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Automatica 49 (2013) 955–959 Contents lists available at SciVerse ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Brief paper Reachable set of open quantum dynamics for a single spin in Markovian environment✩ Haidong Yuan 1 Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong article info Article history: Received 22 November 2011 Received in revised form 15 August 2012 Accepted 23 October 2012 Available online 13 February 2013 abstract In this article, we study the control of a single spin in the Markovian environment. Given the initial state, we compute all the possible states to which the spin can be driven to at any given time, under the assumption that fast coherent control is available. © 2013 Elsevier Ltd. All rights reserved. Keywords: Quantum control Bilinear control system Reachable set 1. Introduction Control of quantum systems has important applications in physics and chemistry. In particular, the ability to steer the state of a quantum system (or an ensemble of quantum systems) from a given initial state to a desired target state forms the basis of spectroscopic techniques such as nuclear magnetic resonance (NMR) (Ernst, Bodenhausen, & Wokaun, 1987), electron spin resonance (ESR) spectroscopy (Schweiger, 1990), laser coherent control (Warren, Rabitz, & Dahleh, 1993) and quantum computing (Cory, Fahmy, & Havel, 1997; Gershenfeld & Chuang, 1997; Wegrzyn, Klamka, Znamirowski, Winiarczyk, & Nowak, 2004). In reality, the systems are always coupled to the environment, which leads to the open system dynamics. In this article, we are interested in finding all possible states that the system can be steered to at any given time. In control language, this is the problem of finding the reachable set of the controlled system at any given time, which has a close connection with time optimal control of the system. Quite a few works have been done on time optimal control of closed quantum systems (Carlini, Hosoya, Koike, & Okudaira, 2006, 2007; Khaneja, Brockett, & Glaser, 2001; Yuan, Glaser, & Khaneja, 2007; Yuan & Khaneja, 2005; Yuan, Zeier, & Khaneja, 2008; Zeier, Yuan, & Khaneja, 2008) and on state transferring of some open ✩ The material in this paper was partially presented at the 48th IEEE Conference on Decision and Control (CDC) and 28th Chinese Control Conference (CCC), December 16–18, 2009, Shanghai, China. This paper was recommended for publication in revised form by Associate Editor James Lam under the direction of Editor Ian R. Petersen. E-mail address: [email protected]. 1 Tel.: +852 34003751; fax: +852 23629045. 0005-1098/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2013.01.005 quantum systems (Altafini, 2004; Bonnard, Chyba, & Sugny, 2009; Bonnard & Sugny, 2009). Some qualitative descriptions of the reachable set have been obtained (Altafini, 2004), while here we will give a platform that enables a quantitative description, provide a frame for general time optimal control problem. This result is also expected to be helpful for quantum error correcting (Shor, 1995) as it gives information on when the errors may reach a certain threshold. The article is organized as follows: Section 2 introduces the dynamical evolution for open quantum systems and the dynamical equation for the spectra of the system; Section 3 gives analytical descriptions to the reachable set of two widely used physical model on the spin system, namely the phase damping and the amplitude damping; Section 4 concludes. 