Identifying quantum change points for Hamiltonians

The identification of environmental changes is crucial in many fields. The present research is aimed at investigating the optimal performance for detecting change points in a quantum system when its Hamiltonian suddenly changes at a specific time. Assume that the Hamiltonians before and after the change are known and that the prior probability of each prospective change point is identical. These Hamiltonians can be time-dependent. The problem considered in this study is an extension of the problem of discriminating multiple quantum processes that consist of sequences of quantum channels. Although it is often extremely difficult to find an analytical solution to such a problem, we demonstrate that the maximum success probability for the Hamiltonian change point problem can be determined analytically and has a simple form.

The detection of transition points is a critical issue that emerges in many fields.In various physical situations, the relevant environment may undergo a sudden change, and determining the precise moment of this transformation can yield valuable insights.This challenge, known as the change point problem, has attracted considerable statistical research interest [1][2][3][4].
In this study, we address the difficulty of identifying the precise instant when the time evolution of the quantum system experiences sudden changes.Figure 1 shows the schematic of this problem.Suppose a quantum system with Hamiltonian H 0 (t) undergoes a sudden change to the Hamiltonian H 1 (t) at time t ⋆ , such as when a magnetic field is suddenly applied at time t ⋆ , or when the frequency of the applied magnetic field undergoes a sudden change.These Hamiltonians may be time-dependent.The exact values of H 0 (t) and H 1 (t) are known but the point of the transition occurrence, i.e., t ⋆ , is unknown; therefore, it is crucial to determine t ⋆ as accurately as possible.Let us assume that there exist a finite number of transition points and that each candidate has an equal chance of becoming a change point.In this situation, we may input a particle with a known initial state into the system and perform a quantum measurement on the output from the system.We can perform any operation allowed by quantum mechanics, such as modifying the Hamiltonian of the system by adding an external magnetic field or replacing a particle in the system with another one at an arbitrary time.
The upper and lower bounds for the maximum success probability in the change point problem for quantum states were established in Ref. [5].In this paper, we discuss the challenge of detecting transitions between Hamiltonians, requiring the identification of quantum processes as a sequence of unitary channels that represent discrete time evolution.It is significantly more difficult to distinguish quantum processes than quantum states because it necessitates optimizing not only output measurements but also input states and channels used during the process.In the context of Ref. [5], the problem is limited to the case of pure states, in which assuming them to be qubit states does not lose generality.Nevertheless, the quantum system under consideration can have dimensions beyond two.To obtain the optimal performance, it is necessary to consider various types of distinguishing strategies, including those that combine entangled input states with ancillary systems and adaptive strategies.
The problem of distinguishing between quantum (memoryless) channels has received significant attention [6][7][8][9][10][11][12][13][14][15][16].Analytical solutions have already been identified for distinguishing between two simple quantum processes or processes with significant symmetry, such as those covariant with respect to unitary operators.However, our circumstance entails a timedependent Hamiltonian, which cannot be categorized as a quantum channel discrimination issue.Moreover, to achieve our objectives, it is necessary to differentiate multiple processes that do not exhibit high symmetry.Thus, obtaining an analytical solution may be difficult.In recent years, researchers have investigated the problem of distinguishing between distinct forms of time-dependent, generalized quantum processes, also known as quantum memory channels or quantum strategies [17][18][19][20][21][22][23].The formulation of the optimal performance as a semidefinite programming problem is known [20], which often proves helpful in obtaining numerical solutions.However, as the dimension of the system and the number of candidates increase, the computational complexity increases exponentially, limiting the feasibility of obtaining a solution to small-scale problems.Also, the conclusions of Ref. [20] cannot be explicitly applied when the candidate Hamiltonians vary continuously.
