolution can produce an orthogonal state at a suitable passage time.
“We shall show that, in fact, there exist infinitely many, although rather special, states |ψ such that ψ|η = 0 for a suitable choice of passage time τ P”
Quote in the paperSubstantial claim - Findingreasoning
For a state that is a superposition of exactly two energy eigenstates with distinct energies E_i and E_j, the minimum time for unitary evolution to produce an orthogonal state along a geodesic is πħ divided by the energy difference E_j−E_i.
“Choosing the smallest value for k and writing E = E j -E i we thus obtain the minimum value τ P of the passage time, given by”
- Also stated in: § 2. Derivation of passage times · plain
“τ P = πh E . (14”
- Also stated in: § 2. Derivation of passage times · plain
“To summarize, when |ψ is transformed into an orthogonal state |η by a one-parameter family of unitary transformations along a geodesic curve, then the time required is given exactly by (14).”
Quote in the paperSubstantial claim - Also stated in: § 2. Derivation of passage times · plain
- Findingreasoning
For a three-energy-eigenstate superposition with α=β=π/4, a necessary condition for orthogonal evolution is that the ratio of energy differences ω_ki/ω_ji equals (2m−1)/(2n−1) for natural numbers m and n.
“Then, (20) implies that a necessary condition for the state |ψ to evolve into an orthogonal state is given by the relation”
- Also stated in: § 3. Fleming's bound · plain
“ω ki ω ji = 2m -1 2n -1(21)”
- Also stated in: § 3. Fleming's bound · plain
“where m, n are natural numbers such that m = n.”
Quote in the paperSubstantial claim - Also stated in: § 3. Fleming's bound · plain
- Findingreasoning
For an initial mixed state that is a probabilistic mixture of two pure states of form (16), with one a superposition of E_i and E_j and the other of E_k and E_l, if the state is to evolve with certainty into an orthogonal state, the required passage time is πħ times the least common multiple of the inverses of the energy differences ω_ji and ω_lk.
“Therefore, if the initial state evolves with certainty into an orthogonal state, then the required passage time is given by”
- Also stated in: § 4. Mixed initial states · plain
“τ P = πh × LCM ω -1 ji , ω -1 lk (27”
Quote in the paperSubstantial claim - Also stated in: § 4. Mixed initial states · plain
- Findingreasoning
For an initial state that is one of many pure states of form (16), the passage time is πħ times the least common multiple of the inverses of the energy differences.
“It is straightforward to generalize this argument to the case where the initial state is one of many states of the form (16).In this case, the passage time is simply given by πh times the least common multiple of the inverses of the energy differences.”
Quote in the paperSubstantial claim
- Findingreasoning
If the three-state constraint is satisfied, the first time the state becomes orthogonal is T = πħ/ω_ji = 3πħ/ω_ki.
“The first time that |ψ becomes orthogonal to |ψ , in particular, is given by”
- Also stated in: § 3. Fleming's bound · plain
“T = πh ω ji = 3πh ω ki . (22”
Quote in the paperSubstantial claim - Also stated in: § 3. Fleming's bound · plain
- Findingown data
For the two-eigenstate superposition (16), the squared energy dispersion is one quarter of the square of the energy difference E_j−E_i.
“By a direct calculation, the energy dispersion in the state (16) is”
- Also stated in: § 3. Fleming's bound · plain
“H 2 = 1 4 (E j -E i ) 2 (17”
Quote in the paperSubstantial claim - Also stated in: § 3. Fleming's bound · plain
- Findingreasoning
The condition e^{-i(E_j−E_i)τ/ħ} = −1 is satisfied by τ = πħ k/(E_j−E_i) for odd positive integers k.
“Then, addition of equations ( 10) and (11) yields the condition”
- Also stated in: § 2. Derivation of passage times · plain
“e -i(E j -E i )τ/h = -1 (12”
- Also stated in: § 2. Derivation of passage times · plain
“which is satisfied if we set”
Quote in the paperSubstantial claim - Also stated in: § 2. Derivation of passage times · plain
- Interpretationreasoning
In general, if a quantum state not of the two-eigenstate form (16) does evolve into an orthogonal state, its passage time is necessarily longer than Fleming's bound.