2. Open quantum system dynamics The density matrix, which is the state of an N-dimensional quantum system, denoted as ρ(ρ ∈ CN ×N ), is semi-positive definite and Tr(ρ) = 1. Its dynamics is described by the Lindblad equation, which takes the form ρ̇ = −i[H (t ), ρ] + L(ρ). (1) Here H (t ) is the Hamiltonian of the system, which can be represented as a Hermitian matrix. −i[H (t ), ρ] is called the unitary part of the evolution, as the solution to ρ̇ = −i[H (t ), ρ] is captured by unitary conjugation, i.e., ρ(t ) = U ρ U Ď , where U ∈ SU (N ). L(ρ) is the dissipative part of the evolution. The term L(ρ) is linear in ρ and is given by the Lindblad form (Alicki & Lendi, 1986; Lindblad, 1976): L(ρ) = αβ aαβ Ď Fα ρ Fβ − 1 2 Ď {Fβ Fα , ρ} , 956 H. Yuan / Automatica 49 (2013) 955–959 where {, } is the anti-commute operation ({A, B} = AB + BA), {Fα } are the Lindblad operators which forms a linear basis for traceless operators on the density matrix, and the coefficients {aαβ }, which indicate the coupling strengths of the system with the environment, can form a Hermitian matrix A = (aαβ ) which is known as the GKS (Gorini, Kossakowski and Sudarshan) matrix (Gorini, Kossakowski, & Sudarshan, 1976). The problem we address in this paper is to compute all the density matrices the system can be steered to under the Lindblad equation of motion given by Eq. (1). Eq. (4) is then Λ̇(t ) = diag(U Ď L(U ΛU Ď )U ) Ď = diag aαβ U Ď Fα U ΛU Ď Fβ U αβ (2) (3) Consider now the change in the spectrum under the evolution of Eq. (3), if the spectrum is not degenerate, the change of the spectrum on the right hand side is, to first order in δ t, just the diagonal, i.e. Λ(t + δ t ) = diag(Λ − i[U Ď H (t )U , Λ(t )]δ t + U Ď L(U ΛU Ď )U δ t ) + O(δ t 2 ). As Λ(t ) is a diagonal matrix, for any matrix P , P Λ(t ) and Λ(t )P have the same diagonal entries, so the diagonal part of −i[U Ď (t ) H (t )U (t ), Λ(t )] is zero. Thus we get Λ̇(t ) = diag(U Ď L(U ΛU Ď )U ). (4) The only density matrix of two-level system which has a degenerate spectrum is ρ = 21 I, at this point, the Lindblad equation (1) becomes ρ̇ = L(ρ), which does not depend on the coherent control and with proper parametrization this point lies at the boundary of the state space and does not affect the dynamics. For a two-level (N = 2) system, we can take the Lindblad operators {Fα } as normalized Pauli spin operators √1 {σx , σy , σz }, 2 where σx := 0 1 1 0 , σy := 0 i −i 0 , and σz := 1 0 0 −1 . The GKS matrix is now a 3 × 3 matrix A= axx ayx azx axy ayy azy axz ayz azz Ď − {U Fβ UU Fα U , Λ} , 2 (6) for the last step we just used the fact that Fβ is a Pauli matrix which is Hermitian. Now substitute U Ď Fα U = γ cαγ Fγ , where C = c cxy cyy czy xx cxz cyz czz ∈ SO(3) is the adjoint representation of U, into 1 ′ Eq. (6), we obtain Λ̇(t ) = diag αβ aαβ Fα ΛFβ − 2 {Fβ Fα , Λ} , where a′αβ = γ ,µ cγ α aγ µ cµβ forms the transformed GKS matrix, ′ ′ i.e., A = (aαβ ) = C T AC . As the trace of Λ(t ) is 1, we can write Λ(t ) = 21 I +λ(t )σz , where λ(t ) ∈ 0, 12 ; from the dynamics of Λ(t ), we obtain the dynamics for λ(t ) in the non-degenerate case: λ̇(t ) = −(a′xx + a′yy )λ(t ) + i (a′xy − a′yx ). 2 To compute the reachable set of λ governed by this dynamics, cyx czx we will just need to know the range of the right side of the above equation, which can be written as Tr(M (t )A′ ) = Tr(M (t )C T AC ), where −λ(t ) M (t ) = i − i 0 2 −λ(t ) 2 0 0 , 0 (7) 0 i.e., λ̇(t ) = Tr(M (t )C T AC ). (8) As we assume the coherent control can generate any unitary operator in negligible time, C then can take all elements in SO(3). Since SO(3) is connected and compact, the range of Tr(M (t ) C T AC ) is thus just an interval [µ(λ(t )), ν(λ(t ))], where µ(λ(t )) = minC ∈SO(3) Tr(M (t )C T AC ) and ν(λ(t )) = maxC ∈SO(3) Tr(M (t )C T AC ). We now just need to describe µ(λ(t )) and ν(λ(t )), as the solution of λ̇(t ) = µ(λ(t )), λ̇(t ) = ν(λ(t )) (9) will thus give the minimum and maximum values of λ(t ) at any given time t under this dynamics. At the degenerate point ρ = 12 I , λ = 0, ρ̇ = L(ρ), since L(ρ) is Hermitian and traceless, we can write it as L(ρ) = x − iy −z z x + iy L(ρ)δ t are λ̇ = 1 2 . It is easy to check that the eigenvalues