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We derive rigorous upper bounds on the distance between quantum states in an open system setting, in terms of the operator norm between the Hamiltonians describing their evolution. We illustrate our results with an example taken from protection against decoherence using dynamical decoupling.
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Shors and Grovers famous quantum algorithms for factoring and searching show that quantum computers can solve certain computational problems significantly faster than any classical computer. We discuss here what quantum computers_cannot_ do, and specifically how to prove limits on their computational power. We cover the main known techniques for proving lower bounds, and exemplify and compare the methods.
Newtonian gravity yields specific observable consequences, the most striking of which is the emergence of a $1/r^2$ force. In so far as communication can arise via such interactions between distant particles, we can ask what would be expected for a theory of gravity that only allows classical communication. Many heuristic suggestions for gravity-induced decoherence have this restriction implicitly or explicitly in their construction. Here we show that communication via a $1/r^2$ force has a minimum noise induced in the system when the communication cannot convey quantum information, in a continuous time analogue to Bells inequalities. Our derived noise bounds provide tight constraints from current experimental results on any theory of gravity that does not allow quantum communication.
We implemented the experiment proposed by Cabello [arXiv:quant-ph/0309172] to test the bounds of quantum correlation. As expected from the theory we found that, for certain choices of local observables, Cirelsons bound of the Clauser-Horne-Shimony-Holt inequality ($2sqrt{2}$) is not reached by any quantum states.
We investigate the theoretical limits of the effect of the quantum interaction distance on the speed of exact quantum addition circuits. For this study, we exploit graph embedding for quantum circuit analysis. We study a logical mapping of qubits and gates of any $Omega(log n)$-depth quantum adder circuit for two $n$-qubit registers onto a practical architecture, which limits interaction distance to the nearest neighbors only and supports only one- and two-qubit logical gates. Unfortunately, on the chosen $k$-dimensional practical architecture, we prove that the depth lower bound of any exact quantum addition circuits is no longer $Omega(log {n})$, but $Omega(sqrt[k]{n})$. This result, the first application of graph embedding to quantum circuits and devices, provides a new tool for compiler development, emphasizes the impact of quantum computer architecture on performance, and acts as a cautionary note when evaluating the time performance of quantum algorithms.
The entanglement content of superpositions of quantum states is investigated based on a measure called {it concurrence}. Given a bipartite pure state in arbitrary dimension written as the quantum superposition of two other such states, we find simple inequalities relating the concurrence of the state to that of its components. We derive an exact expression for the concurrence when the component states are biorthogonal, and provide elegant upper and lower bounds in all other cases. For quantum bits, our upper bound is tighter than the previously derived bound in [Phys. Rev. Lett. 97, 100502 (2006).]
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