Multipartite quantum nonlocality under local decoherence

We study the nonlocal properties of two-qubit maximally entangled and $N$-qubit Greenberger-Horne-Zeilinger states under local decoherence. We show that the (non)resilience of entanglement under local depolarization or dephasing is not necessarily equivalent to the (non)resilience of Bell-inequality violations. Apart from entanglement and Bell-inequality violations, we consider also nonlocality as quantified by the nonlocal content of correlations and provide several examples of anomalous behaviors, both in the bipartite and multipartite cases. In addition, we study the practical implications of these anomalies on the usefulness of noisy Greenberger-Horne-Zeilinger states as resources for nonlocality-based physical protocols given by communication complexity problems.

Figure 1

Entanglement (lower dashed curves) and maximal value of the CHSH polynomial ${\mathcal{I}}_{\text{CHSH}}$ (upper solid curves), for maximally entangled two-qubit states under independent phase damping of noise strength $p$ (in red, upper solid and lower dashed curves) and independent depolarization of noise strength $p/2$ (in blue, middle two curves), as a function of $p$. The horizontal black dashed line represents the local bound ${\mathcal{I}}_{\text{CHSH}}^{\text{L}}=2$, below which there is no more CHSH violation. Decoherence naturally drives the system to dephased states that possess less entanglement than depolarized states but that, at the same time, violate the CHSH inequality more.

Figure 2

Critical probabilities for full separability (in red), distillability (in blue), and violation of the Mermin inequality ($p_c$, in black), for $N$-qubit GHZ states under independent depolarization of strength $p$. Below the red curve the states are entangled; they are distillable though only below the blue curve [11]. Between the red and the blue curves the states are thus bound entangled. The total entanglement (not plotted) decreases exponentially with $N$. Below the black curve in turn the states violate the Mermin inequality—and they do it exponentially with $N$. Hence, for $N>11$, and between the black and the blue curves, decoherence naturally drives the system to states with exponentially small and bound entanglement that yet violate the inequality exponentially.

Figure 3

Distributed computing scenario [8]. $N$ distant users receive each a two-bit input string $\{x_i,y_i\}$, with $1\le i\le N$. Distributed bits $y_i$ are chosen randomly between 1 and $-1$, and each $x_i$ is chosen as 0 or 1 depending on a joint probability distribution $Q(x_1,...,x_N)$. The users are endowed with some preestablished shared correlations, but they can only exchange a restricted amount of public communication. The problem is, for each and all of them, to compute the value $f(x_1,...,x_n,y_1,...,y_n)$ of a given function $f$ with some probability. The minimum number of bits that they must broadcast to do so defines the communicational complexity of the problem. We consider here a specific subclass of such problems with $Q(x_1,...,x_N)\equiv \left|g\right(x_1,...,x_N\left)\right|/{\sum }_{x_1,...,x_N=0}^1\left|g(x_1,...,x_N)\right|$, for $g(x_1,...,x_N)$ some real-valued function, $f$ a boolean function of the form $f=y_1\cdots y_{n}S\left[g(x_1,...,x_n)\right]$, with $f=\pm 1$ and $S\left[g\right]\equiv g/\left|g\right|=\pm 1$ being the sign function, and where each user is allowed to broadcast only a single bit. For this subclass, it was shown in Ref. [9], for a broad family of protocols, that if the shared correlations violate the associated Bell inequality (5) the users can solve the problem with a higher probability than with any classical protocol (assisted by LHV correlations).

Figure 4

Quantum gain $G_Q(p,N)$—for odd $N$—(blue circles) in the efficiency to solve CCPs with $\varrho \left(p\right)$, lower and upper bounds—also for odd $N$—to the nonlocal content ${\tilde{p}}_{\text{NL}}$ in $\varrho \left(p\right)$ (black crosses) and negativity (solid red) in the most robust bipartitions of $\varrho \left(p\right)$, for $p=0.1$ and as a function of $N$. For $3\le N\le 7$ the quantum gain displays a region of growth. However, for large $N$, it very rapidly converges to the universal upper bound ${(1-p)}^N$, and so do all other curves.