To the Editor

Entangled quantum states can be classified into two types: those that can be distilled1 into pure entangled states using local operations and classical communication and those that cannot2. Undistillable entangled states are called bound entangled. A recent claim of experimental bound entanglement3 by Amselem and Bourennane is unfounded. In their work, they aimed to produce the Smolin state4,

in the polarization of four photons, where A to D label the parties and |Ψμ〉 are the Bell states, |ϕ±〉 = (1/√2)(|00〉 ± |11〉) and |Ψ±〉 = (1/√2)(|01〉 ± |10〉). This state is entangled and undistillable4 and is, therefore, bound entangled. Undistillability for the experimental implementation of this state is guaranteed if the partially transposed density matrix is non-negative across each two–two bipartite cut, (AB):(CD), (AC):(BD) and (AD):(BC) (ref. 2). In other words, one considers the partial transposition of the density matrix across each cut and calculates the eigenvalues. If all of the eigenvalues are positive across these three cuts, the state is positive under partial transposition (PPT) and undistillable.

Amselem and Bourennane demonstrated entanglement in their state using a witness5. They applied the PPT test and reported five negative eigenvalues across the three cuts: one instance of −0.02 ± 0.02 and four instances of −0.01 ± 0.01, where the uncertainty is assumed to be one standard deviation.

Just one negative eigenvalue means undistillability has not been demonstrated. Considering the uncertainties and assuming normal distributions, the probability that any one of these five eigenvalues is non-negative is only ½ erfc(1/√2) ≈ 0.159. Although this is already insufficient evidence of undistillability, if we assume the eigenvalues are uncorrelated, the probability that all five are non-negative is (0.159)5, about 1 in 10,000. Furthermore, Amselem and Bourennane also report eleven 0 eigenvalues with uncertainties and another fourteen that are positive by one standard deviation; under the same assumptions, these lower the probability that all eigenvalues are non-negative by an additional factor of 20,000. Correlations could change these estimates considerably, but they cannot be calculated from the published data.

The Smolin state in equation (1) is not an experimentally robust form of bound entanglement. The state and, most importantly, its partial transpose are not full rank and thus the PPT condition is very sensitive to experimental errors. More robust forms of bound entanglement can be created using states with full- or nearly full-rank partial transposes. This approach was used for the observation of pseudo-bound entanglement in liquid-state NMR6 and for the first experimental demonstrations of bound entanglement using optics7 and trapped ions8. Recently, a different approach was used to demonstrate bound entanglement in continuous-variable quantum optics9.

Undistillability is the one property that makes an entangled state bound. Amselem and Bourennane's results fail to show undistillability and thus they cannot claim to have produced bound entanglement.