Time-optimal quantum transformations with bounded bandwidth

In this paper, we derive sharp lower bounds, also known as quantum speed limits, for the time it takes to transform a quantum system into a state such that an observable assumes its lowest average value. We assume that the system is initially in an incoherent state relative to the observable and that the state evolves according to a von Neumann equation with a Hamiltonian whose bandwidth is uniformly bounded. The transformation time depends in an intricate way on the observable’s and the initial state’s eigenvalue spectrum and the relative constellation of the associated eigenspaces. The problem of finding quantum speed limits consequently divides into different cases requiring different strategies. We derive quantum speed limits in a large number of cases, and we simultaneously develop a method to break down complex cases into manageable ones. The derivations involve both combinatorial and differential geometric techniques. We also study multipartite systems and show that allowing correlations between the parts can speed up the transformation time. In a final section, we use the quantum speed limits to obtain upper bounds on the power with which energy can be extracted from quantum batteries.


I. INTRODUCTION
A quantum speed limit (QSL) is a lower bound for the time it takes to transform a quantum state in a certain way under some given conditions. Many QSLs have been derived for both open and closed systems; see [1,2] and the references therein. Several of them are valid under very general conditions and can therefore be applied to virtually any system [3][4][5][6][7]. Extensive applicability is indeed a strength of a QSL, but also means that the QSL can give a rather weak time-bound in specific cases.
In this paper, we take a different approach to deriving QSLs for a family of quantum systems broad enough to include many systems of both practical and theoretical interest but narrow enough for the QSLs to be very sharp. Specifically, we consider a general finite-dimensional system prepared in a state that commutes with a definite but otherwise unspecified observable. For such a system, we examine how long it takes to unitarily transform the state into one where the observable's expectation value is minimal. We allow the Hamiltonian governing the transformation to be time-dependent, but we assume that its energy bandwidth is uniformly bounded. Such an assumption is, in many cases, physically justified [8][9][10][11][12][13][14][15][16][17].
We borrow terminology from thermodynamics and call a state where the expectation value of the observable is minimal passive [18][19][20]. A passivization process is then any unitary process that leaves the system in a passive state. We define the passivization time of the system as the shortest time in which a passivization process that meets the bounded bandwidth condition can drive the system into a passive state. And by a QSL we henceforth mean a lower bound on the passivization time.
After a short preliminary section, we derive a general QSL for systems that conform to the description above. * ole.andersson@fysik.su.se We also describe conditions that ensure that the QSL agrees with the passivization time. In connection with this, we study collective passivization processes of ensembles of identical systems. In such, we allow correlations to develop during the process. We show that allowing correlations reduces the passivization time of the individual system, and we extend the QSL to a lower bound on the collective passivization time.
For many systems, the general QSL is not tight. This is because the passivization time depends in a rather intricate way on the eigenvalue spectra and the eigenspaces of the observable and the initial state. We calculate the passivization time explicitly for completely non-degenerate systems. We also develop a method to derive QSLs for systems where the observable or the state has a degenerate spectrum. The method, which is particularly effective for low-dimensional systems, generates tight QSLs under additional conditions (precisely described in the paper).
In the last section, we use the QSLs to derive bounds on the power with which energy can be extracted from a quantum battery. Here, we follow [21][22][23][24] and define a quantum battery as a closed quantum system whose energy content can be adjusted through cyclic unitary processes. We also use the QSLs for collective passivization processes to obtain bounds on the power of collective energy extraction processes. Similar results but for different constraints can be found in [23][24][25].
The outline of the paper is as follows. In Section II, we set up the problem, introduce terminology, and prove some preliminary results. In Section III, we derive a general QSL and discuss circumstances under which it is tight. In this section, we also consider multipartite systems. Section IV deals with the case when both the observable and the initial state have a non-degenerate spectrum. Section V begins with a general discussion on the characteristics of time-optimal Hamiltonians. Then we develop a method to deal with systems for which the observable or the initial state has a degenerate spectrum. In Section VI, we use the results from previous sections to derive upper bounds on the power of quantum batteries. The paper concludes with a summary and an outlook.

II. PRELIMINARIES
Let A be an observable of an n-dimensional quantum system prepared in a state ρ i . In this paper, we examine how long it takes before the system enters a state where the expectation value of A is minimal. We assume that the system evolves according to a von Neumann equatioṅ ρ(t) = −i[H(t), ρ(t)] with a Hamiltonian satisfying the inequality tr H(t) 2 ≤ ω 2 . (1) The quantity on the left is the bandwidth of the Hamiltonian, and the quantity ω on the right is some fixed positive number. We will refer to the inequality (1) as the bounded bandwidth condition.
The eigenvalue spectrum of a quantum state that evolves according to a von Neumann equation is preserved. We write S(ρ i ) for the space of all states that have the same spectrum as ρ i . Also, we write H for the Hilbert space of the system and denote the group of unitary operators on H by U(H). The unitary group acts on states according to U · ρ = U ρU † . This action preserves and is transitive on S(ρ i ). Since ρ i is assumed to evolve unitarily, S(ρ i ) can be considered the entire state-space of the system. For simplicity, "state" will from now on refer to a member of S(ρ i ) and thus be an abbreviation for "state isospectral to ρ i ." The expectation value function of A on S(ρ i ), is real-valued and continuous. Since S(ρ i ) is compact and connected, the image of E A is a closed and bounded interval. We borrow terminology from thermodynamics and call the states at which E A assumes its minimum value passive and the states at which E A assumes its maximum value maximally active. 1 Then, a more appropriate formulation of the main question addressed in this paper is: What is the shortest time in which ρ i can be transformed into a passive state using a Hamiltonian that satisfies the bounded bandwidth condition?
Remark 1. All results in this paper have a counterpart with an analogous proof for a maximally active final state.

A. Extremal and incoherent states
We call a state incoherent if it commutes with A. 2 An incoherent state is thus a state that preserves the eigenspaces of A. Since H is the direct sum of the eigenspaces of A, the incoherent states decompose into direct sums of operators acting on the eigenspaces. We arrange the different eigenvalues of A in increasing order and write A k for the eigenspace belonging to the eigenvalue number k. A state ρ is then incoherent if, and only if, ρ = ⊕ k ρ k where ρ k is an operator on A k . The operator ρ k is the kth component of ρ.

