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Navigation between quantum states by quantum mirrors

We introduce a technique that allows one to connect any two arbitrary (pure or mixed) superposition states of an N-state quantum system. The proposed solution to this inverse quantum mechanical problem is analytical, exact, and very compact. The technique uses standard and generalized quantum Householder reflections (QHRs), which require external pulses of precise areas and frequencies. We show that any two pure states can be linked by just a single generalized QHR. The transfer between any two mixed states with the same dynamic invariants (e.g., the same density matrix eigenvalues) requires in general N QHRs. Moreover, we propose recipes for synthesis of arbitrary preselected mixed states using a combination of QHRs and incoherent processes (pure dephasing or spontaneous emission).


I. INTRODUCTION
Quantum state engineering in atoms and molecules traditionally uses three basic techniques for transfer of population, complete or partial, from one bound energy state to another, single or superposition state: resonant pulses of precise areas (e.g. π pulses in a two-state system or generalized π pulses for multiple states) [1], adiabatic passage using one or more level crossings [2], or stimulated Raman adiabatic passage (STIRAP) and its extensions [3]. All these techniques require the system to be initially in a single energy state; such a state can be easily prepared experimentally, e.g. by optical pumping. Some of these techniques are "tuned" to a specific initial condition: for example, STIRAP requires a counterintuitive pulse sequence to transfer population from state 1 to 3 in a 1-2-3 linkage, but it is largely irrelevant if the system starts in states 2 or 3 (with some exceptions for state 3) [3]. In other words, STIRAP is (very) useful in producing only one column (the first) of the unitary propagator. Similar conclusions apply, to a large extent, also to the other two techniques using pulse areas and level crossings.
These traditional techniques resolve only a small (although important) part of the general problem of quantum state engineering: given the initial and final states of an N -state system, find a physical set of operations that connect them. This problem requires the construction of the entire propagator, not just a single column or row.
In this paper we introduce a technique for full quantum state engineering, which produces in a systematic manner a propagator that can connect any two preselected superposition states of an N -state quantum system, representing a qunit in quantum information [4]. The two states can be pure as well as mixed, and the latter may have the same or different sets of dynamic invariants (constants of motion). The solution consists of two steps: first, find a propagator that connects the two states, and second, find a physical realization of this propagator.
The first part is the mathematical solution of this inverse problem in quantum mechanics, and the solution is different for three types of problems: (i) pure-to-pure states; (ii) mixed-to-mixed states with the same invariants; (iii) mixed-to-mixed states with different invariants. The case (iii), for instance, contains the important problem of engineering an arbitrary presected mixed state and we pay special attention to it. In this latter respect our exact analytic results are alternative to the (approximate) numeric optimization procedure proposed by Karpati et al [5]; moreover, our approach allows one to engineer any preselected mixed state, whereas the method of Karpati et al [5] can only produce a class of mixed states.
The second part of the solution is the physical realization of the respective propagator. For this we use the recently introduced physical implementation of the quantum Householder reflection (QHR) [6,7] and we show that QHR is a very powerful tool for quantum state engineering. Remarkably, in case (i) only a single QHR is needed to connect two pure states. In case (ii), a general U(N ) propagator is necessary in the general case, which requires N QHRs. In case (iii), some sort of incoherent process is required in order to equalize the different dynamic invariants of the initial and final mixed states, and the remaining coherent U(N ) part is realized by QHRs. We describe the use of two such incoherent processes: pure dephasing and spontaneous emission. The Householder reflection [8] is a powerful and numerically very robust unitary transformation, which has many applications in classical data analysis, e.g., in solving systems of linear algebraic equations, finding eigenvalues of high-dimensional matrices, least-square optimization, QR decomposition, etc. [9]. In its quantum mechanical implementation [6,7] it consists of a single interaction step involving N simultaneous pulsed fields of precise areas and detunings in an N -pod linkage pattern, wherein the N states of our system are coupled to each other via an ancillary excited state, as displayed This paper is organized as follows. In Sec. II we review the standard and generalized QHR gates and their physical implementations. In Sec. III we show how two pure states can be connected by means of standard and generalized QHRs. In Sec. IV we construct the propagator connecting two arbitrary mixed states with the same dynamic invariants. Engineering of an arbitrary preselected mixed qunit state is presented in Sec. V. The conclusions are summarized in Sec. VI.