Surprisingly, in the aforementioned Hamiltonian change point problem, the optimal performance can be obtained ana-FIG.1. Problem of discriminating change points of Hamiltonians.The Hamiltonian of a quantum system suddenly changes from H 0 (t) to H 1 (t) at some time t = t ⋆ .Assuming that the candidate change points are given, we wish to identify t ⋆ as accurately as possible by optimizing the state input to the system, the measurement for the output, and so on.
lytically and expressed in a simple form for an arbitrary number of candidate change points.We should emphasize that even in the change point problem for quantum states, which intuitively seems to be easier to solve, an analytical solution is only found in the limiting case where the number of candidate change points is infinite [5].Starting with a simplified case of determining the sudden change from a unitary channel to the next, we examined the transition between channels.We derive an optimal analytical solution and demonstrate that adaptive strategies and ancillary systems are not required for optimal discrimination.This is in contrast to the discrimination of more than two channels, which generally requires the use of adaptive strategies in conjunction with ancillary systems [8,24].Then, this result is applied to analytically obtain an optimal solution to the Hamiltonian change point problem.
Identification of change points for unitary channels -We first formalize the task of recognizing transition points within unitary channels.We assume a unitary channel, where the first n uses correspond to U 0 , and the remaining uses correspond to U 1 .The channels U 0 and U 1 are known and n can be any integer between 0 to N. The objective is to precisely determine the value of n.Let U n<k be U 1 if n < k, and U 0 otherwise.Let E n be the process consisting of a sequence of N channels (U n<1 , . . ., U n<N ).This problem can be formulated as differentiating between the N+1 processes E 0 , . . ., E N .For example, in the case of N = 2, there is a need to distinguish between the three possible sequences: , and E 2 = (U 0 , U 0 ).Let V k and W k , with equal dimensions, denote the input and output systems, respectively, for the channel U n<k .
The most general discrimination strategy, presented in Fig. 2(a), involves ancillary systems V ′ 1 , . . ., V ′ N .We begin by preparing a bipartite system with initial conditions of V 1 ⊗ V ′ 1 .The first segment V 1 is sent through the channel U n<1 , followed by a channel σ 2 .Subsequently, V 2 is sent through the channel U n<2 , followed by a channel σ 3 , until N steps have been completed.The system W N ⊗ V ′ N is then subjected to a quantum measurement, Π {Π m } N m=0 .A collection (ρ, σ 2 , . . ., σ N , Π) can be used to define any quantum discrimination strategy allowed by quantum mechanics, including an entanglement-assisted and/or adaptive one.This collection of objects is known as a quantum tester [18].We want to find a discrimination strategy that maximizes the success probability.The problem of obtaining the maximum success probability, denoted by P (N) , is an optimization problem over quantum testers, which is formulated as a semidefinite programming problem [20].We assume P (N) < 1, which means that U 0 and U 1 are not perfectly distinguishable with a single evaluation.
Figure 2(b) shows the most general nonadaptive protocol, which can be regarded as a special case of the protocol shown in Fig. 2(a).An initial state, ρ, is prepared for the multipartite system the protocol, where V ′ is an ancillary system.Subsystems V 1 , . . ., V N are then exposed to their respective channels U n<1 , . . ., U n<N and the system . Any nonadaptive discrimination strategy can be described using a collection (ρ, Π).The determination of the optimal performance by using only a nonadaptive strategy is simple; however, the resulting performance could be inferior to those obtained using adaptive strategies.