Proposition 1.
Passive and maximally active states are incoherent. The proposition is known since before [21]. But for the reader's convenience, we have included a proof in Appendix A. In this paper, we assume the following: The initial state ρ i is incoherent. The group of unitary operators commuting with A acts transitively on the passive states. This means, on the one hand, that if ρ is a passive state and U is a unitary that commutes with A, then U ρU † is a passive state, and, on the other hand, that every passive state is of the form U ρU † for some unitary U commuting with A. Since the unitaries commuting with A decompose into direct sums of unitaries operating on the eigenspaces of A, all passive states have isospectral components. We conclude that a state is passive if, and only if, it is incoherent and has components isospectral with those of a passive state.

B. Time-optimal Hamiltonians
We define the passivization time τ pas as the shortest time in which ρ i can be transformed into a passive state using a Hamiltonian that satisfies the bounded bandwidth condition. Below we determine τ pas in several important cases. We also give examples of Hamiltonians that realize such an optimal transformation. We say that a Hamiltonian is time-optimal if it satisfies (1) and transforms ρ i into a passive state in the time τ pas . According to the next proposition, time-optimal Hamiltonians saturate the inequality (1) at all times.

Proposition 2. Time-optimal Hamiltonians saturate the bounded bandwidth condition.
For a proof, see [15] or Appendix B.
The time-evolution operator associated with a Hamiltonian can be considered as a curve in U(H). We equip U(H) with the bi-invariant Riemannian metric g that agrees with the Hilbert-Schmidt inner product on the Lie algebra of U(H). By Proposition 2, the time-evolution operator U (t) associated with a time-optimal Hamiltonian H(t) then has a constant speed ω: Let P(ρ i ) be the set of unitary operators that transform the initial state into a passive state: Proposition 5 below says that P(ρ i ) is a submanifold of U(H). The next proposition, proven in Appendix C, transforms the problem of determining the passivization time into a geometric problem. We write 1 for the identity operator on H.

Proposition 3. The time-evolution operator of a time-
optimal Hamiltonian is a shortest curve from 1 to P(ρ i ).
The shortest curves connecting 1 and P(ρ i ) are pregeodesics (that is, they are geodesics if parameterized such that they have a constant speed [26]). Every geodesic of g emanating from 1 agrees with a one-parameter subgroup of U(H) on its domain of definition. Conversely, every oneparameter subgroup of U(H) is a geodesic of g; see [27].
Thus, a curve of unitaries U (t) such that U (0) = 1 is a geodesic if, and only if, U (t) = e −itH for some Hermitian operator H. Proposition 4 is a direct consequence of Propositions 2 and 3.

Proposition 4. Time-optimal Hamiltonians are timeindependent.
The geodesic distance between two unitary operators is the minimum of the lengths of all smooth curves connecting the two operators. One can express the geodesic distance between two unitaries U and V in terms of the principal logarithm and the Hilbert-Schmidt norm: See Appendices D and E for details. It follows that dist 1, P(ρ i ) = min dist(1, U ) : U ∈ P(ρ i ) = min Log U : U ∈ P(ρ i ) .
The first identity is the definition of the distance between 1 and P(ρ i ). Equation (6) and Propositions 2 and 3 together imply that Unfortunately, the minimum on the right is, in general, difficult to determine. But we will find explicit expressions for τ pas in several important cases.

C. Passivizing unitaries and isotropy groups
As was mentioned above, the unitary group acts on states by left conjugation. We write U(H) ρi for the isotropy group of the initial state: The unitary group also acts on observables by right conjugation, U · B = U † BU , and we write U(H) A for the isotropy group of the observable A: Proposition 5. The set P(ρ i ) is a submanifold of U(H). Moreover, if P is any unitary in P(ρ i ), then The proof is postponed to Appendix F. Proposition 5 implies that if P is any passivizing unitary, then dist(1, P(ρ i )) = min The minimum is over the U s in U(H) A and V s in U(H) ρi . Since ρ i is incoherent, A and ρ i are simultaneously diagonalizable. We fix a common orthonormal eigenbasis |1 , |2 , . . . , |n of A and ρ i and we write a 1 , a 2 , . . . , a n and p 1 , p 2 , . . . , p n for the associated eigenvalues. We furthermore assume that the ordering of the vectors in the basis, hereafter referred to as the computational basis, is such that a 1 ≤ a 2 ≤ · · · ≤ a n . This ordering is consistent with the prior ordering of the different eigenvalues of A in Section II A.

III. A QUANTUM SPEED LIMIT
In this section, we derive a quantum speed limit (QSL) for the time it takes to passivize an incoherent state using a Hamiltonian that satisfies the bounded bandwidth condition. To this end, let δ k be the number of eigenvalues that the kth component of ρ i does not have in common with the kth component of a passive state, and let δ be the sum of all the δ k s. We call δ the discrepancy of the initial state, and we define the QSL τ qsl as Proposition 6. The QSL τ qsl bounds the passivization time from below.
Proof. Let H be a time-optimal Hamiltonian. For each computational basis vector |k define |k(t) = e −itH |k and regard |k(t) as a curve on the unit sphere in H. The speed of |k(t) equals k|H 2 |k 1/2 and Proposition 2 says that the square of these speeds sum to ω 2 . Since the system evolves into a passive state in the time τ pas , at least δ of the basis vectors evolve, in this time, into eigenvectors of A with a different eigenvalue and, thus, into orthogonal states. The spherical distance between any pair of orthogonal states is π/2. Hence, This proves that τ pas ≥ τ qsl .
A natural question is when the passivization time is equal to τ qsl . Below we will see that such is the case if the initial state can be 'permuted' to a passive state with a permutation whose cycles have a length of at most 2. We will also see examples of systems for which the passivization time is greater than τ qsl .
FIG. 1. A graphical representation of (k1 k2 . . . k l ). The number of elements in the cycle is the length of the cycle. A trivial cycle has length 1 and a transposition has length 2.

