Quantum measurement is a fundamental cornerstone of experimental quantum computations. The main issues in current quantum measurement strategies are the high number of measurement rounds to determine a global optimal measurement output and the low success probability of finding a global optimal measurement output. Each measurement round requires preparing the quantum system and applying quantum operations and measurements with high-precision control in the physical layer. These issues result in extremely high-cost measurements with a low probability of success at the end of the measurement rounds. Here, we define a novel measurement for quantum computations called dense quantum measurement. The dense measurement strategy aims at fixing the main drawbacks of standard quantum measurements by achieving a significant reduction in the number of necessary measurement rounds and by radically improving the success probabilities of finding global optimal outputs. We provide application scenarios for quantum circuits with arbitrary unitary sequences, and prove that dense measurement theory provides an experimentally implementable solution for gate-model quantum computer architectures.
The von Neumann measurements are a special case of a more general measurement, the POVM measurement @cite_81 @cite_62 @cite_74 @cite_33 @cite_41 @cite_30 . Without loss of generality, the POVM is a generalized measurement that can be interpreted as a von Neumann measurement that utilizes an additional quantum system (called ancilla). The POVM measurement is mathematically described by a set of positive operators such that their sum is the identity operator @cite_15 @cite_70 @cite_48 . The POVM measurements therefore can be expressed in terms of projective measurements (see also Neumark's dilation theorem @cite_29 @cite_58 @cite_5 ).
@cite_30 @cite_62 @cite_33 @cite_41 @cite_70 @cite_48 @cite_29 @cite_81 @cite_74 @cite_5 @cite_15 @cite_58
1854343180 2649051464 1988174048 2027707976
I construct a positive-operator-valued measure (POVM) which has 2d rank-1 elements and which is informationally complete for generic pure states in d dimensions, thus confirming a conjecture made by Flammia, Silberfarb, and Caves (e-print quant-ph 0404137). I show that if a rank-1 POVM is required to be informationally complete for all pure states in d dimensions, it must have at least 3d-2 elements. I also show that, in a POVM which is informationally complete for all pure states in d dimensions, for any vector there must be at least 2d-1 POVM elements which do not annihilate that vector. It is a fundamental problem to decide how many copies of an unknown mixed quantum state are necessary and sufficient to determine the state. Previously, it was known only that estimating states to error @math in trace distance required @math copies for a @math -dimensional density matrix of rank @math . Here, we give a theoretical measurement scheme (POVM) that requires @math copies to estimate @math to error @math in infidelity, and a matching lower bound up to logarithmic factors. This implies @math copies suffice to achieve error @math in trace distance. We also prove that for independent (product) measurements, @math copies are necessary in order to achieve error @math in infidelity. For fixed @math , our measurement can be implemented on a quantum computer in time polynomial in @math . The most efficient way of obtaining information about the state of a quantum system is not always a direct measurement. It is sometimes preferable to extend the original Hilbert space of states into a larger space, and then to perform a quantum measurement in the enlarged space. Such an extension is always possible, by virtue of Neumark's theorem. The physical interpretation usually given to that theorem is the introduction of an auxiliary quantum system, prepared in a standard state, and the execution of a quantum measurement on both systems together. However, this widespread interpretation is unacceptable, because the statistical properties of the supposedly standard auxiliary system are inseparably entangled with those of the original, unknown system. A different method of preparing the auxiliary system is proposed, and shown to be physically acceptable. The effects of any quantum measurement can be described by a collection of measurement operators l_brace M sub m r_brace acting on the quantum state of the measured system. However, the Hilbert space formalism tends to obscure the relationship between the measurement results and the physical properties of the measured system. In this paper, a characterization of measurement operators in terms of measurement resolution and disturbance is developed. It is then possible to formulate uncertainty relations for the measurement process that are valid for arbitrary input states. The motivation of these concepts is explained from a quantum communication viewpoint. It is shown that the intuitive interpretation of uncertainty as a relation between measurement resolution and disturbance provides a valid description of measurement back action. Possible applications to quantum cryptography, quantum cloning, and teleportation are discussed.