Margolus-Levitin quantum speed limit for an arbitrary fidelity

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I. INTRODUCTION
A quantum speed limit (QSL) is a lower bound on the time it takes to transform a quantum system in some predetermined way.QSLs exist for all sorts of transformations of both open and closed systems.Many involve statistical quantities such as energy, entropy, fidelity, and purity [1][2][3][4][5].
A prominent QSL is attributed to Mandelstam and Tamm [6].The Mandelstam-Tamm QSL states that the time it takes for an isolated quantum system to evolve between two fully distinguishable states is at least π/2 divided by the energy uncertainty, 1,2,3   τ ≥ τ mt (0), τ mt (0) = π 2∆H . ( Mandelstam and Tamm actually showed a more general result: If the fidelity between the initial state and the final state is δ, the evolution time is bounded according to For δ = 0, estimate (2) agrees with estimate (1).Anandan and Aharonov [7] provided the Mandelstam-Tamm QSL with an elegant geometric interpretation when they showed that the Fubini-Study distance between two states with fidelity δ is arccos √ δ and that the Fubini-Study evolution speed equals the energy uncertainty.Inequality (2) is thus saturated for isolated systems where the state evolves along a shortest Fubini-Study geodesic.Anandan and Aharonov's work inspired the writing of Ref. [8], which extends the Mandelstam-Tamm QSL to systems in mixed states in several different ways.
Margolus and Levitin [9] derived another QSL that is often mentioned together with Mandelstam and Tamm's.The Margolus-Levitin QSL states that the time it takes for an isolated quantum system to evolve between two fully distinguishable states is at least π/2 divided by the expected energy shifted by the smallest energy, τ ≥ τ ml (0), τ ml (0 Giovannetti et al. [10] extended Margolus and Levitin's QSL to an arbitrary fidelity.More precisely, they showed that the evolution time of an isolated system is bounded from below according to where α is a function that depends only on the fidelity δ between the initial and final states; see Sec.II.Giovannetti et al. also showed that (4), like (2), is tight, which means that for each δ there is a system for which the inequality in ( 4) is an equality.However, apart from that, the situation is different from the case of the Mandelstam-Tamm QSL: A closed formula for α does not exist, the derivation of (4) rests on numerical estimates, there is no classification of the systems saturating (4), and a geometric interpretation like that of ( 2) is lacking [1,2,11,12].In this paper, we derive the extended Margolus-Levitin QSL analytically and characterize the systems that saturate this estimate (Sec.II).Furthermore, we show that the extended Margolus-Levitin QSL has a symplecticgeometric interpretation and is connected to the Aharonov-Anandan geometric phase [13] (Sec.III).The Margolus-Levitin QSL thus differs fundamentally from most existing QSLs.That the estimates in ( 3) and ( 4) are either invalid or correct but not tight unless we shift the expected energy with the smallest energy suggests that a ground state will play a central role in the derivation of the extended Margolus-Levitin QSL and in its geometric interpretation.
The maximum of the Mandelstam-Tamm and the extended Margolus-Levitin QSL is also a QSL.The characteristics of the maximum QSL differ depending on whether or not the initial state and final state are fully distinguishable.We explain why this is so, and we provide several QSLs that are related to but less sharp than the extended Margolus-Levitin QSL.We also derive a dual version of the extended Margolus-Levitin QSL involving the largest rather than the smallest energy (Sec.IV).The paper concludes with a summary and a comment on the difficulty of extending the Margolus-Levitin quantum speed limit to driven systems (Sec.V).For a more detailed discussion of this difficulty, see Ref. [14].
For each δ, the minimum on the right-hand side is assumed for a unique z. Figure 1 shows the graph of α.Note that α depends only on the fidelity δ between the initial and final states.The fidelity, or overlap, between two pure states ρ a and ρ b is tr(ρ a ρ b ).