A. Systems for which the passivization time equals the quantum speed limit
For some, not necessarily unique, permutation σ of the set {1, 2, . . . , n}, the state ρ σ = k p σ(k) |k k| is passive. We call such a permutation passivizing. Define the permutation operator associated with σ as The operator P σ is unitary and ρ σ = P σ ρ i P † σ . Any permutation of {1, 2, . . . , n} can be uniquely decomposed into disjoint cycles [28]. Each cycle is itself a permutation of a subset of {1, 2, . . . , n}. We denote the cycle which permutes the subset {k 1 , k 2 , . . . , k l } according to k 1 → k 2 → · · · → k l → k 1 by (k 1 k 2 . . . k l ); see Figure  1. The number of elements l is the length of the cycle. A cycle of length 1 will be called trivial, and a cycle of length 2 will be called a transposition. A permutation whose square leaves every element of {1, 2, . . . , n} invariant is called an involution. Being an involution is equivalent to having cycles of length at most 2.

Proposition 7.
If ρ i can be passivized by an involution, then the passivization time equals τ qsl .
Proof. Let σ be a passivizing involution. Reduce σ by replacing each transposition (k 1 k 2 ) of σ for which a k1 = a k2 or p k1 = p k2 holds by a pair of trivial cycles (k 1 )(k 2 ). The reduced σ is also passivizing.
Let c 1 , c 2 , . . . , c m be the transpositions of the reduced σ. Write c j = (k j 1 k j 2 ), with k j 1 < k j 2 , and re-index the transpositions so that every c j for which p k j 1 < p k j 2 holds has a lower index than all those c j s for which p k j satisfies the bounded bandwidth condition and implements the passivizing unitary −iP σ in the time π √ 2m/2ω.
We will show that δ ≥ 2m. The opposite inequality follows from Proposition 6.
For each j, let P j be the unitary operator which interchanges |k j 1 and |k j 2 and leaves all the other computational basis vectors invariant. Set ρ 0 = ρ i and inductively define ρ j = P j ρ j−1 P † j . The sequence of ρ j s starts at ρ i and ends at the passive state ρ σ . Moreover, The second term on the right is positive if The former situation is, however, excluded by the selected order of the transpositions. Otherwise, the final state is not passive. Hence, the sequence of expectation values E A (ρ j ) is monotonically decreasing. This, in turn, means that each p k j 1 belongs to the spectrum of different components of ρ i and ρ σ , and similarly for p k j 2 . We conclude that δ ≥ 2m.
In the proof of Proposition 7, we chose to implement −iP σ rather than P σ because the latter does not belong to the passivizing unitaries closest to the identity. Explicitly, the distance from 1 to P σ equals π √ m, which is √ 2 times greater than the distance from 1 to P(ρ i ). σ be the product of the so obtained transpositions times the trivial cycles whose elements are the indices of the unpaired eigenvalues. The permutation σ is an involution, and the passivization time thus equals τ qsl . The next example is interesting from a thermodynamic perspective. We shall return to this fact in Section VI.
Example 2. If ρ i is maximally active, the sequence of eigenvalues of ρ i is non-decreasing. The state with the reversed spectrum, ρ p = k p n−k+1 |k k|, is passive, and the discrepancy of ρ i equals δ = 2m where m is the greatest integer such that p m < p n−m+1 and a m < a n−m+1 . 3 Defining σ as is time-optimal and implements −iP σ in the time τ qsl . 3 We assume that neither A nor ρ i is proportional to 1 for then ρ i is already passive.

B. Assisted passivization
A passivization catalyst is an auxiliary quantum system used to reduce the passivization time of a system. The catalyst is allowed to transform with the system, but as the system develops into a passive state, the catalyst must return to its original, uncorrelated state. As we will see, allowing correlations between the two in the meantime can reduce the passivization time.
To derive a bandwidth bound of the composite system that allows for a fair comparison between the transformation time of a catalyzed and that of an uncatalyzed passivizing transformation, consider a system in a state ρ i which is coupled to, but uncorrelated with, a catalyst in a state ρ c . Assume that the system and the catalyst in the time τ evolve in parallel to ρ p ⊗ ρ c , with ρ p being passive, according to a von Neumann equation with Hamiltonian . Furthermore, assume that the bandwidth of H s (t) is bounded from above by ω 2 . Then τ is greater than or equal to the system's passivization time τ pas . And only if H s (t) is time-optimal and H c (t) is suitably adjusted, τ can be equal to τ pas . In that case, the bandwidth of H sc (t) is n c ω 2 + n tr(H c (t) 2 ), where n c and n is the dimension of the catalyst and the system, respectively. Here we have used that time-optimal Hamiltonians are traceless; see Section V A. We thus formulate the bounded bandwidth condition for assisted transformations as We define the assisted passivization time τ apas as the shortest time in which a system can be transformed into a passive state in a catalyzed process governed by a Hamiltonian satisfying (19). Moreover, for a system in a state with discrepancy δ we define the assisted QSL as The next two propositions say that the assisted passivization time is at least τ aqsl but not greater than τ pas / √ n c .

Proposition 8. The assisted passivization time is not greater than
Proof. Let H s be a time-optimal Hamiltonian for the system, transforming ρ i into ρ p in the time τ pas , and let |ψ be the pure state of an n c -dimensional catalyst. Define a Hamiltonian for the combined system and catalyst as The Hamiltonian H sc has bandwidth n c ω 2 and transforms ρ i ⊗ |ψ ψ| into ρ p ⊗ |ψ ψ| in the time τ pas / √ n c .

