GROUP-THEORETICAL ASPECTS OF CONTROL OF QUANTUM SYSTEMS S. G. SCHIRMER AND A. I. SOLOMON Quantum Processes Group, The Open University, Milton Keynes, MK7 6AA, UK E-mail:
[email protected],
[email protected] Group-theoretical aspects of control of quantum systems are presented and a concise summary of some important results on controllability of quantum systems is provided.
1
Introduction
The subject of control of quantum-mechanical systems has been a fruitful area of investigation lately. The growing interest of researchers in various aspect of control of quantum systems can be attributed at least partly to the fact that the ability to manipulate quantum systems in a controlled fashion will be an essential prerequisite for many exciting new applications from control of chemical reactions to quantum computing devices. One issue of interest, which we shall address in this paper, is the question which target states can be dynamically reached from a given initial state of the system by application of suitable control fields, given a certain type of interaction of the system with those fields. This paper is organized as follows. In section 2, the concept pure and mixed quantum states is introduced and examples of both are given. Section 3 deals with the evolution of quantum states and the partitioning of states into kinematical equivalence classes arising from the constraint of unitary evolution. In section 4, we discuss the action of the dynamical Lie group on the kinematical equivalence classes and its implications for dynamical reachability of states. In section 5, various degrees of controllability are defined and it is shown how these notions are related to the dynamical Lie group of the system. In section 6, we present some useful results on controllability as well as examples. We conclude with section 7.
2
States of quantum system
The state of a quantum system can be represented by a density operator ρˆ acting on a Hilbert space H. Since a density operator is a positive operator
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with trace one, it has a pure point spectrum with eigenvalues wn satisfying X wn = 1, 0 ≤ wn ≤ 1, (1) n
and a spectral resolution ρˆ =
N X
wn |Ψn ihΨn |,
(2)
n=1
where |Ψn i are the eigenstates and hΨn | their duals. Definition 1 A density operator ρˆ represents a pure state if it has rank one. Otherwise, it represents a mixed state. From the definition, we see immediately that ρˆ represents a pure state if and only if its eigenvalues are 0 and 1 and 0 occurs with multiplicity N − 1, where N is the dimension of the Hilbert space of the system. Therefore, the density matrix associated with a pure state is a projector onto a Hilbert space vector |Ψi, ρˆ = |ΨihΨ|. The element |Ψi, usually called a wavefunction, completely determines the density operator of the state and is therefore an equivalent representation of the state of the system. Example 1 (Pure States vs Mixed States) Consider a simple quantum system whose underlying Hilbert space has dimension four. Choose an arbitrary basis and consider the density matrices 1 1 1000 2 0 0 2 0 0 0 0 0 0 0 0 ρˆ2 = ρˆ1 = 0 0 0 0 . 0 0 0 0, 1 1 0000 2 0 0 2 Both density matrices represent pure states. In case of ρˆ1 this is obvious. To see that ρˆ2 is indeed a pure state, note that 1 1 0 . ρˆ2 = |ΨihΨ| for |Ψi = √ 2 0 1 On the other hand, consider the density matrices 1 1 1 4 0 0 4 2 0 0 0 0 1 0 0 0 1 1 0 4 4 2 ρˆ3 = ρˆ4 = 0 0 0 0, 0 1 1 0 . 4 4 1 1 0 000 4 0 0 4
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as well as
0.4 0 ρˆ5 = 0 0
0 0.3 0 0
0 0 0 0 0.2 0 0 0.1
It can easily be verified that all of these density matrices have eigenvalues between 0 and 1 and therefore represent mixed states of the system. 3
Evolution of quantum states
The evolution of a non-dissipative quantum system is determined by an evoˆ (t, t0 ), which satisfies the Schrodinger equation lution operator U i¯ h
d ˆ ˆ (t)]U ˆ (t, t0 ), U (t, t0 ) = H[f dt
(3)
ˆ (t)] is the total Hamiltonian of the system. If the system is subjected where H[f ˆ depends on a set of time-dependent to external control fields fm (t) then H control functions f (t) = (f1 (t), . . . , fM (t)). However, in any case, conservation of energy and probability requires the evolution operator of the system to be unitary at all times, no matter what fields are applied. ˆ (t, t0 ) of the system is restricted Hence, the time-evolution operator U to the unitary group U (N ). This constraint of unitary evolution imposes restrictions on the quantum states that can be reached from a given initial state. Since these restrictions are independent of the control fields applied, we shall refer to them as kinematical constraints, to distinguish them from dynamical constraints arising from constraints on the control fields. Precisely, given an initial state ρˆ0 and a target state ρˆ1 , the target state is ˆ such kinematically admissible if and only if there exists a unitary operator U that ˆ ρˆ0 U ˆ †. ρˆ1 = U
(4)
(4) defines an equivalence relation, which partitions the set of density operators into kinematical equivalence classes. Definition 2 Two density matrices ρˆ1 , ρˆ2 are kinematically equivalent if ˆ ρˆ1 U ˆ † for some unitary operator they are unitarily equivalent, i.e., if ρˆ2 = U ˆ U. It follows immediately that two density matrices are kinematically equivalent if and only if they have the same eigenvalues.