Although the Margolus-Levitin QSL (3) is quite surprising, its proof is relatively simple [9].Giovannetti et al.'s estimate (4) reduces to the Margolus-Levitin QSL for δ = 0, but the derivation in Ref. [10] for a general δ is rather complicated.Moreover, it is partly based on numerical calculations.In this section, we derive (4) analytically.We also characterize the systems, that is, the states and Hamiltonians that saturate (4).
For simplicity, we write ⟨H⟩ for the expected energy of a system without reference to its state.Furthermore, we write ⟨H − ϵ 0 ⟩ for the difference between ⟨H⟩ and the smallest eigenvalue ϵ 0 of H.We call this quantity the normalized expected energy.In this paper we only consider isolated systems, that is, systems where a timeindependent Hamiltonian governs the dynamics.For such systems, the expected energy and normalized expected energy are conserved quantities.The expected energy and normalized expected energy thus depend on the initial state but do not change as the state evolves.

A. The extended Margolus-Levitin quantum speed limit for a two-dimensional system
In this section we show that Giovannetti et al.'s estimate (4) is valid and tight for qubit systems.Derivations of the statements in this section can be found in Appendix A. Consider a qubit system with Hamiltonian the vectors |0⟩ and |1⟩ being orthonormal.We identify each qubit state ρ with a unit length vector r = (x, y, z) called the Bloch vector of ρ by defining The dynamics induced by H causes the Bloch vectors to rotate about the z axis with a constant inclination and an azimuthal angular speed ϵ 1 − ϵ 0 .Furthermore, a state's expected energy is determined by, and determines, the z coordinate of its Bloch vector: Two states thus have the same expected energy if and only if their Bloch vectors have the same z coordinate.
Let r a and r b be the Bloch vectors of two states with a common z coordinate z and fidelity δ, and suppose that H causes r a to rotate to r b in time τ .Using (7)-( 9) one can show that the inner product between r a and r b relates to the fidelity δ as To determine the distance traveled by the rotating Bloch vector consider the orthogonal projections ra and rb of r a and r b on the xy plane.During the evolution, ra rotates to rb along the peripheral arc of a circular sector in the xy plane of radius The distance traveled by the rotating Bloch vector equals the length of the peripheral arc and is, thus, The speed of the rotating Bloch vector is (ϵ 1 −ϵ 0 )(1−z 2 ) 1/2 , and hence, by (12), the evolution time is Combined with (10), this gives that The relation (11) implies that the z coordinate squared of the Bloch vectors of two states with the same expected energy is less than the fidelity between the states: Conversely, each expected energy level corresponding to a z such that z 2 ≤ δ contains Bloch vectors of states with fidelity δ.We conclude that τ ≥ τ ml (δ) for a qubit.Equality holds if and only if the z coordinate of the Bloch vector of the initial state minimizes the right side of ( 14) over the interval − √ δ ≤ z ≤ √ δ and thus is such that

B. The extended Margolus-Levitin quantum speed limit for systems of arbitrary dimension
Section II A shows that Giovannetti et al.'s estimate (4) is valid and tight for qubit systems.Section II A also shows that for an arbitrary isolated system with Hamiltonian H there is a state that evolves into one with fidelity δ in such a way that (4) is an identity: Let |0⟩ and |1⟩ be eigenvectors of H with eigenvalues ϵ 0 and ϵ 1 , with ϵ 1 > ϵ 0 .Choose a ρ with support in the span of |0⟩ and |1⟩ and such that z defined by (9) satisfies (16).Then ρ will evolve to a state with fidelity δ in time τ = τ ml (δ).Conversely, for any system in a state ρ, a Hamiltonian exists that transforms ρ into a state with fidelity δ in such a way that (4) is saturated: Take an H whose sum of two eigenspaces, one of which corresponds to its smallest eigenvalue ϵ 0 , contains the support of ρ.Adjust H's spectrum so that z defined by ( 9) satisfies (16).Then ρ will evolve to a state with fidelity δ in time τ = τ ml (δ).
