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AIP conference proceedings

How to Derive the Hilbert-Space Formulation of Quantum Mechanics From Purely Operational Axioms

Giacomo Mauro D’Ariano

Published Jan 1, 2006• 18 citations
Publisher

Abstract

In the present paper I show how it is possible to derive the Hilbert space formulation of Quantum Mechanics from a comprehensive definition of physical experiment and assuming experimental accessibility and simplicity as specified by five simple Postulates. This accomplishes the program presented in form of conjectures in the previous paper. Pivotal roles are played by the local observability principle, which reconciles the holism of nonlocality with the reductionism of local observation, and by the postulated existence of informationally complete observables and of a symmetric faithful state. This last notion allows one to introduce an operational definition for the real version of the “adjoint”—i. e. the transposition—from which one can derive a real Hilbert‐space structure via either the Mackey‐Kakutani or the Gelfand‐Naimark‐Segal constructions. Here I analyze in detail only the Gelfand‐Naimark‐Segal construction, which leads to a real Hilbert space structure analogous to that of (classes of generally unbounded) selfadjoint operators in Quantum Mechanics. For finite dimensions, general dimensionality theorems that can be derived from a local observability principle, allow us to represent the elements of the real Hilbert space as operators over an underlying complex Hilbert space (see, however, a still open problem at the end of the paper). The route for the present operational axiomatization was suggested by novel ideas originated from Quantum Tomography.

DOI
10.1063/1.2219356

Related papers

How to Derive the Hilbert-Space Formulation of Quantum Mechanics From Purely Operational Axioms

Giacomo Mauro D’Ariano

In the present paper I show how it is possible to derive the Hilbert space formulation of Quantum Mechanics from a comprehensive definition of "physical experiment" and assuming "experimental accessibility and simplicity" as specified by five simple Postulates. This accomplishes the program presented in form of conjectures in the previous paper quant-ph/0506034. Pivotal roles are played by the "local observability principle", which reconciles the holism of nonlocality with the reductionism of local observation, and by the postulated existence of "informationally complete observables" and of a "symmetric faithful state". This last notion allows one to introduce an operational definition for the real version of the "adjoint"--i. e. the transposition--from which one can derive a real Hilbert-space structure via either the Mackey-Kakutani or the Gelfand-Naimark-Segal constructions. Here I analyze in detail only the Gelfand-Naimark-Segal construction, which leads to a real Hilbert space structure analogous to that of (classes of generally unbounded) selfadjoint operators in Quantum Mechanics. For finite dimensions, general dimensionality theorems that can be derived from a local observability principle, allow us to represent the elements of the real Hilbert space as operators over an underlying complex Hilbert space (see, however, a still open problem at the end of the paper). The route for the present operational axiomatization was suggested by novel ideas originated from Quantum Tomography.

arXiv:quant-ph/0603011•Mar 2, 2006•3 citations

Operational Axioms for Quantum Mechanics

Giacomo Mauro D’Ariano

The mathematical formulation of Quantum Mechanics in terms of complex Hilbert space is derived for finite dimensions, starting from a general definition of physical experiment and from five simple Postulates concerning experimental accessibility and simplicity. For the infinite dimensional case, on the other hand, a C*‐algebra representation of physical transformations is derived, starting from just four of the five Postulates via a Gelfand‐Naimark‐Segal (GNS) construction. The present paper simplifies and sharpens the previous derivation in Ref. [1]. The main ingredient of the axiomatization is the postulated existence of faithful states that allows one to calibrate the experimental apparatus. Such notion is at the basis of the operational definitions of the scalar product and of the transposed of a physical transformation. What is new in the present paper with respect to Ref. [1], is the operational deduction of an involution corresponding to the complex‐conjugation for effects, whose extension to transformations allows to define the adjoint of a transformation when the extension is composition‐preserving. The existence of such composition‐preserving extension among possible extensions is analyzed.

AIP conference proceedings•Jan 1, 2007•41 citations

Operational Axioms for Quantum Mechanics

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The mathematical formulation of Quantum Mechanics in terms of complex Hilbert space is derived for finite dimensions, starting from a general definition of "physical experiment" and from five simple Postulates concerning "experimental accessibility and simplicity". For the infinite dimensional case, on the other hand, a C*-algebra representation of physical transformations is derived, starting from just four of the five Postulates via a Gelfand-Naimark-Segal (GNS) construction. The present paper simplifies and sharpens the previous derivation in version 1. The main ingredient of the axiomatization is the postulated existence of "faithful states" that allows one to calibrate the experimental apparatus. Such notion is at the basis of the operational definitions of the scalar product and of the "transposed" of a physical transformation. What is new in the present paper with respect to quant-ph/0603011 is the operational deduction of an involution corresponding to the "complex-conjugation" for effects, whose extension to transformations allows to define the "adjoint" of a transformation when the extension is composition-preserving.

