Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

Trace Formalism for Quantum Mechanical Expectation Values

John Langerholc

Published Aug 1, 1965• 7 citations
Publisher

Abstract

The trace of a positive bounded operator in a Hilbert space is defined and shown to be independent of the basis used in the definition. The trace of the product WA is defined for bounded A and positive bounded W with unit trace and is also shown to be basis-invariant. An integral representation is derived and used to define Tr(WA) for unbounded A. Linearity, isotony, and continuity with respect to uniform covergence are proved for Tr(WA) as a function of (bounded) A. Several formulas used in statistical mechanics are derived, and it is proved that if A and B are commuting positive operators, then Tr(W(A + B)) = Tr(WA) + Tr(WB). A counter example is given which shows that the commonly used definition of Tr(WA) in terms of an orthonormal basis is not invariant under permutation of the basis vectors even in the case of a very simple unbounded operator A. An expectation-value function M which associates the expectation value M(A) to the operator A is assumed given and subject to certain restrictions. For bounded operators, these include positivity, normalization, additivity, and a condition which may be considered as a requirement of regularity for the probability function induced by M. Modifications of these requirements are imposed for unbounded operators, and von Neumann's statistical formula is proved: there is a unique bounded positive operator W with unit trace for which M(A) = Tr(WA). The requirements placed on M are weaker than those of von Neumann, and are in fact satisfied by Tr(WA) as a function of A.

DOI
10.1063/1.1704762

Related papers

THE PROBLEM OF ENTANGLEMENT OF QUANTUM STATES

On Matem, Atica Argentina

G.A. RAGGIO Abstract. We give a brief and incomplete survey of the problem of entan- glement of of composite quantum systems. 1960's - 1980's there were serious attempts (all of them anticipated by J. von Neu- mann) to do away with the underlying Hilbert space and work directly with the abstract algebraic structure modeled by C � -algebras or W � -algebras (abstract von Neumann algebras)(1, 2, 3). This was particularly fruitful when dealing with sys- tems of infinitely many degrees of freedom (quantum fields, thermodynamic limits, etc.). The problem of entanglement which we want to address here can be formu- lated quite straightforwardly in this algebraic framework of quantum theory. But we will stick to the quantum mechanics of the 1930's and keep the Hilbert space. Mainly because the results which are available concern the finite dimensional case. The states of the quantum system specified by H are associated with the lin- ear functionals f : B(H) → C which are positive (i.e., f(a) ≥ 0 for all positive a ∈ B(H)), normalized (i.e., f(1) = 1) and normal (i.e., f(supn{an}) = supnf(an) for any increasing family of selfadjoint operators {an : n = 1,2, � � � } which is uni- formly bounded). It is this normality condition which guarantees that the are in one-to-one correspondence with positive trace class operators d of unit trace (density operators) via the formula B(H) ∋ a 7→ tr(da). This gives an extremely convenient representation of and one often confuses the state f as a linear functional with the associated density operator d for which f(a) = tr(da). States are automatically continuous, and satisfy f(a � ) = f(a). The complex number f(a) is interpreted probabilistically as the expected value of the observable associated with the operator a when the system is in the state f. The normalization condi- tion is thus necessary for the consistency of this interpretation and the normality condition is seen as a non-commutative version of the �-additivity of probability measures. Clearly states, which we will denote by �(H), form a convex set which is closed

•Jan 1, 2008•0 citations

A probabilistic formulation of quantum theory

Alexander Bach

By means of integration in the Hilbert space H which is associated to a quantum mechanical system we demonstrate that the expectation value of quantum mechanics, EW(A) =tr(WA), can be expressed as EW(A) =ℱH dμW(φ) (1/2) 〈φ‖A‖φ〉 Here μW denotes a probability measure in H which is determined by the statistical operator. We discuss this hidden variables representation of quantum mechanics.