“In general, if a quantum state expressible in the form other than (16) does evolve into an orthogonal state, then the passage time is necessarily longer than Fleming's bound (18).”
Quote in the paperSubstantial claim- Rests on an assumption: A generic Hamiltonian's energy spectrum will not satisfy the ratio condition required for a three-energy-eigenstate superposition to evolve into an orthogonal state.
- Interpretationreasoning
Equation (4) provides the sharpest obtainable bound for the passage time.
“We also show, using the Anandan-Aharonov relation, that (4) actually provides the sharpest obtainable bound for the passage time.”
Quote in the paperSubstantial claim- Rests on an assumption: The Fubini-Study distance between a pair of orthogonal states is π.
- Premisereasoning
Fleming's lower bound on the passage time is πħ divided by twice the energy dispersion ΔH.
“from which we obtain Fleming's bound”
- Also stated in: § 3. Fleming's bound · plain
“τ P = πh 2 H (18”
- Also stated in: § 3. Fleming's bound · hedged
“as indicated in [1].”
Quote in the paperSubstantial claim0 of 2 sources checked- cites [9] Fleming G N, 1973, Nuovo Cimento source
Not checked in the free run.
- cites [1] Schulman L S, 2002
Not checked in the free run.
- Also stated in: § 3. Fleming's bound · plain
- Interpretationreasoning
The Fleming lower bound is typically attained, except in special cases where the energy spectrum has a particularly simple structure.
“We provide an elementary derivation of the passage time, and demonstrate that the known lower bound, due to Fleming, is typically attained, except for special cases in which the energy spectra have particularly simple structures.”
Quote in the paperSubstantial claim - Premisecitation
The Anandan-Aharonov relation states that the speed of evolution of a quantum state is 2ħ^{-1} ΔH.
“This relation is indeed natural if we recall the Anandan-Aharonov relation [12] which states that the 'speed' of the evolution of a given quantum state is given by 2h -1 H .”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [12] Anandan J, 1990, Phys. Rev. Lett. source
Not checked in the free run.
- Interpretationreasoning
Even if each possible pure state becomes orthogonal after the mixed-state passage time, one cannot clearly argue that the density matrix has evolved into another mixed state orthogonal to the initial one.
“Note that, even though each possible pure state will be transformed into an orthogonal state after the system has evolved for the time τ P given in (27), one cannot clearly argue that the density matrix ρ(τ P ) has evolved into another mixed state orthogonal to ρ(0).”
Quote in the paperSubstantial claim - Premisecitation
The Margolus-Levitin inequality becomes technically valid only when the energy scale is fixed so that all energy eigenvalues are nonnegative.
“To avoid this problem, Margolus and Levitin fix the energy scale so that E l 0 for all l = 1, 2, . . ., n.Only then does the inequality (15) become technically valid.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [4] source
Not checked in the free run.
- Interpretationreasoning
In the exceptional cases, the passage times are necessarily larger than the Fleming bound.
“It is also shown, using a geodesic argument, that the passage times for these exceptional cases are necessarily larger than the Fleming bound.”
Quote in the paperSubstantial claim - Interpretationnone given
If an alternative path from |ψ to |η exists, its length is necessarily longer than the geodesic because it is not a geodesic.
“If an alternative path exists, then the length of the trajectory is necessarily longer, since any such path will not be a geodesic.”
Quote in the paperNot a substantial claim - Interpretationnone given
The constraint for orthogonal evolution becomes more severe when the state is a superposition of more than three energy eigenstates.
“The constraint becomes even more severe if |ψ is expressed as a superposition of more than three eigenstates.”
Quote in the paperSubstantial claim- Rests on an assumption: A generic Hamiltonian's energy spectrum will not satisfy the ratio condition required for a three-energy-eigenstate superposition to evolve into an orthogonal state.