An effective qubit for H is a state with support in the sum of two eigenspaces of H.It behaves like a genuine qubit in that its support evolves in the linear span of two energy eigenvectors, one from each eigenspace covering the initial support.The above discussion shows that (4) holds for and can always be saturated by an effective qubit.We say that a state is partly grounded if the eigenspace corresponding to ϵ 0 is not contained in the kernel of the state or, equivalently, if the eigenspace of ϵ 0 is not orthogonal to the support of the state.In Appendix B we show the first main result of this paper: If τ ⟨H − ϵ 0 ⟩ assumes its smallest possible value when evaluated for all Hamiltonians H, states ρ, and τ ≥ 0 such that H transforms ρ into a state with fidelity δ in time τ , then ρ is a partly grounded effective qubit for H and τ ⟨H − ϵ 0 ⟩ = α(δ).The extended Margolus-Levitin QSL (4) thus holds generally and is a tight estimate saturable in all dimensions.We interpret (4) geometrically in Sec.III.There we see, among other things, that one can interpret α(δ) as an extremal dynamical phase.Notice the contrast with the Mandelstam-Tamm QSL, where the numerator is a geodesic distance.
Remark 1.If we select a subset of the spectrum of H and consider only initial states with support in the sum of the eigenspaces of the eigenvalues in the subset, then the support of the evolved state will remain in that sum.The proof in Appendix B shows that the evolution time of each such state satisfies the inequality with ϵ ′ 0 being the smallest eigenvalue in the subset. 5Furthermore, by Sec.II A, inequality ( 17) can be saturated with an effective qubit that evolves in the sum of the eigenspaces corresponding to two eigenvalues in the subset, one of which is ϵ ′ 0 .As a special case, we have that the evolution time is bounded according to where ϵ ′′ 0 is the smallest occupied energy, that is, the smallest eigenvalue of H whose corresponding eigenspace is not annihilated by ρ.Often, the Margolus-Levitin QSL is formulated with the expected energy shifted by the smallest occupied energy rather than the smallest energy.Mathematically, however, there is no difference because we can always reduce the Hilbert space to an effective Hilbert space and consider the smallest occupied energy as the smallest energy.(However, see Remark 2.) Remark 2. The state must evolve in the span of two eigenvectors of H to saturate (4), one of which has eigenvalue ϵ 0 .No requirements are placed on the eigenvalue ϵ 1 of the second eigenvector except that it must differ from ϵ 0 .However, if we want the evolution time to be as short as possible, ϵ 1 must be the largest eigenvalue of H.This follows from (10) since saturation of (4) implies that the quotient 2. An oriented surface in the Bloch sphere bounded by the evolution curve ρt and the shortest geodesics connecting the initial and final states ρ0 and ρτ to the ground state |0⟩⟨0|.
The orientation of the surface is such that its symplectic area is negative.The negative of the symplectic area is minimal if and only if the extended Margolus-Levitin QSL is saturated.
the maximum value of ⟨H − ϵ 0 ⟩, and consequently the minimum value of τ , is obtained for the ϵ 1 maximizing the difference ϵ 1 − ϵ 0 .The observation that the state that saturates (4) with the shortest possible evolution time is an effective qubit with support in the sum of the eigenspaces belonging to the largest and the smallest eigenvalue generalizes the main result in Ref. [15] to arbitrary fidelity; see also Ref. [16].A corresponding statement holds if we restrict the Hilbert space as in Remark 1.
In contrast to the energy uncertainty, we cannot consider the normalized expected energy ⟨H − ϵ 0 ⟩ as a measure of a state's rate of change.Since for each state there exist Hamiltonians H 1 and H 2 , with smallest eigenvalues ϵ 1 0 and ϵ 2 0 , that identically evolve the state but for which ⟨H 1 − ϵ 1 0 ⟩ and ⟨H 2 − ϵ 2 0 ⟩ are different.Thus, unlike most QSLs, the extended Margolus-Levitin QSL is not a quotient of a distance and a speed [1,2,11,12].