Considering the optimal performance of nonadaptive strategies, let Λ n represent the unitary channel composed of N unitary channels U n<N , . . ., U n<1 connected in parallel, i.e., This problem can be expressed as the following optimization problem: where the maximization is taken over all possible input states ρ of the system V N ⊗• • •⊗V 1 ⊗V ′ and over all possible measurements Π of the system W N ⊗ • • • ⊗ W 1 ⊗ V ′ .We denote the optimal value of this problem, i.e., the maximum success probability, by P (N)  na .P (N) na ≤ P (N) clearly holds.For each b ∈ {0, 1}, the channel U b is associated with a unitary matrix, U b , such that U b (ρ) = U b ρU † b .We first consider the simplest case N = 1; then, the problem is reduced to distinguishing two unitary channels U 0 and U 1 in a single trial, and thus P (1) = P (1)  na holds.P (1)  na was obtained in Refs.[25,26]; we here briefly review their results.The maximum success probability for distinguishing two output states U 0 |ψ⟩ and U 1 |ψ⟩ from a pure input state |ψ⟩ is given by The optimal input state |ψ⟩ minimizes the absolute value of the inner product of U 0 |ψ⟩ and U 1 |ψ⟩.Let Γ be the polygon in the complex plane whose vertices are the eigenvalues of the unitary matrix U † 0 U 1 ; then, the distance between the polygon Γ and the origin is equal to the minimum value of | ⟨ψ|U † 0 U 1 |ψ⟩ | [27].If λ 0 and λ 1 are the eigenvalues representing the points at both ends of the polygon Γ closest to the origin, and |λ 0 ⟩ and |λ 1 ⟩ are the corresponding normalized eigenvectors, then | ⟨ψ|U † 0 U 1 |ψ⟩ | attains its minimum value 2. Thus, from Eq. ( 2), we obtain We next consider the case N = 2, where the task is to discriminate between E 0 = (U 1 , U 1 ), E 1 = (U 0 , U 1 ), and E 2 = (U 0 , U 0 ).Let us concentrate on discrimination strategies without any ancillary system.When V 2 ⊗V 1 is prepared in an initial state |ψ⟩, the output states of the three processes are be selected, in which it seems likely that a high success probability would be obtained.Furthermore, we find that when |ψ⟩ is an entangled state represented in the following form, A higher success probability may be obtained using such an entangled state.Also, while the problem with N = 1 can be just reduced to minimizing the absolute value of the inner product of U 0 |ψ⟩ and U 1 |ψ⟩, the problem with N = 2 becomes more complicated.In fact, there is no guarantee that the smaller the values Moreover, the use of an ancillary system may increase the success probability.It would also not be surprising if P (2) were strictly larger than P (2)  na .Considering the above discussion, obtaining an analytical expression of P (N) for N ≥ 2 is challenging.However, we discovered that a nonadaptive strategy can achieve optimal discrimination for any N.The following theorem provides a simple expression for the exact value of P (N) as a function of N and γ.
Theorem 1 In the problem of change point discrimination for unitary channels, we have Proof We present a summary of the proof; for more details, please refer to Appendix C. Consider a nonadaptive discrimination strategy in which pure state |ψ⟩ of the system Assume that the measurement in an orthonormal system {|π m ⟩} N m=0 identifies the pure output state from the channel Λ n , which is represented by [U ⊗(N−n) 1 ⊗ U ⊗n 0 ] |ψ⟩; then, the conditional probability, denoted by p m|n , that the measurement result is m, given that the change point is n is represented as Let us choose |ψ⟩ = with has an even number of elements equal to −1, and 0 otherwise.Note that in the case of N = 2, such |ψ⟩ is in the form of Eq. ( 4).It can be seen that there exists an orthonormal system {|π m ⟩} N m=0 satisfying where is the probability mass function of the discrete Laplace distribution, given by This nonadaptive strategy provides the success probability of which is clearly not greater than P (N) na .In addition, we can demonstrate that the optimal value, denoted by D, of the Lagrange dual of the change point problem is upper bounded by q.As the weak duality inequality P (N) ≤ D holds, we have and thus all these inequalities are equalities.■ In practice, it may be challenging to implement the entangled states described by Eq. (7).Alternatively, consider using a separable state, |+⟩ ⊗N , as an input.In this situation, the problem is to identify the state {(U 1 |+⟩) ⊗(N−n) (U 0 |+⟩) ⊗n } n , and thus reduces to a quantum change point problem for pure states, which has been studied in Ref. [5].However, the use of a separable state results in performance degradation, as shown in Fig. 3.
Identification of change points for Hamiltonians -The above-mentioned discussion can be expanded to address the problem of identifying Hamiltonian change points.Consider a situation where a Hamiltonian H 0 (t) acting on a quantum system changes to H 1 (t) at a particular time, t ⋆ .Assume that the change point t ⋆ is known to be one of the possible candidates t 0 , . . ., t N with equal prior probabilities.We arbitrarily choose a natural number R and time instants b be the unitary channel representing the time evolution with the Hamiltonian H b (t) between the time interval τ (k−1)R+r−1 ≤ t ≤ τ (k−1)R+r , i.e., FIG. 3. Probability of successful identification of change points for λ 0 = 1 and λ 1 = exp(iπ/10).P (N) sep is the maximum success probability when the separable state |+⟩ ⊗N is used as an input, which is obtained by numerically solving a semidefinite programming problem.P (∞) = γ and P (∞)  sep are limits as N → ∞.The analytical solution of P (∞)  sep is given in Ref. [5].