Proposition 9.
The assisted QSL τ aqsl lower bounds the assisted passivization time.
Proof. Let ρ i and ρ p be the initial and a passive state of the system, respectively, and let ρ c be the state of an n c -dimensional catalyst. Since ρ i and ρ p are incoherent relative A, the product states ρ i ⊗ ρ c and ρ p ⊗ ρ c are incoherent relative A ⊗ 1. Moreover, the kth components of ρ i ⊗ρ c and ρ p ⊗ρ c are ρ i;k ⊗ρ c and ρ p;k ⊗ρ c , respectively, where ρ i;k and ρ p;k are the kth components of ρ i and ρ p . Let δ k be the discrepancy between ρ i;k and ρ p;k and let δ c k be the discrepancy between ρ i;k ⊗ ρ c and ρ p;k ⊗ ρ c . Then δ c k ≥ δ k . To see this, let p i1 , p i2 , . . . , p i δ k be the eigenvalues of ρ i;k that are not present in the spectrum of ρ p;k . Each of these eigenvalues is either strictly greater than or strictly smaller than all of ρ p;k 's eigenvalues. Otherwise, ρ p would not be passive. Organize the differing eigenvalues of ρ i;k so that the first l ones are greater than, and the last δ k − l ones are smaller than, all the eigenvalues of ρ p;k . Furthermore, let q 1 be the greatest and These are δ k in number and, hence, δ c k ≥ δ k . Let H sc (t) be a Hamiltonian that satisfies (19) and which, among such Hamiltonians, transforms ρ i ⊗ ρ c into ρ p ⊗ ρ c in the shortest time. (Without loss of generality, we can assume that this time equals τ apas since ρ p is unspecified.) Using arguments identical to the ones used in the uncatalyzed case, one can show that tr(H sc (t) 2 ) = n c ω 2 for all t and that H sc (t) is time-independent.
Write δ c for the sum of all the δ c k s. Let |k l be the product of the kth vector in the computational basis for the system and the lth vector in an eigenbasis of ρ c . The trajectory formed when |k l is affected by H sc has the constant speed k l|H 2 sc |k l 1/2 . Furthermore, for at least δ c such vectors, the trajectory has a length greater than or equal to π/2. Consequently, The proposition follows from the inequality δ c ≥ δ.
By Propositions 8 and 9, τ apas = τ aqsl if τ pas = τ qsl . Such is the case, for example, if the initial state can be passivized by an involution.
Example 3. A system in a maximally active state can be passivized in the time τ qsl and be assisted passivized in the time τ qsl / √ n c using an n c -dimensional catalyst.

C. Collective passivization
Assume that A has a non-degenerate spectrum. Then there is but one passive state ρ p . In this section, we consider N copies of the system prepared in the product shorter than the single copy passivization time τ pas using a Hamiltonian that satisfies the bandwidth condition The right-hand side equals the bandwidth of an N -fold sum of local time-optimal Hamiltonians satisfying (1).
We will see that, as in the case of assisted passivization, allowing correlations between the systems can reduce the passivization time to a value smaller than τ pas . 4 Remark 2. Here we consider transformations of ρ ⊗N i into a specific final state, namely ρ ⊗N p , rather than "some passive state". The state ρ ⊗N p need not be passive for [22] and Example 7.
The computational basis determines a canonical basis in the N -fold tensor product of H. We write |k 1 k 2 . . . k N for |k 1 ⊗ |k 2 ⊗ · · · ⊗ |k N . Then where σ is any permutation that passivizes ρ i . The sums in Equations (23) and (24) are over all the sequences k 1 , k 2 , . . . , k N one can form from the integers 1, 2, . . . , n. Let δ N be the number of such sequences for which p k1 p k2 · · · p k N and p σ(k1) p σ(k2) · · · p σ(k N ) are different. Also, define the collective passivization time τ cpas as the minimum time it takes to transform ρ ⊗N i into ρ ⊗N p using a Hamiltonian that satisfies (22). Then The proof is similar to that of Proposition 6: A Hamiltonian that meets the condition (22) and transforms ρ ⊗N Let H be such a Hamiltonian. Then H transforms each product basis vector |k 1 k 2 . . . k N into an eigenvector of ρ ⊗N p . Furthermore, if p k1 p k2 · · · p k N and p σ(k1) p σ(k2) · · · p σ(k N ) are different, the length of the trajectory of this vector is at least π/2. Since the trajectory has the constant speed The expression on the right-hand side of (25) is the the collective QSL, which we denote by τ cqsl , for short. The collective passivization time equals τ cqsl if σ is an involution. Because if such is the case, the Hamiltonian where the sum is over all sequences k 1 , k 2 , . . . , k N for which p k1 p k2 · · · p k N and p σ(k1) p σ(k2) · · · p σ(k N ) are different, satisfies (22) 2. A plot of τpas/τcpas for an ensemble of N mixed qubits. The trend of the plot tells us that the collective passivization time decreases with N , but the plot also has a noticeable fluctuating appearance indicating a non-monotonic dependence on N ; if N is even, the collective passivization time is smaller for N qubits than for N + 1 qubits, but larger than for N + 2 qubits.
Next, we will calculate the fraction between the single system passivization time and the collective N -fold passivization time for systems prepared in maximally active qubits or qutrits. The fraction can be considered as a measure of the advantage of a collective passivization [24]. For mixed qubits, the fraction depends explicitly on the parity of N , and to simplify the notation we will make use of the parity function Example 4. Suppose that ρ i is a maximally active qubit state. If ρ i is pure, then δ N = 2 and If ρ i mixed, then δ N = 2 N − ℘(N ) N N/2 and The formula for δ N follows immediately from the observation that p k if, and only if, 2k = N . It is apparent that allowing correlations between the qubits during the evolution reduces the passivization time. In Figure 2, we have plotted τ pas /τ cpas against N for a mixed ρ i . The trend says that the more qubits are involved, the smaller the collective passivization time. However, as the plot also indicates, the decrease in collective passivization time is not monotonic in N . Adding a qubit to an ensemble with an even number of qubits increases the passivization time, while adding a qubit to an ensemble with an odd number of qubits reduces the passivization time. Another interesting observation is that the asymptotic behavior of τ pas /τ cpas is very different for a pure and a mixed ρ i .
Example 5. Suppose that ρ i is a maximally active qutrit state. If p 1 = 0, then δ N = 2(2 N − 1) and If ρ i has full rank, then The upper limit N/2 is the greatest integer less than or equal to N/2, and N k,k is the trinomial coefficient N !/k!k!(N − 2k)!. Figure 3 shows a plot of τ pas /τ cpas against N for a full rank ρ i . In this case, the collective passivization time is monotonic in N .
For non-maximally active qutrit states, δ N will depend on the spectrum. This is also the case for higher dimensional systems, even for maximally active initial states. We finish this section with two examples concerning the passivity of ρ ⊗N p . To distinguish the multipartite system's passivization time from that of the single-part system we will call the former the global passivization time. Example 7. Suppose that ρ i is a maximally active qutrit state whose spectrum is such that Furthermore, suppose that the spectrum of A is such that a 2 1 < a 1 a 2 < a 2 2 < a 1 a 3 < a 2 a 3 < a 2 3 .
Then ρ ⊗2 i is a maximally active state for both 2 · A and is not a passive state for either 2 · A or A ⊗2 . The discrepancy of ρ ⊗2 i is 8 and, hence, the global passivization time is π/ω √ 3. Also, δ 2 = 6 and τ cqsl = π/2ω. The collective passivization time is thus smaller than the global passivization time.