III. GEOMETRY OF THE EXTENDED MARGOLUS-LEVITIN QUANTUM SPEED LIMIT
Equations ( 7)-( 9) describe a diffeomorphism between the projective Hilbert space of qubit states and the Bloch sphere.(Thus, we can identify qubit states with their corresponding Bloch vectors.)We push forward the Fubini-Study Riemannian metric and symplectic form using this diffeomorphism. 6The expression in ( 14) is then the negative of the symplectic area of a surface with a triangular boundary in the Bloch sphere.The path traced out by the evolving state and the shortest geodesics connecting the initial and final state to the lowest energy state |0⟩⟨0| form the boundary of the surface; see Fig. 2. Notice that we have oriented the boundary so that the surface has the reverse orientation compared with the standard orientation of the Bloch sphere.In Sec.III B, we show that τ ⟨H − ϵ 0 ⟩ is equal to the negative of the symplectic area of such a triangular surface also in the general case.
The expected energy level to which the initial qubit state ρ belongs is a geodesic sphere centered at |0⟩⟨0|, that is, a sphere made up of all states at a fixed distance from |0⟩⟨0|.The radius of the geodesic sphere is Therefore, z = − cos(2r).If we substitute z for − cos(2r) on the right-hand side of ( 5) we get where r ranges from 1 2 arccos Section III C shows that the expression minimized on the right-hand side is an extremal dynamical phase in a gauge specified by a stationary state with eigenvalue ϵ 0 .First, in Sec.III A, we interpret τ ⟨H − ϵ 0 ⟩ as a dynamical phase.

A. Evolution time times normalized expected energy as a dynamical phase
Consider a quantum system modeled on a finitedimensional Hilbert space H. Let S be the unit sphere in H and P be the projective Hilbert space of orthogonal projection operators of rank 1 on H. 7 The Hopf bundle is the U (1)-principal bundle η that sends each |ψ⟩ in S to the corresponding state |ψ⟩⟨ψ| in P. The Berry connection on the Hopf bundle is defined as on tangent vectors | ψ⟩ at |ψ⟩.
Assume the system has a Hamiltonian H. Choose an eigenstate σ of H with eigenvalue ϵ.Let Ω(σ) be the open neighborhood of σ consisting of all states that are not fully distinguishable from σ. Hypersurfaces Y |ϕ⟩ in S, one for each vector |ϕ⟩ in the fiber over σ, foliate the preimage of Ω(σ) under η; see Fig. 3 and Ref. [17].The hypersurface Y |ϕ⟩ consists of all vectors |ψ⟩ in S that are in phase with |ϕ⟩, that is, are such that ⟨ϕ|ψ⟩ > 0. The gauge group permutes the hypersurfaces, and η maps each hypersurface diffeomorphically onto Ω(σ).We can thus define a gauge potential A σ on Ω(σ) by pushing down the restriction of A to an arbitrary hypersurface Y |ϕ⟩ .The potential depends on σ but not on the choice of |ϕ⟩ in the fiber over σ.The hypersurface Y |ϕ⟩ consists of all the |ψ⟩ in S that are in phase with |ϕ⟩.The gauge group permutes the hypersurfaces, and η maps each hypersurface diffeomorphically onto Ω(σ).We define a potential A σ on Ω(σ) by pushing down the restriction of A to one of the hypersurfaces.The potential depends on σ but not on the choice of hypersurface.
The second main result in the paper reads: If ρ is a state in Ω(σ), and ρ t , where 0 ≤ t ≤ τ , is the evolution curve starting from ρ, then ρ t is contained in Ω(σ) and In other words, τ ⟨H − ϵ⟩ is the dynamical phase of ρ t in a gauge associated with σ. 8 To prove ( 22) select a |ϕ⟩ in the fiber over σ, let |ψ⟩ be the vector over ρ in phase with |ϕ⟩, and let |ψ t ⟩ be the curve that extends from |ψ⟩ and has the velocity field | ψt ⟩ = −i(H − ϵ)|ψ t ⟩.The curve |ψ t ⟩ is in phase with |ϕ⟩ and projects to ρ t : Hence, Notice that ϵ = ϵ 0 if σ is a ground state.