of ± x2 + y2 + z2 1 I 2 + δ t; thus at the degenerate point, x2 + y2 + z 2 . Computing x, y, z from the GKS matrix, we can get . Ď Ď aαβ U Ď Fα U ΛU Ď Fβ U 1 + U L(U ΛU )U δ t )U . αβ ρ(t + δ t ) ≈ ρ(t ) + (−i[H (t ), ρ(t )] + L(ρ(t )))δ t . Substitute ρ(t ) = U (t )Λ(t )U Ď (t ), we get ρ(t + δ t ) ≈ ρ(t ) + (−i[H (t ), ρ] + L(ρ(t )))δ t = U (Λ(t ) − i[U Ď H (t )U , Λ(t )]δ t Ď 2 = diag We assume that the coherent control can be executed on a time scale much shorter than that of dissipation, and the control Hamiltonian H (t ) can produce any unitary transformation U ∈ SU (N ) on the system, i.e., any unitary transformation can be produced on the system in negligible time compared to that of the dissipation. This assumption is widely met in various physical systems, for example, in nuclear magnetic resonance, the time scale for control is ∼10−3 s while the time scale for dissipation is ∼1 s. Under this assumption, the reachable set of the density matrix is completely captured by its diagonal form, or ‘spectrum’, Λ, where ρ = U ΛU Ď . If a spectrum form Λ can be reached at time T , then all the elements on the iso-spectral orbit {U ΛU Ď } can also be reached; thus, we will reformulate the control problem in terms of the spectrum. In Sklarz, Tannor, and Khaneja (2004), this kind of reformulation has been used to find the optimal cooling strategy. In the following, we will give the equation for the spectrum of ρ . For the purpose of this paper, we will confine our attention to twolevel (single spin) systems. The evolution of the density matrix ρ is governed by the Lindblad equation (1), thus Ď Ď Ď − {U Fβ UU Fα U , Λ} 2.1. The dynamics of the spectrum Ď 1 (5) λ̇ = |axy − ayx |2 + |axz − azx |2 + |ayz − azy |2 . (10) H. Yuan / Automatica 49 (2013) 955–959 957 3. Reachable set on single spin systems 3.2. Amplitude damping The range of forms like Tr(MC T AC ), which sometimes is denoted as the C-numerical range, has been studied in mathematics (Gustafson, 1997) and has been recently introduced to solve problems of closed quantum dynamical systems (Dirr, Helmke, Kleinsteuber, Glaser, & Schulte-herbrüggen, 2006; Schulte-herbrüggen, Dirr, Helmke, & Glaser, 2008), where numerical approaches have been discussed. In this section, we will give analytical solutions to the most widely used physics models on single-spin system, namely the phase damping model and the amplitude damping model. The following theorem on C-numerical range will be used in our analysis: If a two level system is initially at the state |1⟩, it will gradually decay to the ground state |0⟩; this process is usually called amplitude damping, or in atomic physics, spontaneous emission (Nielsen & Chuang, 2000). The Lindblad equation for this process is given by Theorem 1 (Brockett, 1988). If A, B ∈ RN ×N are both symmetric matrices, then the range of Tr(BC T AC ), where C ∈ SO(N ), is an interval I = [µ, ν] with µ= N aj b N − j , ν= j=1 N L(ρ) = γ 2 1 σ− ρσ+ − {σ+ σ− , ρ} , (14) 2 where σ+ = σx + iσy = σ− = σx − iσy = 0 0 2 , 0 0 2 0 . 0 (15) (16) In this case, the GKS matrix is aj b j , A=γ j =1 1 −i 0 where a1 ≥ a2 ≥ · · · ≥ aN and b1 ≥ b2 ≥ · · · ≥ bN are the eigenvalues of A and B, respectively. i 1 0 0 0 0 = γ (B1 + iB2 ). (17) When λ = 0, λ̇(t ) = Tr(M (t )C T AC ). 3.1. Phase damping (18) Note that Tr(B1 C B2 C ) = 0, so we have T Phase damping is a decoherence process that results in the loss of coherence between different basis states. It can be caused by random phase shifts of the system due to its interaction with the environment (Nielsen & Chuang, 2000). The Lindblad equation for the phase damping model is ρ̇ = −i[H (t ), ρ] + L(ρ), where the γ dissipative part takes the form L(ρ) = − 4 [σz , [σz , ρ]], in this case 0 the GKS matrix is A = 0 0 0 0 0 0 0 γ 0 0 0 0 1 0 0 0 0 0 , then we can write 2 2 2 c21 + c22 + c23 = 1, λ̇(t ) = Tr(−λ(t )B1 C