IV. THE NON-DEGENERATED CASE
Suppose that A and ρ i have non-degenerate eigenvalue spectra. Let ρ p be the unique passive state and let σ be the unique permutation such that ρ p = k p σ(k) |k k|. In this case, the lengths of the cycles of σ determine the passivization time.
The proposition covers, for example, the case when the observable A is non-degenerate, the initial state is prepared by measuring A, and the outcomes of the measurement are obtained with different frequencies.
Thus τ pas > τ qsl . A time-optimal Hamiltonian that transforms ρ i to ρ p is Proof of Proposition 10. Let c 1 , c 2 , . . . , c m be the cycles of σ and let H j be the linear span of those vectors in the computational basis whose labels are permuted by c j . The H j s are mutually orthogonal and span H. Also, the H j s are invariant for P σ and the isotropy groups of A and ρ i . Let P cj be the restriction of P σ to H j . Every unitary U that commutes with A and every unitary V that commutes with ρ i decomposes as U = ⊕ j U j and V = ⊕ j V j , respectively, with U j and V j being operators of H j . Moreover, According to (7) and Proposition 5, we can determine the passivization time by minimizing the terms on the right-hand side of (38). The non-degeneracy of A and ρ i implies that U j and V j are diagonal relative to the computational basis vectors that span H j . Write c j = (k 1 , k 2 , . . . , k lj ) and let e iαr be the eigenvalue of U j , and e iβr be the eigenvalue of V j , associated with |k r . The minimal polynomial of U j P cj V j is x lj − e iθj where θ j = r (α r + β r ) mod 2π. Hence the eigenvalues of U j P cj V j are λ r = e i(θj +2πr)/lj , where r runs from 0 to l j − 1. It follows that min Uj ,Vj The minimum is attained for θ j = π(1 − l j ), which also meets the requirement that all the phases (θ j + 2πr)/l j belong to the principal branch. We conclude that This proves Proposition 10.
The proof shows that U = ⊕ j e iπ(1−lj )/lj P cj is among the passivizing unitary operators closest to the identity. A time-optimal Hamiltonian that implements U is The operator 1 j is the identity operator on H j and ℘ is the parity function defined in (28).

V. ON DEGENERATED CASES
In this section, we describe some general properties of time-optimal transformations. We also discuss circumstances under which these properties are sufficient to determine a degenerate system's passivization time.

A. Incompatibility and parallelism of time-optimal Hamiltonians
In all the cases considered so far, the specified timeoptimal Hamiltonians are traceless, and the passivizing unitaries lying closest to the identity are special unitary, that is, have determinant equal to 1. As we will see, these observations are consequences of time-optimal Hamiltonians generating shortest curves between 1 and the manifold of passivizing unitaries P(ρ i ).
We say that a Hamiltonian is completely incompatible with A if ΠHΠ = 0 for all the eigenspace projectors Π of A. Moreover, using terminology from the theory of fiber bundles [31][32][33], we say that H is parallel transporting if ΠHΠ = 0 for every eigenspace projector Π of ρ i . Both of these properties separately imply that H is traceless and, hence, the time-evolution operator associated with H is special unitary.

Proposition 11. Time-optimal Hamiltonians are parallel transporting and completely incompatible with A.
Proof. According to Proposition 3, time-optimal Hamiltonians generate shortest curves between 1 and P(ρ i ).
Such a shortest curve has to meet P(ρ i ) perpendicularly [26]. Let H be a time-optimal Hamiltonian and let U be the passivizing unitary generated in the time τ pas . The velocity vector at U of the time-evolution operator of H is −iHU . Let B 1 and B 2 be any Hermitian operators commuting with A and ρ i , respectively. By Proposition 5, −iB 1 U and −iU B 2 are tangent vectors of P(ρ i ) at U . These vectors are perpendicular to −iHU and, hence, Since B 1 and B 2 are arbitrary, the former identity implies that H is completely incompatible with A, and the latter implies that H is parallel transporting.

B. Upper bounds on the passivization time from passivizing permutations
When A or ρ i has a degenerate spectrum, the expression on the right-hand side of (36) need not be equal to the passivization time. However, the expression always is an upper bound for the passivization time. To see this, let σ = c 1 c 2 · · · c m be a passivizing permutation. For any sequence of real numbers θ 1 , θ 2 , . . . , θ m , the operator ⊕ j e iθj P cj is a passivizing unitary. Therefore, by (7), Log(e iθj P cj ) 2 .
Let l j be the length of c j . The eigenvalues of e iθj P cj are the l j th roots of unity multiplied by e iθj . Consequently, It follows that Example 9. Consider an 8-dimensional system for which the spectra of A and ρ i satisfy In this case, there are four passivizing permutations, (52) If we insert the lengths of the cycles of the first permutation into the right-hand side of (46), we find that τ pas ≤ π √ 41/ω √ 18. And if we insert the lengths of the second or the third permutation's cycles, we find that τ pas ≤ π √ 17/3ω. Finally, if we insert the lengths of the fourth permutation's cycles, we find that τ pas ≤ π √ 6/2ω.
The second and the third permutation can be obtained from the first by a division of the first and the last cycle, respectively. The division leads to a reduction of the upper bound in (46). Similarly, the fourth permutation can be obtained by dividing both the first and the last cycle of the first permutation, which leads to an even greater improvement of the upper bound. (In fact, the upper bound π √ 6/2ω equals τ qsl and, hence, τ pas .) A division of the first and last cycle of the first permutation is possible because of the identities p 1 = p 2 and a 6 = a 7 .
Example 9 shows that degeneracies in the spectrum of A or ρ i sometimes make it possible to divide a cycle of a passivizing permutation into two cycles without changing the fact that the permutation is passivizing. Such a division always leads to a lowering of the upper bound in (46). 5 To see this, suppose that c = (k 1 k 2 . . . k l ) is a 5 The authors do not know whether from an arbitrary passivizing permutation, one can always reach the passivization time by means of cycle division.
cycle of a passivizing permutation σ and suppose that for some i < j we have that a ki = a kj or p ki = p kj . Then c can be replaced by (k 1 . . . k i k j+1 . . . k l )(k i+1 . . . k j ), see Figure 4, and σ be redefined accordingly. The resulting permutation is also passivizing. If the lengths of the two new cycles are l j and l j , respectively, then The cycle division thus lowers the upper bound in (46).

