We show that any unitary transformation performed on the quantum state of a closed quantum system describes an inner, reversible, generalized quantum measurement. We also show that under some specific conditions it is possible to perform a unitary transformation on the state of the closed quantum system by means of a collection of generalized measurement operators. In particular, given a complete set of orthogonal projectors, it is possible to implement a reversible quantum measurement that preserves the probabilities. In this context, we introduce the concept of "Truth-Observable", which is the physical counterpart of an inner logical truth.
We show that any unitary transformation performed on the quantum state of a closed quantum system describes an inner, reversible, generalized quantum measurement. We also show that under some specific conditions it is possible to perform a unitary transformation on the state of the closed quantum system by means of a collection of generalized measurement operators. In particular, given a complete set of orthogonal projectors, it is possible to implement a reversible quantum measurement that preserves the probabilities. In this context, we introduce the concept of "Truth-Observable", which is the physical counterpart of an inner logical truth.
A measurement process is logically reversible if the premeasurement density operator of the measured system is uniquely determined from the postmeasurement density operator and the outcome of the measurement. This paper analyzes the necessary and sufficient condition for a measurement process to be logically reversible and discusses specific examples on quantum-nondemolition measurements, quantum counting, and measurement of spin systems. It is shown that for any sharp measurement we can construct a logically reversible measurement that continuously approaches the sharp measurement with a decrease in the measurement error. A general condition for a measurement process to be reversed by another with a nonzero probability of success is given, and the implications of such physical reversibility are discussed. \textcopyright{} 1996 The American Physical Society.
To begin with, it is pointed out that the form of the quantum probabil- ity formula originates in the very initial state of the object system as seen when the state is expanded with the eigen-projectors of the measured ob- servable. Making use of the probability reproducibility condition, which is a key concept in unitary measurement theory, one obtains the relevant coher- ent distribution of the complete-measurement results in the final unitary- measurement state in agreement with the mentioned probability formula. Treating the transition from the final unitary, or premeasurement, state, where all possible results are present, to one complete-measurement result sketchily in the usual way, the well-known probability formula is derived. In conclusion it is pointed out that the entire argument is only formal unless one makes it physical assuming that the quantum probability law is valid in the extreme case of probability-one (certain) events (projectors).
We propose a linear algebraic method, named Eigenlogic, for two-, many-valued and fuzzy logic using observables in Hilbert space. All logical connectives are represented by observables where the truth values correspond to eigenvalues and the atomic input propositional cases, i.e. the “models” of a propositional system, to the respective eigenvectors. In this way propositional logic can be formalized by using combinations of tensored elementary quantum observables. The outcome of a “measurement” of a logical observable will give the truth value of the associated logical proposition, and becomes “interpretable” when applied to vectors of its eigenspace, leading to an original insight into the quantum measurement postulate.Recently the concept of “quantum predicate” has been proposed in leading to similar concepts.We develop logical observables for binary logic and extend them to many-valued logic.For binary logic and truth values {0, 1} logical observables are commuting projector operators.For truth values {+1,−1} the logical observables are isometries formally equivalent to the ones of a composite quantum spin 12 system, these observables are reversible quantum logic gates.The analogy of many-valued logic with quantum angular momentum is then established using a general algebraic method, based on classical interpolation framework suggested by the finite-elementmethod. Logical observables for three-valued logic are formulated using the orbital angular momentum observable Lz with ` = 1. The representative 3-valued 2-argument logical observables for theternary threshold logical connectives Min and Max are then explicitly obtained.Also in this approach fuzzy logic arises naturally when considering vectors outside the eigensystem.The fuzzy membership function is obtained by the quantum mean value (Born rule) of the logical projector observable and turns out to be a probability measure. Fuzziness arises because ofthe quantum superposition of atomic propositional cases, the truth of a proposition being in this case a probabilistic value ranging from completely false to completely true.This method could be employed for developing algorithms in high-dimensional vector spaces for example in modern semantic theories, such as distributional semantics or in connectionist modelsof cognition. For practical implementation, due to the exponential growth of the vector space dimension, adapted logical reduction methods must then be used. LSA (Latent Semantic Analysis)algorithms are often used in quantum-like approaches, this was done using the HAL (Hyperspace Analogue to Language) algorithm.Our approach is also of interest for quantum computation because several of the observables in Eigenlogic are well-known quantum gates (for example CONTROL-Z) and other ones can be derived by unitary transformations. Ternary-logic quantum gates using qu-trits lead to less complex circuits, our formulation of multi-valued logical observables could help the development of new multi-valued quantum gate architectures.
Traditionally, quantum measurements have been associated with irreversible transformations of pure states into mixtures. In this paper we show that this old hypothesis is redundant. A simple model is constructed in which the measuring process is described by a continuous unitary transformation of the state vector of the observed system and the apparatus. The final state is pure, but appears to be a mixture if we ignore the degrees of freedom of the apparatus. This property is most strikingly illustrated by reformulating the problem in the Heisenberg picture: The state vector is fixed, and the dynamical variables are changed by the measurement.
We present a complete classification of all possible sets of classical reversible gates acting on bits, in terms of which reversible transformations they generate, assuming swaps and ancilla bits are available for free. Our classification can be seen as the reversible-computing analogue of Post's lattice, a central result in mathematical logic from the 1940s. It is a step toward the ambitious goal of classifying all possible quantum gate sets acting on qubits. Our theorem implies a linear-time algorithm (which we have implemented), that takes as input the truth tables of reversible gates G and H, and that decides whether G generates H. Previously, this problem was not even known to be decidable. The theorem also implies that any n-bit reversible circuit can be "compressed" to an equivalent circuit, over the same gates, that uses at most 2^n*poly(n) gates and O(1) ancilla bits; these are the first upper bounds on these quantities known, and are close to optimal. Finally, the theorem implies that every non-degenerate reversible gate can implement either every reversible transformation, or every affine transformation, when restricted to an "encoded subspace." Briefly, the theorem says that every set of reversible gates generates either all reversible transformations on n-bit strings (as the Toffoli gate does); no transformations; all transformations that preserve Hamming weight (as the Fredkin gate does); all transformations that preserve Hamming weight mod k for some k; all affine transformations (as the Controlled-NOT gate does); all affine transformations that preserve Hamming weight mod 2 or mod 4, inner products mod 2, or a combination thereof; or a previous class augmented by a NOT or NOTNOT gate. Ruling out the possibility of additional classes, not in the list, requires some arguments about polynomials, lattices, and Diophantine equations.
This chapter is concerned with the measure of quantum states . This requires one to introduce the subtle notion of quantum measurement , an operation that has no counterpart in the classical domain. To this effect, we first need to develop some new tools, starting with Dirac notation , a formalism that is not only very elegant but is relatively simple to handle. The introduction of Dirac notation makes it possible to become familiar with the inner product for quantum states, as well as different properties for operators and states concerning projection , change of basis , unitary transformations , matrix elements , similarity transformations , eigenvalues and eigenstates , spectral decomposition and diagonal representation , matrix trace and density operator or matrix . The concept of density matrix makes it possible to provide a very first and brief hint of the analog of Shannon's entropy in the quantum world, referred to as von Neumann's entropy , to be further developed in Chapter 21. Once we have all the required tools, we can focus on quantum measurement and analyze three different types referred to as basis-state measurements , projection or von Neumann measurements , and POVM measurements. In particular, POVM measurements are shown to possess a remarkable property of unambiguous quantum state discrimination (UQSD), after which it is possible to derive “absolutely certain” information from unknown system states. The more complex case of quantum measurements in composite systems described by joint or tensor states is then considered.