T AC ). (12) When λ = 0, from Eq. (10), we get λ̇ = 0, which is actually the same as Eq. (12), so in this case, Eq. (12) gives a complete description of the dynamics of the spectrum. Applying Theorem 1 and noticing that the eigenvalues of A and −λ(t )B1 are {γ , 0, 0} and {0, −λ(t ), −λ(t )} respectively, it is then easy to see the range of Tr(−λ(t )B1 C T AC ) is [−γ λ(t ), 0]. At a given time T , the minimum value of λ(T ) is then given by the solution of the equation λ̇(t ) = −γ λ(t ), which is λ(T ) = exp(−γ T )λ(0). And the maximum value of λ(T ) is given by the solution of λ̇(t ) = 0, which is λ(T ) = λ(0). So in the case of phase damping, the reachable set of the density matrix at any given time T is ρ(T ) = U 2 I + λ(T )σz + 2 2 2 c11 + c12 + c13 = 1, = −Tr(C T ACB2 ) = −Tr(B2 C T AC ), thus Tr(B2 C T AC ) = 0, and Tr(M (t )C T AC ) = Tr(−λ(t )B1 C T AC ), so = γ [−λ(t )( 2 c11 (11) Tr(B2 C T AC ) = Tr(C T AT CBT2 ) 1 T 2 c12 + 2 c21 2 + c22 ) + (c12 c21 − c11 c22 )]. (19) We now need to find the range of the above expression. First note that as C ∈ SO(3), M = −λ(t )B1 − 2i B2 . As B2 is a skew-symmetric matrix and C T AC is a symmetric matrix in this case, using the fact that Tr(P ) = Tr(P T ), we have i 2 , which is a real symmetric matrix. When λ ̸= 0, from Eq. (8), we get that λ̇(t ) = Tr(M (t )C T AC ). 1 0 0 0 Let B1 = 0 1 0 , B2 = −1 −λ(t )B1 − B2 C γ (B1 + iB2 )C 2 1 T T = γ −λ(t )Tr(B1 C B1 C ) + Tr(B2 C B2 C ) Tr(MC AC ) = Tr T Ď U |λ(T ) 2 2 2 2 1 ≤ c11 + c12 + c21 + c22 ≤ 2, (20) also 1 2 2 2 2 (c11 + c12 + c21 + c22 ) ≥ c12 c21 − c11 c22 2 1 2 2 2 2 + c12 + c21 + c22 ), (21) ≥ − (c11 2 combine 1 the inequalities of (20), (21) and the fact that λ(t ) ∈ 0, 2 , we get 2 1 2 1 2 2 2 2 − λ(t ) ≥ − λ(t ) (c11 + c12 + c21 + c22 ) 2 2 2 2 2 ≥ −λ(t )(c11 + c12 + c21 + c22 ) + (c12 c21 − c11 c22 ) 1 2 2 2 2 ≥ − − λ(t ) (c11 + c12 + c21 + c22 ) 2 ≥ −2 1 2 + λ(t ) , (22) 2 2 2 2 i.e., the range of γ [−λ(t )(c11 + c12 + c21 + c22 ) + (c12 c21 − c11 c22 )] is ∈ [exp(−γ T )λ(0), λ(0)], U ∈ SU (2) . 2 2 and 0 ≤ c13 + c23 ≤ 1, so we have (13) 1 1 + λ(t ) , 2γ − λ(t ) , −2 γ 2 2 (23) 958 H. Yuan / Automatica 49 (2013) 955–959 where the lower bound is achieved by setting c11 = c22 = 1, c12 = c21 = 0 and the upper bound is achieved by setting c11 = c22 = 0, c12 = c21 = 1. When λ = 0, it is at the degenerate point, from Eq. (10) we get λ̇ = 2γ . (24) Now we will calculate the range of λ(T ), which is a subset of 0, 12 . If λ(0) ̸= 0, we start with the dynamics of Eq. (18), in this case we first solve the following equations separately: Thus the reachable set for this system is ρ(T ) = U 1 2 2 2 which gives the minimum and maximum values of λ(T ) respectively under this dynamics. It is easy to get 1 λ(T ) ∈ exp (−2γ T ) + λ(0) 2 1 1 1 − , exp(−2γ T ) λ(0) − + . 2 2 2 If the minimum value 1 2 1 + λ(0) − > 0, 2 1 λ(T ) ∈ exp(−2γ T ) + λ(0) 2 1 1 1 − , exp(−2γ T ) λ(0) − + . 2 2 2 If the minimum value 1 2 1 + λ(0) − ≤ 0, 2 as the derivative of λ(t ) lies in the interval of −2γ 1 2 1 + λ(t ) , 2γ − λ(t ) 2 which contains 0 as λ(t ) ∈ 0, 21 , we can always use coherent control to change the derivative proportionally such that λ(T ) = 0, so the range of λ(T ) would be 0, exp(−2γ T ) λ(0) − 1 2 + 1 2 in this case. Thus the range of λ(T ) is 1 max exp(−2γ T ) + λ(0) − , 0 , 2 2 1 1 exp(−2γ T ) λ(0) − + . 