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Quantum Theory from Hilbert Spaces

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It is not surprising that our language should be incapable of describing processes occurring within atoms, for, as has been remarked, it was invented to describe the experiences of daily life, … Fortunately, mathematics is not subject to this limitation, and it has been possible to invent a mathematical scheme–the quantum theory–which seems entirely adequate for the treatment of atomic processes. Werner Heisenberg, The Physical Principles of the Quantum Theory (1930) In this chapter we introduce the mathematical structure of quantum theory, for a finite number of systems and finite dimension, starting from three simple Hilbert-space postulates, and based on the intuitive notions of state, transformation, and effect, within a causal context. Such notions will be formalized in a general non-causal context in Chapter 3, which is devoted to the informational framework. We will assume the reader has a basic knowledge of matrix analysis and of operators over Hilbert spaces. Comments about the extension to infinite dimensions, continuous outcome-spaces, and infinitely many systems can be found in the notes at the end of the chapter. From the three simple Hilbert-space postulates, the present chapter extracts six principles, which will become the postulates for quantum theory in the axiomatic derivation presented in Part III of the book. In addition, we will take the opportunity to present and explore many relevant features of the theory – including its convex and causal structure, the Choi–Jamiołkowski isomorphism, and relevant notions, such as those of POVM, complete positivity of maps, quantum operation and channels, entangled states, state discrimination, tomography of states, and transformations. The purification principle will open the stage to the dilation theorems of the theory of quantum open systems, along with other consequences, such as the existence of faithful and steering states, the theorem of no information without disturbance, quantum teleportation, and more. We will review the quantum no-cloning and no-programming theorems, and finally provide a thorough derivation of the theory of quantum channel inversion, which is the backbone of the general theory of quantum error correction in quantum information. Additional interesting topics will be classical information theory as a restriction of quantum information theory, and a critical reconsideration of the von Neumann postulate and the notion of observable.

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Where the mathematical structure of Quantum Mechanics coms from

Giacomo Mauro D’Ariano

The mathematical formulation of Quantum Mechanics is derived from purely operational axioms based on a general definition of "experiment" as a set of transformations. The main ingredient of the mathematical construction is the postulated existence of "faithful states" that allows one to calibrate the experimental apparatus. Such notion is at the basis of the operational definitions of the scalar product and of the "adjoint" of a transformation.

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Operational axioms for a C*-algebraic formulation of Quantum Mechanics

Giacomo Mauro D’Ariano

A C*-algebra formulation of Quantum Mechanics is derived from purely operational axioms in which the primary role is played by the "transformations" that the system undergoes in the course of an "experiment". The notion of the {\em adjoint} of a transformation is based on the postulated existence of "faithful states" that allows one to calibrate the experimental apparatus.

