Information In Direct And Indirect Quantum Measurements

Lev B. Levitin Boston University, Boston, USA The information mciency of direct and indirect (generalized) quantum measurements is analyzed. The longstanding controversial problem, whether indirect measurements can yield more information than direct ones, is solved. I t is shown that for any quantum system described by an injinite-dimenswnal separable Hilbert space, the ir$ormations attainable by direct and indirect measurements are exactly equal. The quantum-mechanical generalization of Shannon's measure of information introduced in [ 121 is based on "classical" J. von Neumann's model of quantum measurements. Such measurements are called "direct," since it is assumed that the measuring instrument (classically described) is applied directly to the system whose state we would like to identify (the system that carries information). However, the concept of quantum measurements can be extended by introducing indirect measurements which involve an additional quantum system coupled to the information carrier. Consider a quantum system with an ensemble of macmstates S = (si, pi}, each macrostate q being described by a density matrix ;(')in a Hilbert space H1. Consider an auxilliary quantum system (called in [3] "ancilla") whose state is independent of the state of the system carrying information and described by a density operator i ( O ) in a separable Hilbert space 4. Then the state of the compound system consisting of the two systems together is described by a tensor-product density operator $6) = ,j(i) ~b ~ ( 0 ) in the tensor-product Hilbert space H = HI 60 H2 Denote also i A complete indirect (generalized) measurement performed over the compound system and associated with a certain orthogonal basis K in the space H. Definition 1. Information about the macrostate S of the quantum system for a given state of the ancilia in the outcome of an indirect measurement associated with a basis K in the space HI 63 H2 is the quantity