Journal of Mathematical Physics•Apr 1, 1980•34 citations

Quantum Information Theory

Yorick Hardy, Willi-Hans Steeb

The concepts of classical information theory can be extended to quantum information theory. Since in general measurement yields a result with probability, we may suggest using these probabilities in classical information theory. However the probabilities do not contain phase information, which cannot be neglected. Thus the definitions are given in terms of the density operator. These probabilities depend on the basis used for measurement. A density operator ρ over a n-dimensional Hilbert space H is a positive operator with unit trace. The trace tr(A) is defined as $$tr(A): = \sum\limits_{j = 1}^n {\left\langle {{\beta _j}|A|{\beta _j}} \right\rangle } $$ where β j for j = 1,..., n is any orthonormal basis in HThus tr(P)=1. The eigenvalues of a density operator are greater than zero. By the spectral theorem every density operator can be represented as a mixture of pure states $$\rho = \sum\limits_{j = 1}^n {Pj|} \left\langle {aj} \right.\langle aj|$$ where α j for j = 1,..., n are the orthonormal eigenvectors of ρ (which form a basis in H and $$pj \in R,{p_j}0,\sum\limits_{j = 1}^n {{p_j}} = 1$$

Birkhäuser Basel eBooks•Jan 1, 2001•8 citations

On Extension of Quantum Channels and Operations to the Space of Relatively Bounded Operators

Максим Евгеньевич Широков

We analyse possibility to extend a quantum operation (sub-unital normal completely positive linear map on the algebra $$\mathfrak{B}(\mathcal{H})$$ of bounded operators on a separable Hilbert space $$\mathcal{H}$$ ) to the space of all operators on $$\mathcal{H}$$ relatively bounded w.r.t. a given positive unbounded operator. We show that a quantum operation $$\Phi$$ can be uniquely extended to a bounded linear operator on the Banach space of all $$\sqrt{G}$$ -bounded operators on $$\mathcal{H}$$ provided that the operation $$\Phi$$ is $$G$$ -limited: the predual operation $$\Phi_{*}$$ maps the set of positive trace class operators $$\rho$$ with finite value of $$\textrm{Tr}\rho G$$ into itself. Assuming that $$G$$ has discrete spectrum of finite multiplicity we prove that for a wide class of quantum operations the existence of the above extension implies the $$G$$ -limited property. Applications to the theory of Bosonic Gaussian channels are considered.

arXiv:1903.06086•Lobachevskii Journal of Mathematics•Apr 1, 2020•7 citations

All quantum expectation values as classical statistical mean values

Antonio Cassa

Given a physical quantum system described by a Hilbert H, for any bounded quantum observable (a bounded self-adjoint operator) T it is possible to define several ''hidden observable'' functions f:H->R associated to T and for any quantum mixed state (a density matrix) D it is possible to define several ''hidden mixed states'' (probability measures) m on H associated to D in such a way that the following equality is verified: Trace[ b(T). D] =integral[b(f(psi)).dm(psi) whatever is the continuous function b:R->R. This formula gives a general way to express any expectation value computable in a quantum theory as a classical statistical mean value.

arXiv:0706.2603•Jun 18, 2007•0 citations

The fidelity of density operators in an operator-algebraic framework

Douglas Farenick, Samuel Jaques, Mizanur Rahaman

Josza’s definition of fidelity [R. Jozsa, J. Mod. Opt. 41(12), 2315–2323 (1994)] for a pair of (mixed) quantum states is studied in the context of two types of operator algebras. The first setting is mainly algebraic in that it involves unital C∗-algebras A that possess a faithful trace functional τ. In this context, the role of quantum states (that is, density operators) in the classical quantum-mechanical framework is assumed by positive elements ρ ∈ A for which τ(ρ) = 1. The second setting is more operator theoretic: by fixing a faithful normal semifinite trace τ on a semifinite von Neumann algebra M, we define and consider the fidelity of pairs of positive operators in M of unit trace. The main results of this paper address monotonicity and preservation of fidelity under the action of certain trace-preserving positive linear maps of A or of the predual M∗. Our results in the von Neumann algebra setting are novel in that we focus on the Schrödinger picture rather than the Heisenberg picture, and they also yield a new proof of a theorem of Molnár [Rep. Math. Phys. 48(3), 299–303 (2001)] on the structure of fidelity-preserving quantum channels on the trace-class operators.