C. Invariant subspaces of passivizing unitaries
If, as in Example 9, after repeated cycle division we end up with a passivizing involution, then we know from Proposition 7 that τ pas = τ qsl . But if in Example 9 we replace the identity p 1 = p 2 in (48) by p 1 > p 2 , then there are only two passivizing permutations, namely those in (49) and (51), neither of which are involutions. The inequality in (46) guarantees that in this case, τ pas is not greater than π √ 17/3ω. However, none of the previous propositions certifies that the passivization time equals π √ 17/3ω. In this section and the next, we will develop a method that can be used to prove that this actually is the case. The strategy is to break down the problem of determining τ pas into a number of lower-dimensional problems which can be solved using results from the theory of generalized flag manifolds. Let σ be a passivizing permutation. Consider a decomposition of σ into sub-permutations, σ = σ 1 σ 2 · · · σ m , where each sub-permutation σ j is a cycle or a product of cycles of σ. Define H j as the linear span of the computational basis vectors whose labels are permuted by σ j , and write P σj for the restriction of P σ to H j .

Proposition 12. If each eigenspace of A and each eigenspace of ρ i is contained in an
The Hamiltonian H generates a shortest geodesic from 1 to U and, hence, from 1 to P(ρ i ). Furthermore, H satisfies the bounded bandwidth condition: Thus, H is a time-optimal Hamiltonian. By construction, H preserves the spaces H j .
In the remainder of this section, we assume that the isotropy groups of A and ρ i preserve the spaces H j . We define U(H j ) A as the group of unitary operators on H j that preserve those eigenspaces of A which are contained in H j . Similarly, we define U(H j ) ρi as the group of unitary operators on H j that preserve those eigenspaces of ρ i which are contained in H j . Then where the minima are taken over all the U j s and V j s in U(H j ) A and U(H j ) ρi , respectively. The summands on the right-hand side of (56) are generally difficult to calculate. But the problem simplifies somewhat if one of the isotropy groups contains the other. Because then we only need to minimize over the larger of the two: The right-hand sides of (57) and (58) are geodesic distances in certain homogeneous spaces called generalized flag manifolds. Well-known examples of such are projective Hilbert spaces and Grassmann manifolds. In the next section, we describe how to calculate the distances on the right of (57) under certain circumstances. Notice that the isotropy group of ρ i is contained in that of A if each eigenspace of ρ i is included in a eigenspace of A. This is the case, for example, if ρ i has a non-degenerate spectrum. The case when U(H j ) A is a subgroup of U(H j ) ρi can be treated similarly. Appendix G contains a brief review on generalized flag manifolds.

D. Geodesic distance in generalized flag manifolds
Let 1 j be the identity operator on H j . Regard U(H j ) as a manifold on which U(H j ) A acts from the right by operator pre-composition. If we equip U(H j ) with the biinvariant Riemannian metric that agrees with the Hilbert-Schmidt inner product on the Lie algebra of U(H j ), then the right-hand side of (57) is the geodesic distance between the cosets of 1 j and P σj in the geometry determined by the projected metric on the quotient manifold The quotient manifold U(H j )/U(H j ) A is an example of a generalized flag manifold; see Appendix G. Next, we show how to calculate (59) under special circumstances.

The Grassmann and Fubini-Study distances
That the isotropy group of A preserves H j is equivalent to H j being a direct sum of eigenspaces of A. If H j is a sum of two eigenspaces of A, then U(H j )/U(H j ) A is a Grassmann manifold, and (59) is the Grassmann distance between [1 j ] and [P σj ]; see [34] and Appendix G.
Suppose that H j = A k ⊕ A k . Let n k be the dimension of A k and let Π k be the orthogonal projection of H j onto A k . Furthermore, let s 1 , s 2 , . . . , s n k be the singular values of Π † k P σj Π k . Then, by (G1) in Appendix G, Since P σj is a permutation operator, each singular value is either 0 or 1. The number of 0s equals the number of computational basis vectors in A k that P σj maps into A k . Let δ j be twice this number. Formula (60) yields If one of the eigenspaces of A is 1-dimensional, then the Grassmann manifold U(H j )/U(H j ) A is a projective Hilbert space. In this case, the Grassmann distance function in (60) goes by the name "the Fubini-Study distance." If |k is the computational basis vector that spans the 1-dimensional eigenspace of A, then dist [1 j ], [P σj ] = √ 2 arccos k|σ(k) .
Since k|σ(k) = 0 or k|σ(k) = 1, depending on whether σ leaves k invariant or not, Remark 3. The reader might wonder if the assumption that the isotropy group of A contains the isotropy group of ρ i is essential for the validity of (61). After all, in Example 1, we determined the passivization time for a system with a bivalent A without making assumptions about the structure of U(H) ρi , and the resemblance between (61) and the quantum speed limit (12) is striking. The problem is that if U(H j ) A does not contain U(H j ) ρi , then (61) does not produce a reliable contribution to the distance in (56), in the sense that the right-hand side of (56) is independent of the passivizing permutation. Consider, for example, a qubit system prepared in the maximally mixed state and with a non-degenerated A.