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Example 2 (Kinematical Equivalence of States) • The pure states ρˆ1 and ρˆ2 in example 1 both belong to the kinematical equivalence class of pure states. • The mixed states ρˆ3 and ρˆ4 are both members of the same kinematical equivalence class since they have the same eigenvalues { 12 , 12 , 0, 0}. • The state ρˆ5 , however, is not kinematically equivalent to any of the states above since its eigenvalues are different.
4
Dynamical reachability of states
For any initial state, only target states in the same kinematical equivalence class can possibly be dynamically reached by applying external control fields. However, clearly, not every kinematically admissible target state is necessarily dynamically reachable. The question thus arises what subsets of the kinematical equivalence class are dynamically reachable for a given initial state. The dynamically reachable sets depend on the dynamical Lie group S of the system. If S is the unitary group U (N ) then the dynamically reachable set for every initial state comprises the entire kinematical equivalence class it belongs to, since U (N ) acts transitively on every equivalence class. ˆ such that (4) is satisfied Furthermore, if there exists a unitary operator U ˜ then there always exists a unitary operator U with determinant one such that ˜ . Hence, SU (N ) also acts transitively on every kinematical (4) is satisfied for U equivalence class. However, the only proper subgroups of SU (N ) that act transitively on the class of pure states are Sp( 12 N ) and Sp( 12 N ) ⊗ U (1), where U (1) = eiθ IˆN is a multiple of the identity matrix IˆN . Furthermore, Sp( 12 N ) — and hence Sp( 12 N ) ⊗ U (1) — does not act transitively on many other equivalence classes of mixed states. Example 3 (Non-transitive action of Sp( 12 N ) for mixed states) ˆ =H ˆ 0 + f (t)H ˆ 1 , where Consider a four-level system with H − 32 0 ˆ0 = H 0 0
0 − 12 0 0
0 0 + 12 0
0 0 , 0 + 32
0 +1 0 0 ˆ 1 = +1 0 +1 0 . H 0 +1 0 −1 0 0 −1 0
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ˆ 0 and iH ˆ 1 satisfy x Note that both iH ˆT Jˆ = −Jˆx ˆ for 0 0 0 +1 0 0 +1 0 Jˆ = 0 −1 0 0 . −1 0 0 0 Therefore, the dynamical Lie algebra of the system is a subalgebra of sp(2)a ˆ in the dynamical Lie group must preserve Jˆ in the sense that and any U ˆ T JˆU ˆ = J. ˆ U Thus, two kinematically equivalent states ρˆ0 and ρˆ1 are dynamically reachable ˆ that preserves Jˆ such from each other only if there exists a unitary operator U that ˆ ρˆ0 U ˆ †. ρˆ1 = U Combining the last two equations shows that, in order to be dynamically equivalent, ρˆ0 and ρˆ1 must satisfy ˆ∗=U ˆ (Jˆ† ρˆ0 J) ˆ ∗U ˆ† (Jˆ† ρˆ1 J) This shows that 1