B. Evolution time times normalized expected energy as the negative of a symplectic area
We equip the projective Hilbert space P with the Fubini-Study Riemannian metric g and symplectic form ω. For tangent vectors ρa and ρb at ρ, 9   g( ρa , ρb ) = 1 2 tr( ρa ρb ), ( 26) 8 Some authors call the negative of the right-hand side of ( 22) the dynamical phase of ρt. 9 We normalize g and ω asymmetrically as this gives rise to cleaner formulas.
4. On the left, the σ closure ρ σ t made up of the shortest geodesic γ 1 t from σ to the initial state ρ0, the evolution curve ρt, and the shortest geodesic γ 2 t from the final state ρτ to σ.On the right, a Seifert surface Σ for ρ σ t .The circular arrow indicates the orientation of Σ compatible with that of ρ σ t .
The geodesic distance function associated with the Fubini- Suppose the system is in a state ρ and a Hamiltonian H governs its dynamics.Let ρ t , 0 ≤ t ≤ τ , represent the evolving state.Since H is time-independent, the distances between ρ t and the eigenstates of H are preserved.Let σ be an eigenstate with eigenvalue ϵ located at a distance r < π/2 from ρ. Furthermore, let γ 1 t and γ 2 t be the shortest unit speed geodesics from σ to ρ 0 and from ρ τ to σ, respectively.The σ closure of ρ t is the concatenation The left part of Fig. 4 illustrates the σ closure.Below we show the third main result of the paper: where Σ is any Seifert surface for ρ σ t that is homologous to a Seifert surface for ρ σ t in Ω(σ).A Seifert surface for ρ σ t is an oriented surface Σ in P whose boundary is parametrized by ρ σ t as illustrated in the right part of Fig. 4. For example, the ruled surface obtained by connecting σ and each ρ t with the shortest arclength-parametrized geodesic is such a Seifert surface.In Appendix C we provide formulas for the geodesics γ 1 t and γ 2 t and explicitly construct a ruled Seifert surface for ρ σ t .The pull-back of ω to S by the Hopf projection is exact and equals the negative of the Berry curvature, η * ω = −dA [17].We construct a lift |ψ σ t ⟩ of ρ σ t to S as follows.Let |ϕ⟩ be any vector in the fiber over σ, let |ψ t ⟩ be the lift of ρ t which is in phase with |ϕ⟩, and for 0 ≤ t ≤ r define |ϕ The first identity is a consequence of Σ − Σ ′ being a homological boundary and that ω is closed; cf.Remark 3 below.
The second identity results from η being a diffeomorphism from Y |ϕ⟩ onto Ω(σ) and −dA being the pull-back of ω.
The third identity follows from |ψ σ t ⟩ parametrizing the boundary of the lift of Σ ′ .
A direct calculation shows that the Berry connection annihilates the velocity fields of |ϕ 1 t ⟩ and |ϕ 2 t ⟩: Thus we have that Equations ( 25), (33), and (35) yield the third main result (30).Note that the calculations rely on ρ not being fully distinguishable from σ, cf.Remark 1. Remark 3. The homology class of the projectivization of any two-dimensional subspace of H, that is, a Bloch sphere, generates the second singular homology group of P with integer coefficients [18].We choose such a Bloch sphere that we orient so that its symplectic area is positive.The area is 2π, and ω/2π is thus of integral class.The difference between two Seifert surfaces for ρ σ t is a 2-cycle.The homology class of such a difference is an integer multiple of the homology class for the Bloch sphere.It follows that the difference of the symplectic areas of two Seifert surfaces for ρ σ t is an integer multiple of the symplectic area of the Bloch sphere.Hence, for an arbitrary Seifert surface Σ for ρ σ t , We can equivalently express this as τ ⟨H − ϵ⟩ being equal to the Aharonov-Anandan geometric phase [13] of the σ closure of ρ t modulo 2π.This connection to the Aharonov-Anandan phase is utilized in Ref. [19] where QSLs for cyclic systems are derived.