A. Definition
The standard QHR is defined as where I is the identity operator and |v is an Ndimensional normalized complex column-vector. The QHR (1) is both hermitian and unitary, M = M † = M −1 , which means that M is involutary, M 2 = I. In addition, det M = −1. For real |v the Householder transformation (1) has a simple geometric interpretation: reflection with respect to an (N −1)-dimensional plane with a normal vector |v . In general, the vector |v is complex and it is characterized by 2N − 2 real parameters (with the normalization condition and the unimportant global phase accounted for).
The generalized QHR is defined as where ϕ is an arbitrary phase. The standard QHR (1) is a special case of the generalized QHR (2) for ϕ = π: M(v; π) ≡ M(v). The generalized QHR is unitary,

B. Physical implementation
We have shown recently [6,7] that the standard and generalized QHR operators can be realized physically in a coherently coupled N -pod system shown in Fig. 1. The N degenerate [in the rotating-wave approximation (RWA) sense [1]] ground states |n (n = 1, 2, . . . , N ), which represent the qunit, are coupled coherently by N external fields to an ancillary, excited state |e ≡ |N + 1 [6]. The excited state |e can generally be off resonance by a detuning ∆ (t) [6], which must be the same for all fields. The Rabi frequencies Ω 1 (t), . . . , Ω N (t) of the couplings between the ground states and the excited state have the same pulse-shaped time dependence f (t), but possibly different phases β n and amplitudes χ n , Ω n (t) = χ n f (t) e iβn (n = 1, 2, . . . , N ). ( The qunit+ancilla RWA Hamiltonian reads The exact solution to the Schrödinger equation for the propagator U(t), can be found in [6].
The standard QHR M(v) is realized on exact resonance (∆ = 0), for any pulse shape f (t), and for root-meansquare (rms) pulse area where with . Then the transition probabilities to the ancilla state vanish and the propagator within the qunit space is given exactly by the standard QHR M(v) (1). The components of the Ndimensional normalized complex vector |v are the Rabi frequencies, with the accompanying phases [7], . Hence the qunit propagator represents a physical realization of the standard QHR in a single interaction step. Any QHR vector (8) can be produced on demand by appropriately selecting the peak couplings χ n and the phases β n of the external fields.
The generalized QHR M(v; ϕ) can be realized in the same N -pod system, but for specific detunings off resonance. Again the transition probabilities to the ancilla state must vanish; the corresponding rms pulse areas (7) in general depend on the pulse shape and differ from the resonance values (6). The propagator within the qunit space is the generalized QHR (2), wherein the phase ϕ depends on the interaction parameters. Although the parameters (i.e. the rms area and the detuning) of any needed generalized QHR can be found numerically for essentially any pulse shape, it is very convenient to use a hyperbolic-secant pulse shape, for which there is a simple exact analytic solution: the Rosen-Zener model [11], For this pulse shape, the rms area (7) is A = πχT . A generalized QHR transformation M(v; ϕ) (2) is realized when the interactions satisfy again Eq. (8), and the pulse area and the detuning obey [6,7] A = 2πl (l = 1, 2, . . .) , (10a) For any given ϕ, there are l values of ∆ 0 , which satisfy Eq. (10b) [6]. This is also the case for ϕ = π, i.e. for the standard QHR, for which one of the solutions is ∆ 0 = 0. Hence the standard QHR M(v) can be realized both on and off resonance, whereas the generalized QHR M(v; ϕ) can only be realized for nonzero ∆ 0 . The advantage of tuning off resonance is the lower transient population in the ancilla excited state, which would reduce the population losses if the lifetime of this state is short compared to the interaction duration. This implementation is particularly suited for a qutrit (N = 3) formed of the magnetic sublevels of an atomic level with angular momentum J = 1; then the ancilla state should be a J = 0 state. The three pulsed fields can be delivered from the same laser by using beam splitters and polarizers, which would authomatically ensure that all of them have the same detuning and pulse shape. Moreover, with femtosecond pulses it would be possible to use pulse shapers [12], which can easily deliver pulses with the desired areas. Of course, the use of femtosecond pulses offers another advantage: decoherence is irrelevant on such time scales.