where T is the time-ordered operator.Also, let E n be the sequence expressed by The problem is then reduced to distinguishing quantum processes E 0 , . . ., E N similar to the problem of unitary channels.However, the main difference in this case is that the Hamiltonians can be time-dependent and we can use arbitrarily short time intervals.As a result, this problem is challenging to solve analytically.Let µ max (t) and µ min (t), respectively, be the maximum and minimum eigenvalues of H 1 (t) − H 0 (t).Also, let where For N = 1, the problem is to identify two processes E 0 and E 1 .In this problem, the maximum success probability is known as (γ 1 + 1)/2 [25,28], which is obtained as limit R → ∞.In addition, some experiments have been conducted using this result [29].We find that by extending the proof of Theorem 1, an analytical expression of the ultimate performance for each N is obtained, as stated in the following theorem (the proof is given in Appendix C 3).

Theorem 2
The maximum success probability in the change point problem for two Hamiltonians H 0 (t) and H 1 (t) with any integer N ≥ 1 is given by In the limit of large N, it follows from Eq. ( 15) that the maximum success probability tends to the average of γ 1 , γ 2 , . . ., γ N (i.e., N k=1 γ k /N).Conclusions -In this paper, we investigated the difficulty of identifying a precise moment when the Hamiltonian suddenly changes, and we presented an analytical expression of the maximum success probability.We first discussed the quantum change point problem for unitary channels, as a simpler problem.The objective of this task is to accurately identify the exact moment when a unitary channel changes to another.We demonstrated that the maximum success probability can be expressed in a simple analytical form by using only the number of possible change points and a parameter reflecting the ease of recognizing the channels before and after the change, assuming identical prior probabilities.The proposed method was then applied to derive the optimal performance for the problem of discriminating change points for Hamiltonians.
This work lays the foundation for future research on related topics, including the estimation of a continuous-valued change point and the detection of multiple change points.In addition, it can facilitate research on the change point problem for channels in open systems (i.e., non-unitary channels) and the optimization with other criteria such as unambiguous or Neyman-Pearson.We anticipate that our results will provide a solid starting point for addressing these challenges.
We thank for O. Hirota, M. Sohma, T. S. Usuda, and K. Kato for insightful discussions.This work was supported by the Air Force Office of Scientific Research under award number FA2386-22-1-4056.

Appendix A: Notation
Let us set up some notation.Let R, R ≥0 , and C be, respectively, the sets of all real numbers, all nonnegative real numbers, and all complex numbers.For a unitary matrix U, let Ad U be the unitary channel determined by Ad U (ρ) = UρU † .We call a completely positive map a single-step process.Pos(V, W) and Chn(V, W), respectively, denote the sets of all single-step processes and quantum channels (i.e., completely positive trace-preserving maps) from a system V to a system W. Also, let Pos V and Den V be, respectively, the sets of all positive semidefinite matrices and states (i.e., density matrices) of a system V.Let Cone(X) be the convex cone spanned by a set X. V V ′ denotes that the dimensions of the systems V and V ′ are the same.Let I V be the identity matrix on V and we write 1 where systems are depicted by labeled wires.Wires representing one-dimensional systems will often be omitted.Single-step processes can be composed sequentially or in parallel.Let us consider the concatenation of T single-step processes, which we call a T -step process, We write this T -step process by 1) , where ⊛ denotes the concatenation.A T -step process ) is called a quantum comb [30] if E (1) , . . ., E (T ) are quantum channels.In particular, this manuscript often focuses on a T -step process generated by T unitary channels, which is depicted by where {U (t) ∈ Chn(V t , W t )} T t=1 are unitary channels.Given a T -step process written by Eq. (A3), we consider a collection of (T + 1)-step processes, denoted by {D m } M m=0 is called a tester if D (1) , . . ., D (T ) are quantum channels [which implies that D (1) is a state] and {Π m } M m=0 is a quantum measurement.We will depict tester elements in blue.