1
000 0 1 0 0 2 ρˆ0 = 0 0 0 0 0 000 2
000 0 0 0 0 ρˆ1 = 0 0 0 0 0 0 0 12 2
and
are not dynamically reachable from each other since ˆ ∗ = ρˆ1 (Jˆ† ρˆ1 J)
but
ˆ ∗ 6= ρˆ0 (Jˆ† ρˆ0 J)
and it is impossible to find a unitary transformation that maps both ρˆ0 and ˆ ∗ to ρˆ1 . (Jˆ† ρˆ0 J) This example shows that there is no proper subgroup of SU (N ) that acts transitively on all the kinematical equivalence classes. 5
Degrees of Controllability
Let us now turn our attention to various degrees of controllability. a It
can actually be verified that the dynamical Lie algebra is sp(2)
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Definition 3 (Complete controllability) A quantum system is completely controllable if any unitary evolution is dynamically realizable, i.e., if given any ˆ , there exists tF > 0 and an admissible control-trajectory unitary operator U ˆ (t, t0 )) such that U ˆ =U ˆ (tF , t0 ) pair (f , U Definition 4 (Density matrix controllability) A quantum system is density matrix controllable if given any two density matrices ρˆ0 , ρˆ1 in the same kinematical equivalence class, there exists tF > 0 and an admissible ˆ (t, t0 )) such that ρˆ1 = U ˆ (tF , t0 )ˆ ˆ (tF , t0 )† . control-trajectory pair (f , U ρ0 U Definition 5 (Pure-state controllability) A quantum system is purestate controllable if for any two pure states (wavefunctions) |Ψ0 i, |Ψ1 i, there ˆ (t, t0 )) such that exists tF > 0 and an admissible control-trajectory pair (f , U ˆ |Ψ1 i = U (tF , t0 )|Ψ0 i. From our considerations in the previous section, it is obvious that only a system with dynamical Lie group U (N ) is completely controllable. A system with dynamical Lie group SU (N ) or U (N ) is density matrix controllable since these groups act transitively on every equivalence class of density matrices. Furthermore, a system with dynamical Lie group Sp( 21 N ), Sp( 21 N ) ⊗ U (1), SU (N ) or U (N ) is pure-state controllable since these groups act transitively on the equivalence class of pure states. 6
Important results on controllability and examples
For a quantum system with Hamiltonian of the form ˆ (t)] = H ˆ0 + H[f
M X
ˆ m, fm (t)H
(5)
m=1
the dynamical Lie group is determined by the dynamical Lie algebra L genˆ m and we can use the erated by the skew-Hermitian operators (matrices) iH observations made above to derive Lie algebraic criteria for various degrees of controllability. Theorem 1 1,2 A quantum system with Hamiltonian (5) is • completely controllable if and only if L ' u(N ). • density matrix controllable if and only if L ' su(N ) or L ' u(N ). • pure-state controllable if and only if L is isomorphic to either u(N ), su(N ), or if N = 2`, sp(`), sp(`) ⊕ u(1).
