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Optics and Spectroscopy

Entangling quantum measurements

B. A. Grishanin, Victor N. Zadkov

Published Jun 8, 2004• 3 citations
Publisher

Abstract

The concept of an entangling measurement as the most general form of ideal quantum measurement is introduced. The mathematical definition of an entangling measurement is given in the form of a superoperator transformation, which projects the initial states of an object-apparatus system on the corresponding entangled states appearing after a measurement. The mathematical properties of the superoperator of an entangling measurement are considered. The spectrum of superoperator eigenstates is studied analytically on the example of a two-level system. The algebraic relation ℳ 2 =ℳ AM between the superoperators of entangling and standard measurements is ascertained. The coherent information obtained in the course of measurement is analyzed and the relevant potentialities of entangling measurements are discussed.

DOI
10.1134/1.1753631

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The concept of an entangling measurement as the most general form of ideal quantum measurement is introduced. The mathematical definition of an entangling measurement is given in the form of a superoperator transformation, which projects the initial states of an object-apparatus system on the corresponding entangled states appearing after a measurement. The mathematical properties of the superoperator of an entangling measurement are considered. The spectrum of superoperator eigenstates is studied analytically on the example of a two-level system. The algebraic relation ℳ 2 =ℳ AM between the superoperators of entangling and standard measurements is ascertained. The coherent information obtained in the course of measurement is analyzed and the relevant potentialities of entangling measurements are discussed.
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The concept of generalized quantum measurement is introduced as a transformation that sets a one-to-one correspondence between the initial states of the measured object system and final states of the object-meter system with the help of a classical informational index, unambiguously linked to a classically compatible set of quantum states. It is shown that the generalized quantum measurement concept covers all key types of quantum measurement---standard projective, entangling, fuzzy, and generalized measurements with a partial or complete destruction of initial information associated with the object. A special class of soft quantum measurements as a basic model for the fuzzy measurements widespread in physics is introduced and its information properties are studied in detail. Also, a special class of partially destructive measurements mapping all states of the Hilbert space of a finite-dimensional quantum system onto the basis states of an infinite-dimensional quantum system is considered.
A concept of the generalized quantum measurement is introduced as the transformation, which establishes a correspondence between the initial states of the object system and final states of the object--measuring device (meter) system with the help of a classical informational index, unambiguously linked to the classically compatible set of states of the object--meter system. It is shown that the generalized measurement covers all the key known quantum measurement concepts--standard projective, entangling, fuzzy and the generalized measurement with the partial or complete destruction of the initial information contained in the object. A special class of partially-destructive measurements that map the continual set of the states in finite-dimensional quantum systems to that one of the infinite-dimensional quantum systems is considered. Their informational essence and some information characteristics are discussed in detail.
We use the term ‘measurement’ to refer to the interaction between an object and an apparatus on the basis of which information concerning the initial state of the object may be obtained from information on the resulting state of the apparatus. The quantum theory of measurement is a quantum theoretic investigation of such interactions in order to analyse the correlations between object and apparatus that measurement must establish. Although there is a sizeable literature on quantum measurements there appear to be just two sorts of interactions that have been employed. There are the ‘disturbing’ interactions consistent with the analysis of Landau and Peierls (8) as developed by Pauli (11) and by Landau and Lifshitz (7), and there are the ‘non-disturbing’ interactions explicitly set out by von Neumann ((10), chs. 5, 6), and that dominate the literature. In this paper we shall investigate the most general types of interactions that could possibly constitute measurements and provide a precise mathematical characterization (section 2). We shall then examine an interesting subclass, corresponding to Landau's ideas, that contains both of the above sorts of measurements (section 3). Finally, we shall discuss von Neumann measurements explicitly and explore the purported limitations suggested by Wigner(12) and Araki and Yanase (2). We hope, in this way, to provide a comprehensive basis for discussions of quantum measurements.
We analyze a quantum measurement where the apparatus is initially in a mixed state. We show that the amount of information gained in a measurement is not equal to the amount of entanglement between the system and the apparatus, but is instead equal to the degree of classical correlations between the two. As a consequence, we derive an uncertainty-like expression relating the information gain in the measurement and the initial mixedness of the apparatus. Final entanglement between the environment and the apparatus is also shown to be relevant for the efficiency of the measurement.