C. α(δ) as an extremal dynamical phase
Let σ be any state and S(σ, r) be the geodesic sphere of radius r centered at σ.The geodesic sphere consists of all states at distance r from σ. Fix a fidelity δ and choose r such that arccos √ δ ≤ 2r ≤ arccos(− √ δ).The geodesic sphere S(σ, r) is then contained in Ω(σ) and includes states between which the fidelity is δ.
Write Γ(σ, r, δ) for the space of smooth curves in S(σ, r) that extend between two states with fidelity δ.Define J σ r,δ to be the functional that assigns the dynamical phase to each ρ t in Γ(σ, r, δ) in the gauge specified by σ, In Appendix D we show that ρ t is an extremal for J σ r,δ if and only if for every |ϕ⟩ in the fiber over σ, the lift |ψ t ⟩ of ρ t which is in phase with |ϕ⟩ splits orthogonally as for some function Λ t that vanishes for t = 0.The corresponding extreme value is The constraint on Λ t arising from the assumption that ρ t extends between two states with fidelity δ reads Thus, According to (20), α(δ) is equal to the smallest positive extreme value of J σ r,δ minimized over the interval This observation is the fourth main result of the paper.Remark 4. If ρ t is an extremal curve for J σ r,δ , then so is ρ τ −t , and . Hence the ± in (41).This observation is related to the dual QSL derived and discussed in Sec.IV B.

IV. RELATED QUANTUM SPEED LIMITS
The maximum of the Mandelstam-Tamm and the extended Margolus-Levitin QSLs is a new QSL.Interestingly, the maximum QSL behaves differently for fully and not fully distinguishable initial and final states.We describe this difference in Sec.IV A. In Sec.IV B we derive a QSL that extends the dual Margolus-Levitin QSL studied in Ref. [20], and in Sec.IV C we provide three QSLs that are less sharp but easier to calculate than the extended Margolus-Levitin QSL.These three QSLs are not new but can be found in the cited papers.

A. The maximum quantum speed limit
According to Anandan and Aharonov [7], the QSL of Mandelstam and Tamm ( 2) is saturated if and only if the evolving state follows a shortest Fubini-Study geodesic.Furthermore, by Brody [21], such a state is an effective qubit that follows the equator of the Bloch sphere associated with the two eigenvectors that have nonzero fidelity with the initial state; cf.Sec.II A. 10 Levitin and Toffoli [16] showed that if we require the initial and final states to be fully distinguishable, the same holds for the Margolus-Levitin QSL (3).However, in that case, one of the eigenvectors must belong to the smallest eigenvalue (or ϵ 0 be replaced by the smallest occupied energy, cf.Remark 1).Thus, if (3) is saturated, so is (1), and the reverse holds if the support of the initial state is not perpendicular to the eigenspace belonging to the smallest eigenvalue.These observations led Levitin and Toffoli to conclude that the maximum of τ mt (0) and τ ml (0) is a tight lower bound for the evolution time only reachable for states such that ⟨H − ϵ 0 ⟩ = ∆H.We can draw the same conclusion from the discussion in Sec.II.
Remark 5.The state does not follow a geodesic in a system that saturates the extended Margolus-Levitin QSL for a nonzero fidelity.However, the state follows a sub-Riemannian geodesic in a geodesic sphere centered at a ground state.See Appendix C for details.
Remark 6.In quite a few papers it is claimed that arccos √ δ/⟨H − ϵ 0 ⟩ is a QSL for isolated systems.Figure 5 and the fact that the extended Margolus-Levitin QSL is tight show that this is not true in general. 10None of the eigenvectors need to be associated with the smallest eigenvalue of the Hamiltonian.