- May combine two claims. 8
May combine two claims.
FindingreasoningThe diagonal elements of the initial and time-evolved density matrices in the energy basis are identical, so expectation values of any observable commuting with the Hamiltonian are also identical.
“Indeed, the diagonal elements of ρ(0) and ρ(τ P ), when expressed in the energy basis, are identical, and therefore the expectation values of any observable commuting with the Hamiltonian will also be identical.”
Quote in the paperSubstantial claimMay combine two claims.
FindingreasoningFor the spin-1 example with eigenvalues -1, 0, +1 and state (23), applying U(πħ) produces an orthogonal state (24), giving a cyclic evolution that interchanges orthogonal states.
“The application of Û(πh) yields”
- Also stated in: § 3. Fleming's bound · plain
“and we have ψ|η = 0. Likewise, the action of Û(πh) on |η yields |ψ , hence, we have a cyclic evolution that interchanges a pair of orthogonal states |ψ and |η .”
Quote in the paperSubstantial claim- Also stated in: § 3. Fleming's bound · plain
May combine two claims.
FindingreasoningIn the spin-1 example, the time T=πħ required to interchange the orthogonal states is longer than the Fleming bound; specifically T=√2 τ_P.
“However, because the trajectory Û(t)|ψ in P does not correspond to a geodesic curve, the time required to interchange these states, given by T = πh, is longer than the Fleming bound.”
- Also stated in: § 3. Fleming's bound · plain
“Indeed, we have T = √ 2τ P in this example”
Quote in the paperSubstantial claim- Also stated in: § 3. Fleming's bound · plain
May combine two claims.
FindingreasoningIn the three-state case, T is larger than Fleming's passage time; specifically T = √6 τ_P.
“However, since in this case Û(t)|ψ does not describe a geodesic path, T will be larger than Fleming's passage time τ P given in (18).”
- Also stated in: § 3. Fleming's bound · plain
“Then, it is straightforward to verify that T = √ 6τ P .”
Quote in the paperSubstantial claim- Also stated in: § 3. Fleming's bound · plain
May combine two claims.
Findingown dataIn the spin-1 example, the squared energy dispersion is 1/2 and the expectation value of the Hamiltonian is 0, so ΔH^2=1/2.
“because in the state (23) we have H 2 = 1 2 and H = 0 so that H 2 = 1 2 .”
Quote in the paperSubstantial claimMay combine two claims.
InterpretationreasoningFleming's bound can be derived directly from the Anandan-Aharonov relation because the evolution speed is constant under unitary action and the minimum distance between orthogonal states is π.
“Since the velocity 2h -1 H of the quantum state is a constant under the action of the unitary group, while the minimum distance of the trajectory joining a pair of orthogonal states is always π, it follows that the Fleming bound can be derived directly from the Anandan-Aharonov relation.”
Quote in the paperSubstantial claim- Rests on an assumption: The Fubini-Study distance between a pair of orthogonal states is π.
May combine two claims.
InterpretationreasoningIf a state is a superposition of two energy eigenstates, its unitary trajectory remains on the projective line joining those eigenstates, so no alternative path exists.
“If |ψ is expressible as a superposition of |E i and |E j , then the trajectory of Û(t)|ψ never leaves the projective line that joins these two states, and hence there does not exist any alternative path.”
Quote in the paperSubstantial claimMay combine two claims.
InterpretationreasoningThe Margolus-Levitin bound is in general not physically viable and is never sharper than the right-hand side of (14).
“However, this inequality is in general not physically viable, and it is in fact never sharper than the right-hand side of (14).”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [4] source
Not checked in the free run.
- Rests on an assumption: The physical properties of a quantum system do not change when all energy eigenvalues are shifted by the same constant.
The paper relies on this without support.
The physical properties of a quantum system do not change when all energy eigenvalues are shifted by the same constant.
“This is because the physical characteristics of quantum systems are invariant under an overall shift of the energy spectrum”
The paper relies on this without support.
