A non-distillability criterion for secret correlations

Within entanglement theory there are criteria which certify that some quantum states cannot be distilled into pure entanglement. An example is the positive partial transposition criterion. Here we present, for the first time, the analogous thing for secret correlations. We introduce a computable criterion which certifies that a probability distribution between two honest parties and an eavesdropper cannot be (asymptotically) distilled into a secret key. Nothing is known about the existence of non-distillable correlations with positive secrecy cost, also known as bound information. This criterion may be the key for finding such correlations, but we ignore whether this is the case. If this is not the case, and hence the criterion is useless, this implies a very interesting result: Any correlation with positive secrecy cost can increase the secrecy content of another. In other words, all correlations with positive secrecy cost constitute a useful resource.