The modern tools of quantum mechanics (A tutorial on quantum states, measurements, and operations)

Abstract

This tutorial is devoted to review the modern tools of quantum mechanics, which are suitable to describe states, measurements, and operations of realistic, not isolated, systems in interaction with their environment, and with any kind of measuring and processing devices. We underline the central role of the Born rule and and illustrate how the notion of density operator naturally emerges, together the concept of purification of a mixed state. In reexamining the postulates of standard quantum measurement theory, we investigate how they may formally generalized, going beyond the description in terms of selfadjoint operators and projective measurements, and how this leads to the introduction of generalized measurements, probability operator-valued measures (POVM) and detection operators. We then state and prove the Naimark theorem, which elucidates the connections between generalized and standard measurements and illustrates how a generalized measurement may be physically implemented. The "impossibility" of a joint measurement of two non commuting observables is revisited and its canonical implementations as a generalized measurement is described in some details. Finally, we address the basic properties, usually captured by the request of unitarity, that a map transforming quantum states into quantum states should satisfy to be physically admissible, and introduce the notion of complete positivity (CP). We then state and prove the Stinespring/Kraus-Choi-Sudarshan dilation theorem and elucidate the connections between the CP-maps description of quantum operations, together with their operator-sum representation, and the customary unitary description of quantum evolution. We also address transposition as an example of positive map which is not completely positive, and provide some examples of generalized measurements and quantum operations.

Tutorial. 26 pages, 1 figure. Published in a special issue of EPJ - ST devoted to the memory of Federico Casagrande

References in corpus (5)

• Experimental Quantum State Tomography of Optical Fields and Ultrafast Statistical Sampling
• Unknown Quantum States and Operations, a Bayesian View
• Universal Uncertainty Principle in the Measurement Operator Formalism
• Covariant Mappings for the Description of Measurement, Dissipation and Decoherence in Quantum Mechanics
• Guide to Mathematical Concepts of Quantum Theory

Cited by in corpus (25)

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• Flexible resources for quantum metrology
• Computing partial traces and reduced density matrices
• Ultimate Limits of Thermal Pattern Recognition
• The canonical Naimark extension for the Pauli quantum roulette wheel
• Qubit systems subject to unbalanced random telegraph noise: quantum correlations, non-Markovianity and teleportation
• Reality variation under monitoring with weak measurements
• Naimark extension for the single-photon canonical phase measurement
• An effective iterative method to build the Naimark extension of rank-n POVMs
• Defect production due to time-dependent coupling to environment in the Lindblad equation
• Quantum computation using arrays of N polar molecules in pendular states
• Effects of the free evolution in the Arthurs-Kelly model of simultaneous measurement and in the retrodictive predictions of the Heisenberg uncertainty relations
• Employing weak-field homodyne detection for quantum communications
• A dynamical model for Positive-Operator Valued Measures
• Quantum Coding with Finite Thermodynamic Resources
• Quantum communications in continuous variable systems
• Creation of quantum coherence with general measurement processes
• Activation of entanglement in generalized entanglement swapping
• Deterministic carving of quantum states with Grover's algorithm
• Experimental demonstration of generalized quantum fluctuation theorems in the presence of coherence
• The data aggregation problem in quantum hypothesis testing
• Nonlocal Variable-Strength Measurements of N Qubits Using GHZ-like Entanglement
• Detection of the genuine non-locality of any three-qubit state