arXiv:quant-ph/0701219•Jan 29, 2007•2 citations
In the present paper I show how it is possible to derive the Hilbert space formulation of Quantum Mechanics from a comprehensive definition of physical experiment and assuming experimental accessibility and simplicity as specified by five simple Postulates. This accomplishes the program presented in form of conjectures in the previous paper. Pivotal roles are played by the local observability principle, which reconciles the holism of nonlocality with the reductionism of local observation, and by the postulated existence of informationally complete observables and of a symmetric faithful state. This last notion allows one to introduce an operational definition for the real version of the “adjoint”—i. e. the transposition—from which one can derive a real Hilbert‐space structure via either the Mackey‐Kakutani or the Gelfand‐Naimark‐Segal constructions. Here I analyze in detail only the Gelfand‐Naimark‐Segal construction, which leads to a real Hilbert space structure analogous to that of (classes of generally unbounded) selfadjoint operators in Quantum Mechanics. For finite dimensions, general dimensionality theorems that can be derived from a local observability principle, allow us to represent the elements of the real Hilbert space as operators over an underlying complex Hilbert space (see, however, a still open problem at the end of the paper). The route for the present operational axiomatization was suggested by novel ideas originated from Quantum Tomography.
In the present paper I show how it is possible to derive the Hilbert space formulation of Quantum Mechanics from a comprehensive definition of "physical experiment" and assuming "experimental accessibility and simplicity" as specified by five simple Postulates. This accomplishes the program presented in form of conjectures in the previous paper quant-ph/0506034. Pivotal roles are played by the "local observability principle", which reconciles the holism of nonlocality with the reductionism of local observation, and by the postulated existence of "informationally complete observables" and of a "symmetric faithful state". This last notion allows one to introduce an operational definition for the real version of the "adjoint"--i. e. the transposition--from which one can derive a real Hilbert-space structure via either the Mackey-Kakutani or the Gelfand-Naimark-Segal constructions. Here I analyze in detail only the Gelfand-Naimark-Segal construction, which leads to a real Hilbert space structure analogous to that of (classes of generally unbounded) selfadjoint operators in Quantum Mechanics. For finite dimensions, general dimensionality theorems that can be derived from a local observability principle, allow us to represent the elements of the real Hilbert space as operators over an underlying complex Hilbert space (see, however, a still open problem at the end of the paper). The route for the present operational axiomatization was suggested by novel ideas originated from Quantum Tomography.
The mathematical formulation of Quantum Mechanics in terms of complex Hilbert space is derived for finite dimensions, starting from a general definition of physical experiment and from five simple Postulates concerning experimental accessibility and simplicity. For the infinite dimensional case, on the other hand, a C*‐algebra representation of physical transformations is derived, starting from just four of the five Postulates via a Gelfand‐Naimark‐Segal (GNS) construction. The present paper simplifies and sharpens the previous derivation in Ref. [1]. The main ingredient of the axiomatization is the postulated existence of faithful states that allows one to calibrate the experimental apparatus. Such notion is at the basis of the operational definitions of the scalar product and of the transposed of a physical transformation. What is new in the present paper with respect to Ref. [1], is the operational deduction of an involution corresponding to the complex‐conjugation for effects, whose extension to transformations allows to define the adjoint of a transformation when the extension is composition‐preserving. The existence of such composition‐preserving extension among possible extensions is analyzed.
The mathematical formulation of Quantum Mechanics in terms of complex Hilbert space is derived for finite dimensions, starting from a general definition of "physical experiment" and from five simple Postulates concerning "experimental accessibility and simplicity". For the infinite dimensional case, on the other hand, a C*-algebra representation of physical transformations is derived, starting from just four of the five Postulates via a Gelfand-Naimark-Segal (GNS) construction. The present paper simplifies and sharpens the previous derivation in version 1. The main ingredient of the axiomatization is the postulated existence of "faithful states" that allows one to calibrate the experimental apparatus. Such notion is at the basis of the operational definitions of the scalar product and of the "transposed" of a physical transformation. What is new in the present paper with respect to quant-ph/0603011 is the operational deduction of an involution corresponding to the "complex-conjugation" for effects, whose extension to transformations allows to define the "adjoint" of a transformation when the extension is composition-preserving.
It is not surprising that our language should be incapable of describing processes occurring within atoms, for, as has been remarked, it was invented to describe the experiences of daily life, … Fortunately, mathematics is not subject to this limitation, and it has been possible to invent a mathematical scheme–the quantum theory–which seems entirely adequate for the treatment of atomic processes. Werner Heisenberg, The Physical Principles of the Quantum Theory (1930) In this chapter we introduce the mathematical structure of quantum theory, for a finite number of systems and finite dimension, starting from three simple Hilbert-space postulates, and based on the intuitive notions of state, transformation, and effect, within a causal context. Such notions will be formalized in a general non-causal context in Chapter 3, which is devoted to the informational framework. We will assume the reader has a basic knowledge of matrix analysis and of operators over Hilbert spaces. Comments about the extension to infinite dimensions, continuous outcome-spaces, and infinitely many systems can be found in the notes at the end of the chapter. From the three simple Hilbert-space postulates, the present chapter extracts six principles, which will become the postulates for quantum theory in the axiomatic derivation presented in Part III of the book. In addition, we will take the opportunity to present and explore many relevant features of the theory – including its convex and causal structure, the Choi–Jamiołkowski isomorphism, and relevant notions, such as those of POVM, complete positivity of maps, quantum operation and channels, entangled states, state discrimination, tomography of states, and transformations. The purification principle will open the stage to the dilation theorems of the theory of quantum open systems, along with other consequences, such as the existence of faithful and steering states, the theorem of no information without disturbance, quantum teleportation, and more. We will review the quantum no-cloning and no-programming theorems, and finally provide a thorough derivation of the theory of quantum channel inversion, which is the backbone of the general theory of quantum error correction in quantum information. Additional interesting topics will be classical information theory as a restriction of quantum information theory, and a critical reconsideration of the von Neumann postulate and the notion of observable.