B. The dual extended Margolus-Levitin quantum speed limit
Let ρ a and ρ b be states with fidelity δ, and suppose that H evolves ρ a to ρ b in time τ .Then −H evolves ρ b to ρ a in time τ .The smallest eigenvalue of −H is −ϵ max , where ϵ max is the largest eigenvalue of H, and according to the extended Margolus-Levitin QSL, This estimate generalizes the main result in Ref. [20] to an arbitrary fidelity between initial and final states.We adhere to the terminology in Ref. [20] and call τ * ml (δ) the dual extended Margolus-Levitin QSL.Note that for some states, τ * ml (δ) is greater than τ ml (δ) and τ mt (δ).Since the estimate in ( 42) is a consequence of applying the extended Margolus-Levitin QSL to evolution generated by −H, it follows from Appendix B that (42) can be saturated in all dimensions and that, when so, the state is an effective qubit for −H whose support is not orthogonal to the eigenspace corresponding to −ϵ max .But then the state is also an effective qubit for H whose support is not orthogonal to the eigenspace corresponding to ϵ max .We call such an effective qubit partly maximally excited.To summarize, if τ ⟨ϵ max − H⟩ assumes its smallest possible value when evaluated for all Hamiltonians H, states ρ, and τ ≥ 0 such that H transforms ρ into a state with fidelity δ in time τ , then ρ is a partly maximally excited effective qubit for H and τ ⟨ϵ max − H⟩ = α(δ).
Remark 7. If we restrict the set of states as in Remark 1, ϵ max can be replaced by the largest occupied energy.
The discussion in Sec.III, where we deliberately did not specify the eigenvalue ϵ, tells us that where σ is an eigenstate of H with eigenvalue ϵ max that has nonzero fidelity with the initial state.The surface Σ is an arbitrary Seifert surface for the σ closure ρ σ t in Ω(σ).In Fig. 6 we have illustrated such a Seifert surface in the Bloch sphere for the same evolution as in Fig. 2. In this case, a concatenation of the evolution curve with the shortest geodesic connecting the initial and final states to the excited state |1⟩⟨1| parametrizes the boundary of the Seifert surface.Furthermore, the orientation of the boundary is such that the Seifert surface has a positive symplectic area, which is consistent with equation (43).The z coordinate of a qubit state that saturates the dual extended Margolus-Levitin QSL satisfies the equation For δ ̸ = 0, the z coordinate of a saturating evolution is strictly positive in contrast to an evolution that saturates the "original" extended Margolus-Levitin QSL, in which case the z coordinate is strictly negative, as in Fig. 2.This > FIG. 6.A Seifert surface in the Bloch sphere for the |1⟩⟨1| closure of an evolving qubit.The symplectic area of the surface is positive and equal to τ ⟨ϵmax −H⟩.Note that the surface is, in a sense, dual to the Seifert surface in Fig. 2. If the dual extended Margolus-Levitin QSL is saturated, the surface assumes its smallest possible symplectic area.For not fully distinguishable initial and final states, the evolving state then has a strictly positive z coordinate.This is in contrast to the case when the extended Margolus-Levitin QSL is saturated, in which case the z coordinate is strictly negative.For fully distinguishable initial and final states, both QSLs can be saturated simultaneously.
The magnitudes of the symplectic areas of the corresponding Seifert surfaces are then equal.
observation lets us conclude that for not fully distinguishable initial and final states, the extended Margolus-Levitin QSL and its dual can never saturate simultaneously.Because if that were the case, the state would be an effective qubit with support in the span of a ground state and a highest energy state.In the projectivization of the span of these eigenstates, the z coordinate of the Bloch vector of the system's state would be strictly positive and strictly negative, which is contradictory.For δ = 0, however, the QSLs can saturate simultaneously.The evolving state then follows a geodesic and C. Approximations of the extended Margolus-Levitin quantum speed limit Consider a quantum system in a state ρ with Hamiltonian H. Suppose the system evolves into a state with fidelity δ relative to ρ in time τ .Then where the requirement that y = 1 − βx is a tangent line to cos x specifies β (β ≈ 0.724).Also, Derivations of ( 46) and (47) are found in Refs.[22,23], and (48) is mentioned in Ref. [10].Where this latter QSL comes from is still unclear to the authors.Figure 7 displays the graphs of the extended Margolus-Levitin QSL τ ml (δ) as well as the QSLs τ 1 (δ), τ 2 (δ), and τ 3 (δ) multiplied by ⟨H − ϵ 0 ⟩.Apparently, τ 1 (δ), τ 2 (δ), and τ 3 (δ) are weaker than τ ml (δ), and τ 2 (δ) is weaker than τ 1 (δ) and τ 3 (δ).However, which of τ 1 (δ) and τ 3 (δ) is the stronger QSL depends on the fidelity δ, with τ 1 (δ) being greater than τ 3 (δ) for large values of δ and τ 3 (δ) being greater than τ 1 (δ) for small values of δ.

