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arXiv:quant-ph/9705045 · Physical Review A

Dissipation and decoherence in a quantum register

Paolo Zanardi A

Published May 1, 1998• 123 citations
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Abstract

A model for a quantum register $\mathcal{R}$ made of $N$ replicas of a $d$-dimensional quantum system (cell) coupled with the environment is studied by means of a Born-Markov master equation (ME). Dissipation and decoherence are discussed in various cases in which a subdecoherent encoding can be rigorously found. For the quantum-bit case $(d=2)$ we have solved, for small $N,$ the ME by numerical direct integration and studied, as a function of the coherence length ${\ensuremath{\xi}}_{c}$ of the bath, fidelity and decoherence rates of states of the register. For large enough ${\ensuremath{\xi}}_{c}$ the singlet states of the global su(2) pseudo spin algebra of the register (noiseless at ${\ensuremath{\xi}}_{c}=\ensuremath{\infty})$ are shown to have a much smaller decoherence rates than the rest of the Hilbert space.

DOI
10.1103/physreva.57.3276

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A model for a quantum register R\mathcal{R}R made of NNN replicas of a ddd-dimensional quantum system (cell) coupled with the environment is studied by means of a Born-Markov master equation (ME). Dissipation and decoherence are discussed in various cases in which a subdecoherent encoding can be rigorously found. For the quantum-bit case (d=2)(d=2)(d=2) we have solved, for small N,N,N, the ME by numerical direct integration and studied, as a function of the coherence length \ensuremathξc{\ensuremath{\xi}}_{c}\ensuremathξc​ of the bath, fidelity and decoherence rates of states of the register. For large enough \ensuremathξc{\ensuremath{\xi}}_{c}\ensuremathξc​ the singlet states of the global su(2) pseudo spin algebra of the register (noiseless at \ensuremathξc=\ensuremath∞){\ensuremath{\xi}}_{c}=\ensuremath{\infty})\ensuremathξc​=\ensuremath∞) are shown to have a much smaller decoherence rates than the rest of the Hilbert space.
A model for a quantum register dissipatively coupled with a bosonic thermal bath is studied. The register consists of NNN quantum bits (qubits) the bath is described by Nb{N}_{b}Nb​ bosonic modes. The register-bath coupling is chosen in such a way that the total number of excitations is conserved. The Hilbert space splits, allowing the study of the dynamics separately in each sector. Assuming that the coupling with the bath is the same for all qubits, the excitation sectors have a further decomposition according the irreducible representations of the su(2) spin algebra. The stability against environment-generated noise of the information encoded in a quantum state of the register depends on its SU(2) symmetry content. At zero temperature we find that states belonging to the vacuum symmetry sector have, for long time, vanishing fidelity, whereas each lowest spin vector is decoupled from the bath and therefore is decoherence free. Numerical results are shown in the one-excitation space in the case of qubit-dependent bath-system coupling.
In this paper we study a model quantum register RRR made of NNN replicas (cells) of a given finite-dimensional quantum system S. Assuming that all cells are coupled with a common environment with equal strength we show that, for NNN large enough, in the Hilbert space of RRR there exists a linear subspace CN{C}_{N}CN​ which is dynamically decoupled from the environment. The states in CN{C}_{N}CN​ evolve unitarily and are therefore decoherence-dissipation free. The space CN{C}_{N}CN​ realizes a noiseless quantum code in which information can be stored, in principle, for an arbitrarily long time without being affected by errors.
The dynamical evolution of a quantum register of arbitrary length coupled to an environment of arbitrary coherence length is predicted within a relevant model of decoherence. The results are reported for quantum bits (qubits) coupling individually to different environments (``independent decoherence'') and qubits interacting collectively with the same reservoir (``collective decoherence''). In both cases, explicit decoherence functions are derived for any number of qubits. The decay of the coherences of the register is shown to strongly depend on the input states: We show that this sensitivity is a characteristic of both types of coupling (collective and independent) and not only of the collective coupling, as has been reported previously. A nontrivial behavior (``recoherence'') is found in the decay of the off-diagonal elements of the reduced density matrix in the specific situation of independent decoherence. Our results lead to the identification of decoherence-free states in the collective decoherence limit. These states belong to subspaces of the system's Hilbert space that do not get entangled with the environment, making them ideal elements for the engineering of ``noiseless'' quantum codes. We also discuss the relations between decoherence of the quantum register and computational complexity based on the dynamical results obtained for the register density matrix.
In order to eliminate disturbing effects of decoherence, encoding of quantum information in decoherence-free subspaces has been suggested. We analyze the benefits of this concept for a quantum register that is realized in a spin chain in contact with a common bosonic bath. Within a dissipationless model, we provide explicit analytical results for the average fidelity of plain and encoded quantum registers. For the investigation of dissipative spin-boson couplings, we employ a master equation of Bloch-Redfield type.
We present a rigorous analysis of the phenomenon of decoherence for general N -level quantum systems coupled a reservoir modelled by a thermal bosonic quantum environment. We present an explicit form of the dominant reduced dynamics of open systems. We give explicit results for a spin 1/2 (qubit), including decoherence and thermalization times. Our approach is based on a dynamical theory of quantum resonances. It yields the exact reduced dynamics of the small system and does not involve master equation or van Hove limit approximations. This approach is suitable for a wide variety of systems which are not explicitly solvable, including systems of interacting spins (registers of interacting qubits), for which the coupling between the system and the environment is fixed but small.