Engineering open quantum systems

The effects of an infinite environment on a finite physical system under interest are usually deciphered with equilibrium or non-equilibrium thermodynamics, where the central concept are temperature and entropy as well as Onsager-type relations. However, there exist more elaborate and accurate methods, which are essential in engineering and control. The pioneering work along this line is represented by Boltzman's equation and his H-theorem. In that treatment, the effects are carefully accounted in terms of scattering. In recent years, some general schemes, such as master equation approach and Langevin approach, have been developed to take care of the real time dynamics of open systems. These methods are widely used in laser design and quantum de-coherence and quantum measurement. The latter delves fundamentally in quantum computing. On the other hand, the coupling to environment may be utilized to good ends. Common sense may say that such couplings are not easily manipulated and usually detrimental to quantum control. However, this situation changes and these couplings becomes tractable and can be engineered. Look at this work [Nature, 470:486(2011)]:

The dynamics of an open quantum system S coupled to an environment E can be described by the unitary transformation , with ρ_SE the joint density matrix of the composite system S + E. Thus, the reduced density operator of the system will evolve as ρ_S = Tr_E(Uρ_SEU^†). The time evolution of the system can also be described by a completely positive Kraus map

with E_k operation elements satisfying , and initially uncorrelated system and environment³¹. If the system is decoupled from the environment, the general map (1) reduces to , with U_S the unitary time evolution operator acting only on the system.

Control of both coherent and dissipative dynamics is then achieved by finding corresponding sequences of maps (1) specified by sets of operation elements {E_k} and engineering these sequences in the laboratory. In particular, for the example of dissipative quantum-state preparation, pumping to an entangled state |ψ reduces to implementing appropriate sequences of dissipative maps. These maps are chosen to drive the system to the desired target state irrespective of its initial state. The resulting dynamics have then the pure state |ψ as the unique attractor, . In quantum optics and atomic physics, the techniques of optical pumping and laser cooling are successfully used for the dissipative preparation of quantum states, although on a single-particle level. The engineering of dissipative maps for the preparation of entangled states can be seen as a generalization of this concept of pumping and cooling in driven dissipative systems to a many-particle context. To be concrete, we focus on dissipative preparation of stabilizer states, which represent a large family of entangled states, including graph states and error-correcting codes³².

We start by outlining the concept of Kraus map engineering for the simplest non-trivial example of ‘pumping’ a system of two qubits into a Bell state. The Hilbert space of two qubits is spanned by the four Bell states defined as and . Here, |0 and |1 denote the computational basis of each qubit, and we use the short-hand notation |00 = |0₁|0₂, for example. These maximally entangled states are stabilizer states: the Bell state |Φ⁺, for instance, is said to be stabilized by the two stabilizer operators Z₁Z₂ and X₁X₂, where X and Z denote the usual Pauli matrices, as it is the only two-qubit state that is an eigenstate of eigenvalue +1 of these two commuting observables, that is, Z₁Z₂|Φ⁺ = |Φ⁺ and X₁X₂|Φ⁺ = |Φ⁺. In fact, each of the four Bell states is uniquely determined as an eigenstate with eigenvalues ±1 with respect to Z₁Z₂ and X₁X₂. The key idea of pumping is that we can achieve dissipative dynamics which pump the system into a particular Bell state, for example , by constructing two dissipative maps, under which the two qubits are irreversibly transferred from the +1 into the −1 eigenspaces of Z₁Z₂ and X₁X₂.

The dissipative maps are engineered with the aid of an ancilla ‘environment’ qubit^{25, 33} and a quantum circuit of coherent and dissipative operations. The form and decomposition of these maps into basic operations are discussed in Box 1. The pumping dynamics are determined by the probability of pumping from the +1 into the −1 stabilizer eigenspaces, which can be directly controlled by varying the parameters in the employed gate operations. For pumping with unit probability (p = 1), the two qubits reach the target Bell state—regardless of their initial state—after only one pumping cycle, that is, by a single application of each of the two maps. In contrast, when the pumping probability is small (p 1), the process can be regarded as the infinitesimal limit of the general map (1). In this case, the system dynamics under a repeated application of the pumping cycle are described by a master equation³⁴:

Here H_S is a system Hamiltonian, and c_k are Lindblad operators reflecting the system–environment coupling. For the purely dissipative maps discussed here, H_S = 0. Quantum jumps from the +1 into the −1 eigenspace of Z₁Z₂ and X₁X₂ are mediated by a set of two-qubit Lindblad operators (see Box 1 for details); here the system reaches the target Bell state asymptotically after many pumping cycles.