$\mathcal{PT}$-symmetric Hamiltonians and their application in quantum information

We discuss the prospect of $\mathcal{PT}$-symmetric Hamiltonians finding applications in quantum information science, and conclude that such evolution is unlikely to provide any benefit over existing techniques. Although it has been known for some time that $\mathcal{PT}$-symmetric quantum theory, when viewed as a unitary theory, is exactly equivalent to standard quantum mechanics, proposals continue to be put forward for schemes in which $\mathcal{PT}$-symmetric quantum theory can outperform standard quantum theory. The most recent of these is the suggestion to use $\mathcal{PT}$-symmetric Hamiltonians to perform an exponentially fast database search, a task known to be impossible with a quantum computer. Further, such a scheme has been shown to apparently produce effects in conflict with fundamental information-theoretic principles, such as the impossibility of superluminal information transfer, and the invariance of entanglement under local operations. In this paper we propose three inequivalent experimental implementations of $\mathcal{PT}$-symmetric Hamiltonians, with careful attention to the resources required to realize each such evolution. Such an operational approach allows us to resolve these apparent conflicts, and evaluate fully schemes proposed in the literature for faster time evolution and state discrimination.