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Complete controllability implies density matrix controllability and the latter implies pure-state controllability. The converse is clearly false. For a system that is not density matrix controllable, there exist initial states for which the set of dynamically accessible target states does not comprise the entire kinematical equivalence class, i.e., at least some of the kinematical equivalence classes are partitioned into disjoint subsets of dynamically reachable states. If the system is not pure-state controllable then there exists such a partition for the equivalence class of pure states. However, lack of controllability does not necessarily imply that the dynamical Lie group of the system does not act transitively on any kinematical equivalence class. For instance, every dynamical Lie group clearly acts transitively on the equivalence class of completely random ensembles, which consists of the single element ρˆR = N1 IˆN , where IˆN is the identity matrix of dimension N . ˆ =H ˆ 0 + f (t)H ˆ 1, Theorem 2 3,4 A quantum system with Hamiltonian H ˆ 0 = diag(E1 , . . . , EN ), H
ˆ1 = H
N −1 X
dn (|nihn + 1| + |n + 1ihn|)
(6)
n=1
is density matrix controllable if dn 6= 0 for 1 ≤ n ≤ N − 1 and either 1. there exists p such that ωn 6= ωp for n 6= p where ωn = En+1 − En ; or 2. ωn = ω for all n but there exists p such that vn 6= vp for n 6= p, where vn ≡ 2d2n − d2n−1 − d2n+1 . If N = 2p then d2p−k 6= d2p+k for some k 6= 0 is required as well. ˆ 0 ) 6= 0 then the system is completely controllable. If in addition Tr(H This theorem can be used to show that many systems of physical interest are indeed completely controllable. Example 4 (Completely controllable systems) The following systems are completely controllable: • N -level Morse oscillators with arbitrary (non-zero) dipole moments • N -level harmonic oscillators with non-uniform, non-zero dipole moments • N -level one-electron atoms with arbitrary (non-zero) dipole moments • A confined particle such as an electron in a box (quantum well) Pure-state controllability can also be related to connected graphs
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Theorem 3 5,6 Consider a quantum system with N non-degenerate energy levels En and transitions |ni → |mi with transition frequency ωmn and corresponding dipole moment dmn . Let the energy levels be the vertices of a graph and the transitions between them be the edges. Eliminate the edges corresponding to any degenerate transitions, i.e., transition pairs with ωmn = ωm0 n0 . Then the system is pure-state controllable if the resulting graph is connected. This theorem provides an useful tool for quick identification of pure-state controllable systems. Example 5 (Pure-state controllable systems) Each of the following transition diagrams correspond to pure-state controllable systems
provided that the energy levels and transition frequencies are non-degenerate. Note, however, that the converse of the theorem is not true, i.e., if the graph is disconnected the system may still be controllable. Example 6 Consider a sequentially coupled N -level system with equally spaced energy levels and equal transition probabilities. This system does not satisfy the hypothesis of theorem 3 since there is only one transition frequency. Therefore, removing the edges corresponding to the degenerate transition frequency leaves us with a completely disconnected graph. However, it can be shown that the dynamical Lie algebra of this system is • sp( 12 N ) or sp( 12 N ) ⊕ u(1) if N = 2` • so(N ) or so(N ) ⊕ u(1) if N = 2` + 1 Therefore, the system is pure-state controllable if N is even. 7
Conclusion
We have shown that the kinematical constraint of unitary evolution partitions the states of a quantum system into kinematical equivalence classes. The dynamically reachable sets are subsets of the kinematical equivalence classes determined by the action of the dynamical Lie group of the system. If the dynamical Lie group acts transitively on all the equivalence classes then the dynamically reachable sets correspond to the kinematical equivalence classes.
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Acknowledgments We sincerely thank J. V. Leahy (University of Oregon), V. Ramakrishna (University of Texas at Dallas), H. Rabitz (Princeton University) and G. Turinici (University of Paris) for helpful discussions and comments. AIS acknowledges the hospitality of the Laboratoire de Physique Th´eorique des Liquides, University of Paris IV, where he is currently a visiting faculty member. References 1. F. Albertini and D. D’Alessandro. Notions of controllability for quantummechanical systems. quant-ph/0106128 (2001) 2. S. G. Schirmer, J. V. Leahy, and A. I. Solomon. Degrees of controllability for quantum systems and applications to atomic systems. quantph/0108114 (2001) 3. H. Fu, S. G. Schirmer, and A. I. Solomon. Complete controllability of finite-level quantum systems. J. Phys. A 34, 1679 (2001) 4. S. G. Schirmer, H. Fu, and A. I. Solomon. Complete controllability of quantum systems. Phys. Rev. A 63, 063410 (2001) 5. G. Turinici. Controllable quantities for bilinear quantum systems in 39th IEEE CDC Proceedings (Causal Productions, Adelaide, Australia) 2000. 6. G. Turinici and H. Rabitz. Quantum wavefunction controllability. Chem. Phys. 267, 1–9 (2001)
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