The Choi-Jamiołkowski representations [30][31][32] of processes will be denoted by the same letter in the Fraktur font.The Choi-Jamiołkowski representation, E, of the T -step process E written by Eq. (A2) is the state of the system where , where U (t) is the Choi-Jamiołkowski representation of U (t) , i.e., 1) , ∀t ∈ {2, . . ., T }, A T -step process E written in the form of Eq. (A2) is a quantum comb if and only if its Choi-Jamiołkowski representation 1) , ∀t ∈ {2, . . ., T }, Also, a collection of (T + 1)-step processes {D m } M m=0 written in the form of Eq. (A4) is a tester if and only if W T ,V T ;...;W 1 ,V 1 , the probability that the tester {D m } M m=0 performed on E gives the outcome m is diagrammatically depicted by , (A8) where T is the transpose).We can easily verify M m=0 Tr(D T m E) = 1.

Appendix B: Problem Formulation
For each k ∈ {0, . . ., N} and b ∈ {0, 1}, let Let us concentrate on the case in which V k,r W k,r V 1,1 holds for any k and r.We choose a unitary matrix b is written by where b .Let us consider the following NR-step process: where ⟦n < k⟧ is 1 if n < k, and 0 otherwise.E (N) n is the process such that n processes U (1) 0 , U (2) 0 , . . ., U (n) 0 are applied and then N − n processes U (n+1) 1 , . . ., U (N) 1 are applied.The problem of discriminating quantum processes E (N) 0 , . . ., E (N) N can be seen as a generalization of the problem presented in the main paper.We find that the change point problem for unitary channels described in the main paper can be viewed as a special case of this problem with R = 1 and U (1)  b = ).Also, this problem with R → ∞ corresponds to the change point problem for Hamiltonians.The Choi-Jamiołkowski representation, E (N)  n , of the process E (N)  n is given by For example, in the case of N = 2, E (2) 0 , and 0 hold.The problem of discriminating quantum combs E 0 , . . ., E N with equal prior probabilities can be formulated as the following semidefinite programming problem: (P) Let P (N) denote the optimal value of this problem.The Lagrange dual problem is given by [20] minimize η(X) where and η is the function defined as satisfying X = tχ are uniquely determined by t = η(X) and χ = X/t.The optimal value of the dual problem coincides with the optimal value, P (N) , of the primal problem [20].
Γ (k,r) denotes the polygon in the complex plane whose vertices are the eigenvalues of Ũ(k,r) 1 .Let λ (k,r) 0 and λ (k,r) 1 be the ends of a side of Γ (k,r) that is closest to the origin (see Fig. 4).Note that such a side may not be unique.By swapping λ (k,r) 0 and λ (k,r) 1 if necessary, we assume 0 ≤ arg[λ (k,r)  1 /λ (k,r) 0 ] ≤ π.Let |λ (k,r) 0 ⟩ and |λ (k,r) 1 ⟩ be, respectively, the normalized eigenvectors of Ũ(k,r) 1 corresponding to the eigenvalues λ (k,r) 0 and λ (k,r) 1 .For each k ∈ {1, . . ., N}, let γ k be 1 if at least one of the polygons Γ (k,1) , . . ., Γ (k,R) includes the origin or R r=1 arg[λ (k,r) 1 /λ (k,r) 0 ] ≥ π holds, and otherwise.Also, let γ 0 1 and γ N+1 1 for convenience.In the case of N = 1, Problem (P) reduces to the problem of discriminating U (1)  0 and U (1)  1 with equal prior probabilities in a single shot, whose optimal value is known to be (γ 1 + 1)/2 [25,26,28].It is easily seen that for each k, the maximum success probability for discriminating U (k)  0 and U (k) 1 with equal prior probabilities in a single shot is (γ k + 1)/2, and thus γ k = 1 holds if and only if U (k)  0 and U (k) 1 are perfectly distinguishable.Our main theorem is the following: Theorem 1 For any integer N ≥ 1, we have Theorem 1 in the main paper is a special case of this theorem with In order to prove Theorem 1, we first show that there exists a tester such that the success probability is P ⋆ ( N k=1 γ k + 1)/(N + 1) in Sec.C 1.This immediately implies P (N) ≥ P ⋆ .In Sec.C 2, we next show that there exists a feasible solution, X, to Problem (D) satisfying η(X) = P ⋆ , which implies P (N) ≤ P ⋆ and thus P (N) = P ⋆ .In Sec.C 3, we prove Theorem 2 in the main paper by applying Theorem 1 to the problem of discriminating change points for Hamiltonians.