C. Householder decomposition of a U(N ) propagator
The standard QHR M(v) and the generalized QHR M(v; ϕ) can be used for U(N ) decomposition [7]. Any N -dimensional unitary matrix U (U −1 = U † ) can be expressed as a product of N − 1 standard QHRs M (v n ) (n = 1, 2, . . . N − 1) and a phase gate, as or as a product of N generalized QHRs, (13)

III. TRANSITION BETWEEN PURE STATES
The designed recipe for constructing a general U(N ) transformation makes it possible to solve the important quantum mechanical problem of transfering an N -state quantum system from one arbitrary preselected initial superposition state to another such state, i.e. the inverse problem of quantum-state engineering. The cases of pure and mixed states require separate analyses.
A. Transition by standard QHR

General case
A pure qunit state is described by a state vector |Ψ = N n=1 c n |n , where the vectors |n represent the qunit basis states, and c n is the complex-valued probability amplitude of state |n . Given the preselected initial state |Ψ i and the final state |Ψ f of the qunit, we wish to find a propagator U, such that We shall show that one of the possible solutions of Eq. (14) reads where M(v i ) and M(v f ) are standard QHRs. Here D is an N -dimensional unitary matrix, which, when acting upon a single qunit basis state |n , only shifts its phase, For example, D can be an arbitrary N -dimensional phase gate (11). Alternatively, D can be an arbitrary Ndimensional generalized QHR M (v; ϕ) with vector |v orthogonal to the qunit state |n , v |n = 0. Finally D can be the identity D = I. In order to prove Eq. (15) we first define the vector where |n is an arbitrarily chosen basis qunit state, ϕ αn = arg[Ψ α ] n and α = i, f . The QHR M(v in ) acting upon |Ψ i reflects it onto the single qunit state |n , The action of D upon |n only shifts its phase, see Eq.
(16). The action of M(v f n ) upon |n reflects this vector onto the final state, Equations (18), (16) and (19) imply that which, up to an unimportant phase, proves Eq. (15). The arbitrariness in the choice of the unitary matrix D, and the intermediate basis state |n , means that the solution (15) for U is not unique. However, what is important is that it always exists. In fact the availability of multiple solutions offers some flexibility for a physical implementation. In particular we can always choose D = I; then the physical realization of the propagator U requires only two standard QHRs.

Special cases
In several important special cases only a single standard QHR is needed for pure-to-pure transition.
1. If the qunit is in a single initial basis state |Ψ i = |n then, as follows from Eq.

B. Transition by generalized QHR
Generalized QHR is ideally suited for a pure-to-pure transition because, as it is easily seen, only one generalized QHR is sufficient to reflect state |Ψ i onto |Ψ f , where In comparison with (15) the solution (21) is unique; there is no arbitrariness in the choice of the QHR vector |v (up to an unimportant global phase) and the phase ϕ.

C. Examples
We consider a qutrit (N = 3), for which the QHR implementation is particularly suitable. As a first example, the transition from a single state to a superposition state, is performed by a single QHR M(v), with (24) Figure 2 shows the corresponding time evolutions of the populations and the state mismatch D. The latter is defined as the distance between the qutrit state vector |Ψ(t) and the target state |Ψ f , where ρ mn are the elements of the qutrit density matrix ρ. This definition of D applies to pure and mixed states as well. The behavior of D allows us to verify that not only the populations but also the phases of the probability amplitudes of the target state |Ψ f are produced by the QHR. Indeed, as time progresses, D approaches zero, which implies that |Ψ(t) aligns with |Ψ f . In another example, we transfer a two-state superposition to a three-state superposition, and ϕ = 0.574π. Figure 3 shows the time evolution of the populations and the state mismatch (25) for a standard-QHR implementation, and Fig. 4 for generalized QHR. In both cases, the mismatch D vaniches, indicating the creation of the desired superposition (26). The generalized-QHR implementation is clearly superior because it creates the target state in a single step.
In conclusion of this section, we have demonstrated that any two pure superposition qunit states can be connected by just a single generalized QHR, or by two standard QHRs. This suggests that QHR, and particularly the generalized version, is the most convenient and efficient tool for pure-to-pure state navigation in Hilbert space. We now turn our attention to the problem of connecting two arbitrary mixed states.

