Individual clicks can be used to unambiguously compare quantum devices

Imagine we are given a pair of unknown quantum devices and our goal is to decide whether they operate in the same way, or not. A natural solution would be to completely reconstruct the action of both devices separately and then compare the results. However, it turns out that in quantum case such comparison procedure is very inefficient. It requires enormous amount of experimental and computational resources. On the other hand since the comparison is a binary decision problem it seems to be possible to design an experiment such that individual outcomes can provide us with unambiguous error-free conclusions.

One of the basic lessons of quantum theory is that predictions and conclusions we can make are intrinsically nondeterministic. This feature gives the quantum theory the hallmark of a statistical theory describing large ensembles of systems. Because of their randomness, the individual outcomes of the experiment are considered to be of a very little use and sense. However, with the development of quantum information theory and related experimental progress the individual experimental clicks are becoming of interest and potential use. Under very specific circumstances the uncertainty of predictions of individual outcomes and conlusions made out of individual clicks can be reduced to zero. QAP researchers have investigated whether this is the case for the problem of comparison of quantum devices.

The research was focused on the comparison of unitary channels [1] and projective measurements [2]. These specific families of quantum processes are of particular interest in quantum information processing. The projective measurements represent the final read-out stage of quantum information processing and both of them can be used to perform individual steps of quantum algorithms (quantum gates). The statistical nature of quantum theory implies that the perfect comparison of unitary channels and projective measurements is not possible. However, if we allow imperfections in the form of inconclusive outcomes, than we can make an experiment unambiguously concluding the difference. It turns out it is impossible to unambiguously conclude that unknown unitary channels, or unknown projective measurements, are identical. In summary, it is possible to design a two-outcome comparison experiment with one inconclusive outcome and one outcome impliying with certainty that the devices are different.

Optimal solutions using a minimal number of usages of the unknown devices are illustrated on figures. Let us note that the concept of unknown measurement apparatus can be interpreted in two ways. Either its outcomes are labeled (known), or not. We found that for measurements with unlabeled outcomes each of the devices must be employed at least twice. Otherwise, for labeled measurement devices and for unitary channels, a single usage of each apparatus is sufficient to make the unambiguous conlusion. Moreover, the frequency of inclonclusive outcome serves as an operational measure of the difference between the devices.

Fig.1 Comparison of unitary channels. Optimal experiment consists of the preparation of a bipartite state ρ_asym having support on the antisymmetric subspace and of the symmetry measurement S deciding whether the state is symmetric, or antisymmetric. The outcome symmetric is used to unambiguously conclude that the unitary channels A and B are different. The average success probability equals P=(d+1)/(2d), where d is the dimension of the system.

Fig.2 Comparison of labeled projective measurements. Optimal experiment consists of the preparation of a bipartite state ρ_asym. If each of the measurement apparatuses A and B gives the same outcome, then we can unambiguously conclude that they are different. This happens with the average success probability P=1/d, where d is the dimension of the system.

Fig.3 Comparison of unlabeled projective measurements. The optimal solution was found only for the case of qubit observables and consists of the preparation of a specific four-qubit state φ. In this case each of the measurement apparatuses A and B is used twice. If the pair of outcomes observed on one of the apparatuses are the same, whereas the ones observed on the second measurement device are different, then we can unambiguously conclude that A and B are different. For qubits the optimal average success probability equals P=4/9.

References:

M.Sedlák, M.Ziman: Unambiguous comparison of unitary channels, Phys.Rev.A 79, 012303 (2009) [arXiv:0809.4401]
M.Ziman, T.Heinosaari, M.Sedlák: Unambiguous comparison of quantum measurements, Phys. Rev. A 80, 052102 (2009) [arXiv:0905.4445]