1 2 2 If λ(0) = 0, it will deviate from 0 a little bit under the dynamics of Eq. (24), then it evolves according to Eq. (18), and the maximum value λ(T ) takes the same form exp(−2γ T ) λ(0) − 21 + 12 , as the integral does not change by the derivative on a single point. The minimum value is 0, as after it deviates from 0, we can control to always use coherent drive it back. So the range is 0, exp(−2γ T ) λ(0) − 12 + 21 , and it can also be written as 1 + λ(0) − , 0 , 2 2 1 1 exp(−2γ T ) λ(0) − + . max exp(−2γ T ) 1 2 2 (25) The intuitive interpretation of this is as follows. For times short compared with γ1 , the reachable spectra are close to the original spectrum. For longer times, however, one can play the tendency of the system to relax off against the ability to perform unitary control to manipulate the spectrum in any desired faction so that for T ≫ 1 , essentially all possible states can be reached. This is different γ from the case of phase damping: there the time evolution is unital and tends inevitably toward the fully mixed state; coherent control can only delay the process and achieve a variety of less than fully mixed states at various times along the way. The presence of relaxation in addition to decoherence allows a richer set of states to be attained by playing decoherence (which drives the system to a fully mixed state) and relaxation (which drives the system to a pure state) off against each other. 4. Conclusion then the range of λ(T ) is just exp(−2γ T ) 2 1 1 exp(−2γ T ) − + λ(0) + . 2 1 λ̇(t ) = −2γ + λ(t ) , 2 1 λ̇(t ) = 2γ − λ(t ) , I + λ(T )σz U Ď |U ∈ SU (2), λ(T ) 1 1 ∈ max 0, exp(−2γ T ) + λ(0) − , exp(−2γ T ) 2 Control of open quantum systems is an important problem for a wide variety of physics, chemistry and engineering applications. This paper analyzed the problem of controlling open quantum systems in cases where fast coherent control of the system is available. This coherent control can be used to ‘present’ various aspects of the system’s state to the environmental interaction. Because of the presence of fast coherent control, the quantity of interest under control is the spectrum of the density matrix. 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A, 52, 2493; Steane, A. M. (1996). Physical Review Letters, 77, 793; Preskill, J. (1998). Proceedings of the Royal Society of London, 454, 385. Sklarz, S. E., Tannor, D. J., & Khaneja, N. (2004). Optimal control of quantum dissipative dynamics: analytic solution for cooling the three-level Λ system. Physical Review. A, 69, 053408. Warren, W. S., Rabitz, H., & Dahleh, M. (1993). Coherent control of quantum dynamics: the dream is alive. Science, 259, 1581. Wegrzyn, S., Klamka, J., Znamirowski, L., Winiarczyk, R., & Nowak, S. (2004). Nano and quantum systems of informatics. Bulletin of the Polish Academy of Sciences. Technical Sciences, 52(1). Yuan, H., Glaser, S. J., & Khaneja, N. (2007). Geodesics for efficient creation and propagation of order along Ising spin chains. Physical Review. A, 76, 012316. Yuan, H., & Khaneja, N. (2005). Time optimal control of coupled qubits under nonstationary interactions. Physical Review. A, 72, 040301(R). 959 Yuan, H., Zeier, R., & Khaneja, N. (2008). Elliptic functions and efficient control of Ising spin chains with unequal couplings. Physical Review. A, 77, 032340. Zeier, R., Yuan, H., & Khaneja, N. (2008). Time-optimal synthesis of unitary transformations in a coupled fast and slow qubit system. Physical Review. A, 77, 032332. Haidong Yuan received his B.E. in Electrical Engineering from Tsinghua University in 2001, M.S. in Engineering Science in 2002 and Ph.D. in Applied Mathematics in 2006, both from Harvard University. After completing his Ph.D., Dr. Yuan worked as postdoctoral fellow at Massachusetts Institute of Technology and Harvard University. Starting from August 2012, he is with the Department of Applied Mathematics at The Hong Kong Polytechnic University as an Assistant Professor. His research interests are in the areas of dynamical systems and control theory, information, optimization, signals and systems, current interests are on control and system problems in the context of quantum science and technology, including control and optimization problems in NMR (Nuclear Magnetic Resonance) spectroscopy, magnetic resonance imaging, atomic system and superconducting quantum interference devices.