VI. ON THE POWER OF ENERGY EXTRACTION FROM QUANTUM BATTERIES
In this final section, we use results from previous sections to derive bounds on the power with which energy can be reversibly extracted from a quantum battery. We follow [21][22][23][24] and define a quantum battery as a closed n-dimensional quantum system that can release energy through a controllable process causing the battery state to change according to a von Neumann equation of the Here, H is the battery's internal Hamiltonian and V (t) is a time-dependent potential. We limit our considerations to cyclic processes and, thus, assume that the potential vanishes outside a finite time interval [0, τ ], the final time τ being the duration of the process. Also, to connect with previous sections we assume that the initial state of the battery is incoherent relative to H and that the available resources are limited in such a way that the bandwidth of the potential cannot exceed a given value: The internal Hamiltonian H here plays the role of the observable A, and minimality of E H characterizes the passivity of states. We prefer to denote the eigenvalues of H by k rather than a k , but apart from that, we follow the standard set in Section II C.

A. Ergotropy and the power of complete discharge processes
The maximal amount of (average) energy that can be cyclically extracted from a battery state ρ i is called the ergotropy of the battery [21]. The ergotropy equals the difference in energy of ρ i and that of a passive state: Since all the passive states have the same energy content, the ergotropy only depends on the initial state and the internal Hamiltonian. In terms of the internal energies and the eigenvalues of ρ i , the ergotropy reads Here, σ is any passivizing permutation.
No energy can be extracted through a cyclic unitary process from a battery in a passive state [21]. Therefore, we call an energy extraction process that leaves the battery in a passive state a complete discharging of the battery. The (average) power of a complete discharging of duration τ is W (ρ i )/τ . We define τ pas as the passivization time of the battery determined by the bandwidth condition (1) with the same right-hand side as in (90). Note that this definition of the passivization time does not take any characteristics of the internal Hamiltonian H into account, even though H affects the battery's dynamics. If the bandwidth of the internal Hamiltonian greatly exceeds ω 2 , then no Hamiltonian of the form H +V (t) is even close to being time-optimal. Nevertheless, according to the next proposition, the duration of a complete discharging is at least τ pas . The power of such a process is, thus, bounded from above by W (ρ i )/τ pas . Proposition 13. The duration of a complete discharge process is greater than the passivization time.
Proof. Let V (t) be a potential that satisfies (90) and completely discharges the battery in the time τ . We regard the potential as a perturbation and go over to the interaction picture. In the interaction picture, the state of the battery evolves according toρ Example 12. Consider a fully charged battery. That is, consider a battery in a maximally active state ρ i . Let P be the power of a complete discharge process with a potential whose bandwidth is bounded by ω 2 . Then where m is the greatest integer for which p m < p n−m+1 and m < n−m+1 hold.
Example 13. Suppose that the internal Hamiltonian and the initial state of a battery are non-degenerate. Let P be the power of a complete discharging of the battery with a potential whose bandwidth is bounded by ω 2 . Then Here, l 1 , l 2 , . . . , l m are the lengths of the cycles of the unique permutation σ that passivizes the initial state.
Since we require that the discharge processes are cyclic, their duration can never be as small as the passivization time. But there are discharge processes whose duration comes arbitrarily close to the passivization time.

Proposition 14. The passivization time is a tight bound on the duration of complete discharge processes.
Proof. Let > 0 be arbitrary and let u(t) be a smooth function which equals 0 for t ≤ 0 and τ pas for t ≥ τ pas + and whose derivative u (t) takes values between 0 and 1 for all t. Let V I be a time-optimal Hamiltonian and define a potential as V (t) = u (t)e −itH V I e itH . The potential vanishes outside [0, τ pas + ] and satisfies the bounded bandwidth condition (90): tr(V (t) 2 ) = u (t) 2 ω 2 ≤ ω 2 . Furthermore, the solution ρ(t) to the von Neumann equation with Hamiltonian H + V (t) which extends from ρ i is passive at In an assisted discharging, a catalyst is used in the discharge process. The catalyst is ultimately unchanged but may interact and become correlated with the battery during the discharge process. According to Proposition 9, the power P with which a fully charged battery can be discharged with the help of an n c -dimensional catalyst is bounded from above as Here, δ is the discrepancy of the fully charged battery.
We define a battery ensemble to be a system built up of several identical batteries. In a collective complete discharge process, the states of all the batteries are collectively transformed into uncorrelated passive states.
Let P be the supremum of all powers of collective complete discharge processes directed by global potentials whose bandwidth is bounded from above by ω 2 N n N −1 , and let P be the supremum of all powers of parallel complete discharge processes directed by local potentials fulfilling (90). The fraction P /P is a measure of the advantage of using a collective complete discharge process over a parallel complete discharge process; see [24]. According to Proposition 13 and (25), P ≤ W (ρ ⊗N i )/τ cqsl , and by Proposition 14, P = W (ρ ⊗N i )/τ pas . The advantage is thus upper bounded by the fraction of the single battery passivization time and the collective quantum speed limit: Example 14. If the batteries of the ensemble are prepared in mixed and maximally active qubit states, then, by Example 4, the power P with which energy can be extracted in a complete discharge process governed by a potential satisfying (22) is bounded as Furthermore, the advantage of a collective discharging is If the batteries are identically prepared maximally active full rank qutrits, then, by Example 5, the power with which energy can be extracted in a complete discharge process is upper bounded according to In this case, the advantage of a collective discharging is We saw in Example 7 that the product of two passive qutrit states need not be globally passive. In the next example, we compare the power of an optimal collective passivization process with the power of an optimal global passivization process for such a qutrit battery.
Example 15. Suppose that H is non-degenerate with energies satisfying and suppose that ρ i is a maximally active qutrit state with a spectrum that satisfies Each battery has a unique passive state ρ p . The product ρ ⊗2 p , however, is not a passive state; see Example 7. The energy one extracts from ρ ⊗2 i in a collective passivization process is less than the energy one extracts in a global passivization process. To be precise, the difference in energy content of ρ ⊗2 p and that of a globally passive state is ( 1 + 3 −2 2 )(p 1 p 3 −p 2 2 ). There are collective passivization processes with a duration arbitrarily close to π/2ω and global passivization processes with a duration arbitrarily close to (but not less than) π/ω √ 3. Thus, the difference in 'optimal power' of a collective passivization process and a global passivization process is Due to (101) and (102), the expression in (103) is positive. Hence, an optimal collective passivization process is more 'powerful' than a global passivization process.