IV. COHERENT NAVIGATION BETWEEN MIXED STATES
A mixed qunit state can be described by its density matrix ρ, whose spectral decomposition reads ρ = N n=1 r n |ψ n ψ n | . (29) The eigenvalues r n of ρ satisfy N n=1 r n = 1, and |ψ n are the orthonormalized ( ψ k |ψ n = δ kn ) complex eigenvectors of ρ. The density matrix is hermitian; hence it can be parameterized by N 2 − 1 real parameters.
A hermitian Hamiltonian induces unitary evolution between an initial mixed state ρ i and a final state ρ f , A unitary evolution does not change the eigenvalues {r n } N n=1 , which are therefore dynamic invariants, which is easily seen from Eq. (30) (as an equivalent set of dynamic invariants one can take the set {Trρ n } N n=1 ). Therefore, a unitary propagator U can only connect mixed states with the same set of invariants {r n } N n=1 . In order to connect mixed states with different invariants we need an incoherent process; we shall return to this problem in the next section. Here we shall find the solution to the problem of linking two mixed states with the same invariants.
Because the eigenvalues {r n } N n=1 of ρ i and ρ f are the same, we should have where the unitary matrices R i and R f diagonalize respectively ρ i and ρ f , and ρ 0 =diag{r 1 , r 2 , . . . , r N }. By replacing Eq. (31) into Eq. (30) we find Because D is a unitary matrix we find ρ 0 D = Dρ 0 . Since ρ 0 is diagonal, D must be diagonal too. There are no other restrictions on D; hence D is an arbitrary diagonal matrix. It follows from Eq. (32b) that the solution for the unitary propagator in Eq. (30) is given by The propagator (33) is not unique; it depends on the choice of the diagonal matrix D. In particular, we can always choose D = I. Hence, the transfer between two mixed states requires a general U(N ) propagator. The latter can be expressed as a product of N − 1 standard QHRs M(v n ) (n = 1, 2, . . . , N − 1) and a phase gate Φ (φ 1 , φ 2 , . . . , φ N ), Eq. (12), or by N generalized QHRs M(v n ; ϕ n ) (n = 1, 2, . . . , N ), Eq. (13) [7]. We take as an example a qutrit, with the arbitrarily chosen initial and final density matrices (34b) These density matrices can be connected by the unitary propagator (33) with D = I: U = R f R † i . The latter can be expressed as a product of two standard QHRs and one phase gate or by three generalized QHRs, Figure 5 shows the respective time evolution of the populations and the state mismatch (25) for the generalized-QHR realization (36).
The first QHR M(v 3 ; ϕ 3 ) does not cause population changes because it is in fact a phase gate. As time progresses, the mismatch decreases and the target density matrix (34b) is approached.

V. SYNTHESIS OF ARBITRARY PRESELECTED MIXED STATES
As it was shown in the previous sections, by applying one or more QHRs one can connect any two arbitrary pure states, or two arbitrary mixed states with the same dynamic invariants {r n } N n=1 . Mixed states with different invariants cannot be connected by coherent hermitian evolution because these invarants are constants of motion. Hence in order to connect mixed states with different invariants we need a mechanism with non-hermitian dynamics, which can alter the dynamic invariants.
In this section we shall describe two techniques for engineering an arbitrary mixed state, starting from a single pure state. This is the most interesting special case of the general problem of connecting two arbitrary mixed states, because the initial state can be prepared routinely by optical pumping. Moreover, the general mixedto-mixed problem can be reduced to the single-to-mixed problem by optically pumping the initial mixed state into a single state.
The two techniques use a combination of coherent and incoherent evolutions. The coherent evolution uses QHRs, whereas the incoherent non-hermitian evolution is induced either by pure dephasing or spontaneous emission. We shall consider the two techniques separately.