A generic Hamiltonian's energy spectrum will not satisfy the ratio condition required for a three-energy-eigenstate superposition to evolve into an orthogonal state.
“Because the spectrum of a generic Hamiltonian Ĥ will not satisfy (21), a state |ψ will never evolve into a state orthogonal to |ψ .”
The paper relies on this without support.
The Fubini-Study distance between a pair of orthogonal states is π.
“The Fubini-Study distance between a pair of orthogonal states is given by π”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
Both the Mandelstam-Tamm and Margolus-Levitin bounds are valid, as shown by elementary proofs and by proofs for mixed and entangled states.
“That both bounds, ( 12) and (16), hold true, however, was illustrated by further elementary proofs by, e.g.Uffink [19], Brody [54], Andrecut and Ali [55], and Kosiński and Zych [56], and by more eleborated proofs for mixed and entangled states [57][58][59].”
From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [54]
All three approaches to survival probability have the common lower bound (2).
“All three approaches, therefore, have the common lower bound (2).”
From Kind of entanglement that speeds up quantum evolution (2012) · cites this paper as [21]
For pure states, equality in the bound holds if and only if the initial state is an equally weighted superposition of exactly two eigenstates of H.
“For pure states, it has been shown (e.g., by Refs. [15, 21, 22]) that equality holds if and only if the initial state is an equally weighted superposition of exactly two eigenstates of H .”
From Kind of entanglement that speeds up quantum evolution (2012) · cites this paper as [21]
The above expression generalizes in a unified manner previous results derived in Refs. [7-22].
“The above expression generalizes in an unified manner previous results derived in Refs. [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22].”
From Energy–time uncertainty relation for driven quantum systems (2013) · cites this paper as [10]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
The mixed-state waiting-time formula can miss orthogonality
MajorEquation (13) requires an odd multiple of each pure state's first passage time, whereas equation (27) takes an unrestricted least common multiple. For the paper's energy levels −1, 0 and +1, pairs with gaps 1 and 2 give inverse gaps 1 and 1/2; their least common multiple is 1, at which the gap-2 state has returned to itself rather than become orthogonal.
“τ P = πh × LCM ω -1 ji , ω -1 lk (27 )”
“τ = πhk E j -E i (k = 1, 3, 5, . . .), (13”
Density-matrix evolution runs opposite to the stated ket evolution
MinorFrom |η⟩ = U(τ)|ψ⟩ and ρ = |ψ⟩⟨ψ|, the evolved density matrix is U(t)ρU†(t). Equation (26) instead gives U†(t)ρU(t), which corresponds to the opposite unitary action on the state.
“ρ(t) = Û † (t) ρ Û(t). (26”
“|η = Û(τ )|ψ (3”
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- Authors
- Brody, D. C.
- Affiliations
- Imperial College London (GB)
- Corresponding author
- Brody, Dorje C
- ORCID
- Brody, Dorje C
- Venue
- Journal of Physics A: Mathematical and General
- Publisher
- IOP Publishing
- Published
- 2003-05-08
- Volume
- 36
- Issue
- 20
- Pages
- 5587-5593
- ISSN
- 0305-4470 (ISSN-L)
- 1361-6447
- OpenAlex
- W2087409509
- Semantic Scholar
- 8a0ddffd5557e284dbb3fa766d77e0bedb25d3d3
- Cited by
- 99
- References
- 8
- Citations by year
- 2026: 5 · 2025: 4 · 2024: 4 · 2023: 4 · 2022: 5 · 2021: 1 · 2020: 6 · 2019: 2 · 2018: 3 · 2017: 4 · 2016: 6 · 2015: 9 · 2014: 3 · 2013: 8 · 2012: 7
- FWCI
- 1.69
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Journal of Physics A Mathematical and General (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Royal Society
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Mathematics › Mathematical Physics › Spectral Theory in Mathematical Physics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Quantum chaos and dynamical systems
- Keywords
- passage time, unitary evolution, lower bounds, energy spectrum, mixed states, quantum state evolution
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