Feasible solution to the primal problem
We assume that for each k ∈ {1, . . ., N}, U (k) 0 and U (k) 1 are not perfectly distinguishable, i.e., γ k < 1 holds, unless otherwise stated.Let S be the set of all testers where and ).In such a tester {D m } N m=0 ∈ S, we first prepare the composite system V N,1 ⊗• • •⊗V 1,1 in an initial state ρ.Just after applying the channel U (k,r)  ⟦n<k⟧ , the channel ) is applied.For each r ∈ {1, . . ., R − 1}, the output of Q (k,r) is sent through the channel U (k,r+1)  ⟦n<k⟧ .We have This can be explained as follows.Q (k,r) transforms the channels U (k,r) 0 and U (k,r) then, it follows from Eq. (C6) that the channel Let Note that if λ (k) 0 and λ (k) 1 are independent of k, in which case , and ν k are also independent of k, and thus we have ν k = γ.Let |ψ ′ n ⟩ be the state just prior to the measurement {Π m } N m=0 when the change point is n; then, we can see ) be the Gram matrix of the collection of the states |ψ ′ 0 ⟩ , . . ., |ψ ′ N ⟩, whose (n + 1)-th row and (n ′ + 1)-th column is G n,n ′ ⟨ψ ′ n |ψ ′ n ′ ⟩.First, we prove the following two lemmas.
Proof First, we consider the case in which, for any k ∈ {1, . . ., N}, U (k) 0 and U (k) 1 are not perfectly distinguishable, i.e., γ k < 1 holds.Let us consider the tester {D m } N m=0 ∈ S that is determined by the input state |ψ⟩ given by Eq. (C11) and the measurement ] From Lemmas 2 and 3, the conditional probability that the measurement result is m given that the change point is n is given by | ⟨π m |ψ ′ n ⟩ | 2 = p m|n .Thus, the average probability is Next, we consider the case in which there exists k satisfying γ˜k = 1.In the following discussion, we assume for simplicity that γ k < 1 holds for each k k, but the same discussion can be easily applied to any other case.Since U ( k) 0 and U ( k) 1 are perfectly distinguishable, we can distinguish n < k and n ≥ k with certainty.Thus, Problem (P) can be divided into the problem of discriminating k processes {U ( k−1) ⟦n< k−1⟧ ⊛ • • • ⊛ U (1)  ⟦n<1⟧ } k−1 n=0 and the problem of discriminating N + 1 − k processes {U (N)  ⟦n<N⟧ ⊛ • • • ⊛ U ( k+1) ⟦n< k+1⟧ } N n= k.As mentioned above, it follows that, for the former and latter problems, there exist testers with the success probabilities ( k−1 n=1 γ n + 1)/ k and ( N n= k+1 γ n + 1)/(N + 1 − k), respectively.Thus, for Problem (P), there exists a tester whose success probability is k This problem can be seen as the Lagrange dual of the problem of discriminating the two processes U (n) 0 and U (n) 1 with equal prior probabilities.As already mentioned at the beginning of this section, the optimal value of this primal problem is (γ n + 1)/2.Since the duality gap is zero, we have η[Y (n) ] = (γ n + 1)/2.Let (C34) then, we have the following lemma.
Lemma 5 X (N) is a feasible solution to Problem (D).

FIG. 2 .
FIG. 2. (a) The most general protocol of change point discrimination for unitary channels.Each process E n consists of a sequence of N channels (U n<1 , . . ., U n<N ).Any discrimination strategy is expressed as a collection of a state ρ, channels σ 2 , . . ., σ N , and a measurement {Π m } N m=0 .(b) The most general nonadaptive protocol, which consists of a state ρ and a measurement {Π m } N m=0 .
an R-step process that consists of R unitary channels U (k,R) b , . . ., U (k,1) b .The input and output systems of U (k,r) b are, respectively, denoted by V k,r and W k,r .U (k) b is diagrammatically represented as .(B2)