Entanglement properties of multipartite entangled states under the influence of decoherence

We investigate entanglement properties of multipartite states under the influence of decoherence. We show that the lifetime of (distillable) entanglement for GHZ-type superposition states decreases with the size of the system, while for a class of other states -namely all graph states with constant degree- the lifetime is independent of the system size. We show that these results are largely independent of the specific decoherence model and are in particular valid for all models which deal with individual couplings of particles to independent environments, described by some quantum optical master equation of Lindblad form. For GHZ states, we derive analytic expressions for the lifetime of distillable entanglement and determine when the state becomes fully separable. For all graph states, we derive lower and upper bounds on the lifetime of entanglement. To this aim, we establish a method to calculate the spectrum of the partial transposition for all mixed states which are diagonal in a graph state basis. We also consider entanglement between different groups of particles and determine the corresponding lifetimes as well as the change of the kind of entanglement with time. This enables us to investigate the behavior of entanglement under re-scaling and in the limit of large (infinite) number of particles. Finally we investigate the lifetime of encoded quantum superposition states and show that one can define an effective time in the encoded system which can be orders of magnitude smaller than the physical time. This provides an alternative view on quantum error correction and examples of states whose lifetime of entanglement (between groups of particles) in fact increases with the size of the system.Comment: 27 pages, 11 figure

A Quantum to Classical Phase Transition in Noisy Quantum Computers

The fundamental problem of the transition from quantum to classical physics is usually explained by decoherence, and viewed as a gradual process. The study of entanglement, or quantum correlations, in noisy quantum computers implies that in some cases the transition from quantum to classical is actually a phase transition. We define the notion of entanglement length in $d$-dimensional noisy quantum computers, and show that a phase transition in entanglement occurs at a critical noise rate, where the entanglement length transforms from infinite to finite. Above the critical noise rate, macroscopic classical behavior is expected, whereas below the critical noise rate, subsystems which are macroscopically distant one from another can be entangled. The macroscopic classical behavior in the super-critical phase is shown to hold not only for quantum computers, but for any quantum system composed of macroscopically many finite state particles, with local interactions and local decoherence, subjected to some additional conditions. This phenomenon provides a possible explanation to the emergence of classical behavior in such systems. A simple formula for an upper bound on the entanglement length of any such system in the super-critical phase is given, which can be tested experimentally.Comment: 15 pages. Latex2e plus one figure in eps fil

Stability of macroscopic entanglement under decoherence

We investigate the lifetime of macroscopic entanglement under the influence of decoherence. For GHZ-type superposition states we find that the lifetime decreases with the size of the system (i.e. the number of independent degrees of freedom) and the effective number of subsystems that remain entangled decreases with time. For a class of other states (e.g. cluster states), however, we show that the lifetime of entanglement is independent of the size of the system.Comment: 5 pages, 1 figur

Finite-time quantum-to-classical transition for a Schroedinger-cat state

The transition from quantum to classical, in the case of a quantum harmonic oscillator, is typically identified with the transition from a quantum superposition of macroscopically distinguishable states, such as the Schr\"odinger cat state, into the corresponding statistical mixture. This transition is commonly characterized by the asymptotic loss of the interference term in the Wigner representation of the cat state. In this paper we show that the quantum to classical transition has different dynamical features depending on the measure for nonclassicality used. Measures based on an operatorial definition have well defined physical meaning and allow a deeper understanding of the quantum to classical transition. Our analysis shows that, for most nonclassicality measures, the Schr\"odinger cat dies after a finite time. Moreover, our results challenge the prevailing idea that more macroscopic states are more susceptible to decoherence in the sense that the transition from quantum to classical occurs faster. Since nonclassicality is prerequisite for entanglement generation our results also bridge the gap between decoherence, which appears to be only asymptotic, and entanglement, which may show a sudden death. In fact, whereas the loss of coherences still remains asymptotic, we have shown that the transition from quantum to classical can indeed occur at a finite time.Comment: 9+epsilon pages, 4 figures, published version. Originally submitted as "Sudden death of the Schroedinger cat", a bit too cool for APS policy :-

Quantum speedup of classical mixing processes

Most approximation algorithms for #P-complete problems (e.g., evaluating the permanent of a matrix or the volume of a polytope) work by reduction to the problem of approximate sampling from a distribution $\pi$ over a large set $\S$. This problem is solved using the {\em Markov chain Monte Carlo} method: a sparse, reversible Markov chain $P$ on $\S$ with stationary distribution $\pi$ is run to near equilibrium. The running time of this random walk algorithm, the so-called {\em mixing time} of $P$, is $O(\delta^{-1} \log 1/\pi_*)$ as shown by Aldous, where $\delta$ is the spectral gap of $P$ and $\pi_*$ is the minimum value of $\pi$. A natural question is whether a speedup of this classical method to $O(\sqrt{\delta^{-1}} \log 1/\pi_*)$, the diameter of the graph underlying $P$, is possible using {\em quantum walks}. We provide evidence for this possibility using quantum walks that {\em decohere} under repeated randomized measurements. We show: (a) decoherent quantum walks always mix, just like their classical counterparts, (b) the mixing time is a robust quantity, essentially invariant under any smooth form of decoherence, and (c) the mixing time of the decoherent quantum walk on a periodic lattice $\Z_n^d$ is $O(n d \log d)$, which is indeed $O(\sqrt{\delta^{-1}} \log 1/\pi_*)$ and is asymptotically no worse than the diameter of $\Z_n^d$ (the obvious lower bound) up to at most a logarithmic factor.Comment: 13 pages; v2 revised several part