D. Energy and power fluctuations of discharge processes
We finish this paper with some observations concerning energy fluctuations of discharge processes. We consider a battery prepared in a state of definite energy through a stochastic preparation procedure. Such a statistical state can be modeled by a density operator that is incoherent relative to the internal Hamiltonian. The average energy extracted from such a battery in a complete discharge process, which leaves the battery in a passive statistical state, is equal to the ergotropy.
To derive an expression for the variation in transferred energy in a complete discharge process, let ρ i be the prepared statistical state, let H be the internal Hamiltonian, and let E 1 , E 2 , . . . , E r be the different, possibly degenerate, eigenvalues of H. Also, let Π k be the orthogonal projection onto the eigenspace corresponding to E k . The variation in transferred energy is where p(l, k) is the probability that a battery starts out in a state with energy E k and ends up in a state with energy E l in a complete discharge process. If U is the unitary implemented by the discharge process, then If we insert this expression into (104) and simplify, we obtain the following expression for the variation Here, ∆ 2 H(ρ) is an abbreviation for E H 2 (ρ) − E 2 H (ρ) and ρ p = U ρ i U † is the final statistical state of the battery. Since U does not commute with H, unless ρ i is passive, the last term is process dependent (while the other terms are not). Thus, the variation in the amount of energy extracted from a battery may differ for different complete discharge processes. We nuance this statement with a proposition and an example: Proposition 15. If the unitaries in the isotropy group of the initial state commute with H, then all complete discharge processes have the same variation in the transferred energy.
Proof. Let P be any passivizing unitary and let σ be any passivizing permutation. By Proposition (5), P can be decomposed as P = U P σ V , where U commutes with H and V commutes with ρ i . By assumption, V also commutes with H. A straightforward calculation yields tr P † HP Hρ i = tr P † σ HP σ Hρ i .
This shows that the value of the final term in (106) is independent of P and, hence, that the variation in transferred energy is process independent.
According to Proposition 15, all complete discharge processes of a battery in a non-degenerate statistical state have the same variation in the energy transfer. The next example shows that this need not be the case if the state has a degenerate spectrum.
Example 16. Consider a qutrit battery whose internal Hamiltonian is non-degenerate with spectrum 1 = 0 < 2 < 3 , and whose initial statistical state is degenerate with spectrum p 1 = p 2 < p 3 . Relative to the computational basis, a general passivizing unitary has the matrix where α, β, γ, and θ are arbitrary and 0 ≤ a ≤ 1. The variation in transferred energy is independent of the phases: Also, ∆ 2 W (ρ i ) is maximal for a = 0 and minimal for a = 1. We find time-optimal processes among those which implement a U for which a = 0. But none of the processes which implement a U for which a = 1 is time-optimal. Example 16 shows that there may be a trade-off between being time-optimal and having small fluctuations in transferred energy for complete discharge processes. Further investigation is required.

VII. SUMMARY AND OUTLOOK
In this paper we have considered, and to some extent answered, the following question: Suppose that a finitedimensional quantum system is prepared in a state that is incoherent relative to an observable. In how short a time can the state be transformed into a passive state for the observable provided that the Hamiltonian responsible for the transformation has bounded bandwidth?
We began by deriving a general QSL for the transformation time, which we also expanded to a lower time-bound for collective passivization processes. We then showed that for some systems, such as systems prepared in maximally active states, the QSL is equal to the passivization time, that is, the least possible time in which the system can be transformed into a passive state under the specified conditions. But we also showed that for some systems, the QSL is not tight. We calculated the passivization time explicitly for systems such that the observable and the initial state are non-degenerate. Then we developed a method to determine the passivization time for degenerate systems. The method presupposes that the eigenspaces of the observable and the state are in a particular relative constellation, which means that the method does not apply to all conceivable systems that match the description above -other approaches are required.
The problem discussed in this paper is an example of a brachistochrone problem [8-10, 12, 15, 17]. For constrained closed quantum systems, such problems are typically reformulated as one or more time-local relations for the Hamiltonian using the calculus of variations [8,9,15,17]. However, such relations are generally of little help in determining the shortest possible transformation time. For example, in the case dealt with here, calculus of variations would generate the results in Propositions 4 and 11. From there, there is still a long way to go.
In the last section, we applied the results from previous sections to quantum batteries. Specifically, we derived upper bounds on the power with which energy can be extracted from a quantum battery through a cyclic unitary process. Here, we only considered complete discharge processes, that is, processes that leave the battery in a passive state relative to the battery's internal Hamiltonian. From such a state, no more energy can be extracted through a cyclic unitary process.
Recently, the interest in quantum batteries has grown considerably, not least because of their predicted practical significance [36]. Here we have assumed that the battery is initially in an incoherent state relative to the internal Hamiltonian. The next step is to extend the results to quantum batteries in coherent states. The recent paper [37] is possibly the first step in such a direction.

VIII. ACKNOWLEDGMENTS
The authors thank Supriya Krishnamurthy for the many inspiring discussions and Ingemar Bengtsson for suggesting improvements to the text. eigenvalues of U and select the phases of these in the principal branch (−π, π]. Also, let e −itH be any geodesic from 1 to U such that U = e −iH . Write 1 , 2 , . . . , n for the eigenvalues of H. Possibly after a re-indexing, e −i j = e iθj and, hence, j = −θ j mod 2π. This implies that θ 2 j ≤ 2 j . The length of e −itH equals H and the length of e −it Log U equals Log U . The inequality shows that e −it Log U is not longer than e −itH .

Appendix F: Proof of Proposition 5
That U P V belongs to P(ρ i ) for each U in U(H) A and each V in U(H) ρi follows from U(H) A leaving the set of passive states invariant. Conversely, that every passivizing unitary has the form U P V for some U in U(H) A and V in U(H) ρi follows from U(H) A acting transitively on the set of passive states. Indeed, if W is any passivizing unitary, then W ρ i W † = U P ρ i P † U † for some U in U(H) A due to the transitivity of the action of U(H) A . But then That P(ρ i ) is a submanifold of U(H) follows from P(ρ i ) being the orbit of P under the action (U, V ) · W = U W V † of U(H) A × U(H) ρi on U(H): Since the product of the isotropy groups is compact, the action is proper and, hence, the orbits are submanifolds of U(H).