A. Using dephasing
We assume that the qunit is initially in the single qunit state ρ i = |i i|, and we wish to transform the system to an arbitrary mixed state ρ f . Let us denote the eigenvalues of ρ f by r n (n = 1, 2, . . . , N ). We proceed as follows.
• First, using the prescription from Sec. III, we apply a single QHR to transfer state |i to a pure superposition state, in which the populations are equal to the eigenvalues of ρ f : ρ nn = r n (n = 1, 2, . . . , N ). The phases of this superposition are irrelevant.
• In the second step we switch the dephasing on and let all coherences decay to zero. This can be done, for example, by using phase-fluctuating faroff-resonance laser fields. In the end of this process, the density matrix will be diagonal, with the eigenvalues r n of ρ f on the diagonal, which implies that it will have the same dynamic invariants as ρ f .
• The third step is to connect this intermediate state to the desired state ρ f by a sequence of QHRs, as explained in the previous Sec. IV.
In summary, we need three steps: a single QHR, a dephasing process, and a sequence of N QHRs. Figure  6 shows the evolution of the populations and the state mismatch (25) during the engineering of the mixed state (34b) by the dephasing technique. The first step is the single QHR M(v), with QHR vector which transfers the single initial state |1 to the pure superposition state The second step is the pure dephasing process, which nullifies all coherences and leaves the density matrix in a diagonal form, The third step is a sequence of two generalized QHRs, which transfer ρ 2 into the desired final density matrix ρ f , Eq. (34b). The QHR components read |v 2 = 0, 0.793e 0.740πi , 0.609e 0.025πi T , (40b) ϕ 1 = 0.950π, ϕ 2 = −0.760π. (40c)

B. Using spontaneous emission
In the method, which uses spontaneous emission, we start again in a single qunit state ρ i = |i i|, and the target is the arbitrary mixed state ρ f . The procedure now consists of only two steps: incoherent and coherent. It is particularly well suited for a qutrit, which we shall describe, although it is readily extended to more states. This method requires a closed qunit-ancilla transition; if the ancilla state can decay to other levels then the fidelity will be reduced accordingly.
It is possible here to apply directly the incoherent step, which produces a density matrix with the desired final dynamic invariants, without the need to prepare first a coherent qunit superposition, as in the dephasing method above. The idea is to use laser-induced spontaneous emission from the ancilla excited state to prepare a completely allows for some flexibility in the choice of most convenient path. The physical realization uses an N -pod configuration of N lower states, forming the qunit, and an ancillary upper state. It is particularly convenient for a qutrit, where the N = 3 states are the magnetic sublevels of a J = 1 level and the ancilla state is a J = 0 level. Then only a single tunable laser is needed to provide the necessary polarized laser pulses.
The hermitian part uses a sequence of sets of short coherent laser pulses with appropriate pulse areas and detunings. For each set, the propagator of the N -pod represents a quantum Householder reflection (QHR). A sequence of at most N suitably chosen QHRs can synthesize any desired unitary propagator.
We have shown that two arbitrary preselected pure superposition states can be connected by a single QHR only, because the respective propagator has exactly the QHR symmetry. Two mixed states, with the same set of dynamic invariants, require a general U(N ) transformation, which can be realized by at most N QHRs. This is a significant improvement over the existing setups involving O(N 2 ) operations, which can be crucial in making quantum state engineering and operations with qunits experimentally feasible.
The most general case of two arbitrary mixed states with different dynamic invariants requires an incoherent step, which equalizes the invariants of the initial den-sity matrix to those of the final density matrix. We have demonstrated how this can be done by using pure dephasing or spontaneous decay of the ancillary upper state. Once the invariants are equalized, the problem is reduced to the one of connecting two mixed states with the same invariants, which, as explained above, can be done by at most N QHRs. This method has been described for a qutrit, but it is easily generalized to an arbitrary qunit.
The present results can have important applications in the storage of quantum information. For example, a qubit can encode two continuous parameters: the population ratio of the two qubit states and the relative phase of their amplitudes. A qunit in a pure state can encode 2(N −1) parameters (N −1 populations and N −1 relative phases), i.e. by using qunits information can be encoded in significantly fewer particles than with qubits. Moreover, a mixed qunit state can encode as many as N 2 − 1 real parameters. This may be particularly interesting if the number of particles that can be used is restricted, e.g., due to decoherence [4].