V. SUMMARY AND OUTLOOK
Giovannetti et al. [10] showed, partly numerically, that if an isolated system evolves between two states with fidelity δ, then the evolution time is bounded from below by the extended Margolus-Levitin QSL (4).Giovannetti et al. also showed that this QSL is tight in all dimensions.
In this paper, we have derived the extended Margolus-Levitin QSL analytically and characterized the systems for which this QSL is saturated.Furthermore, we have interpreted the extended Margolus-Levitin QSL geometrically as an extremal dynamical phase in a gauge specified by the system's ground state.We have also shown that the maximum of the Mandelstam-Tamm and the extended Margolus-Levitin QSLs is a tight QSL that behaves differently depending on whether or not the initial state and the final state are fully distinguishable.In addition, we have derived a tight dual version of the extended Margolus-Levitin QSL using a straightforward time-reversal argument.The dual QSL has similar properties as the extended Margolus-Levitin QSL and saturates under similar circumstances but involves the largest rather than the smallest occupied energy.We showed that the two QSLs can only be saturated simultaneously if the start and end states are fully distinguishable.We concluded the paper by reproducing three QSLs related to, but slightly weaker than, the extended Margolus-Levitin QSL.
A recent paper on evolution time estimates for closed systems [14] suggests that the Margolus-Levitin QSL does not straightforwardly extend to systems whose dynamics are governed by time-dependent Hamiltonians, at least not without limitations on the width of the energy spectrum.
The geometric analysis performed here shows that extended Margolus-Levitin QSL is closely related to the Aharonov-Anandan geometric phase [13].This observation is further elaborated in Ref. [19] where QSLs for cyclically evolving systems are derived.
Mixed state QSLs resembling the Margolus-Levitin QSL exist [22][23][24], and Giovannetti et al. [10] showed that also the extended Margolus-Levitin QSL can be generalized to a QSL for systems in mixed states, with δ being the Uhlmann fidelity between the initial and final states [25].Whether this generalization has a symplectic-geometric interpretation similar to the one presented here is an open question.So is the question whether the generalized extended Margolus-Levitin QSL connects to a geometric phase for mixed states [23,[26][27][28][29].The authors intend to investigate these questions in a forthcoming paper.

Fully distinguishable initial and final states
If δ = 0 we must modify the previous section's arguments slightly.We start by replacing g with the two functions

FIG. 1 .
FIG.1.The graph of α as a function of the fidelity δ between the initial state and the final state.If δ = 0, the initial and final states are fully distinguishable, and α(δ) = π/2.In this case we recover the Margolus-Levitin QSL (3).If δ = 1, the initial and final states are the same, and α(δ) = 0.This is reasonable since it takes no time to remain in the initial state.

FIG. 3 .
FIG.3.The preimage under the Hopf projection η of the set Ω(σ) of states having nonzero fidelity with σ is foliated by hypersurfaces Y |ϕ⟩ ; one for each |ϕ⟩ in S being projected to σ.The hypersurface Y |ϕ⟩ consists of all the |ψ⟩ in S that are in phase with |ϕ⟩.The gauge group permutes the hypersurfaces, and η maps each hypersurface diffeomorphically onto Ω(σ).We define a potential A σ on Ω(σ) by pushing down the restriction of A to one of the hypersurfaces.The potential depends on σ but not on the choice of hypersurface.