arXiv:1602.08177•Journal of Mathematical Physics•Oct 1, 2016•6 citations
The trace of a positive bounded operator in a Hilbert space is defined and shown to be independent of the basis used in the definition. The trace of the product WA is defined for bounded A and positive bounded W with unit trace and is also shown to be basis-invariant. An integral representation is derived and used to define Tr(WA) for unbounded A. Linearity, isotony, and continuity with respect to uniform covergence are proved for Tr(WA) as a function of (bounded) A. Several formulas used in statistical mechanics are derived, and it is proved that if A and B are commuting positive operators, then Tr(W(A + B)) = Tr(WA) + Tr(WB). A counter example is given which shows that the commonly used definition of Tr(WA) in terms of an orthonormal basis is not invariant under permutation of the basis vectors even in the case of a very simple unbounded operator A. An expectation-value function M which associates the expectation value M(A) to the operator A is assumed given and subject to certain restrictions. For bounded operators, these include positivity, normalization, additivity, and a condition which may be considered as a requirement of regularity for the probability function induced by M. Modifications of these requirements are imposed for unbounded operators, and von Neumann's statistical formula is proved: there is a unique bounded positive operator W with unit trace for which M(A) = Tr(WA). The requirements placed on M are weaker than those of von Neumann, and are in fact satisfied by Tr(WA) as a function of A.
G.A. RAGGIO Abstract. We give a brief and incomplete survey of the problem of entan- glement of of composite quantum systems. 1960's - 1980's there were serious attempts (all of them anticipated by J. von Neu- mann) to do away with the underlying Hilbert space and work directly with the abstract algebraic structure modeled by C � -algebras or W � -algebras (abstract von Neumann algebras)(1, 2, 3). This was particularly fruitful when dealing with sys- tems of infinitely many degrees of freedom (quantum fields, thermodynamic limits, etc.). The problem of entanglement which we want to address here can be formu- lated quite straightforwardly in this algebraic framework of quantum theory. But we will stick to the quantum mechanics of the 1930's and keep the Hilbert space. Mainly because the results which are available concern the finite dimensional case. The states of the quantum system specified by H are associated with the lin- ear functionals f : B(H) → C which are positive (i.e., f(a) ≥ 0 for all positive a ∈ B(H)), normalized (i.e., f(1) = 1) and normal (i.e., f(supn{an}) = supnf(an) for any increasing family of selfadjoint operators {an : n = 1,2, � � � } which is uni- formly bounded). It is this normality condition which guarantees that the are in one-to-one correspondence with positive trace class operators d of unit trace (density operators) via the formula B(H) ∋ a 7→ tr(da). This gives an extremely convenient representation of and one often confuses the state f as a linear functional with the associated density operator d for which f(a) = tr(da). States are automatically continuous, and satisfy f(a � ) = f(a). The complex number f(a) is interpreted probabilistically as the expected value of the observable associated with the operator a when the system is in the state f. The normalization condi- tion is thus necessary for the consistency of this interpretation and the normality condition is seen as a non-commutative version of the �-additivity of probability measures. Clearly states, which we will denote by �(H), form a convex set which is closed
The concepts of classical information theory can be extended to quantum information theory. Since in general measurement yields a result with probability, we may suggest using these probabilities in classical information theory. However the probabilities do not contain phase information, which cannot be neglected. Thus the definitions are given in terms of the density operator. These probabilities depend on the basis used for measurement. A density operator ρ over a n-dimensional Hilbert space H is a positive operator with unit trace. The trace tr(A) is defined as tr(A):=∑j=1n⟨βj∣A∣βj⟩tr(A): = \sum\limits_{j = 1}^n {\left\langle {{\beta _j}|A|{\beta _j}} \right\rangle } tr(A):=j=1∑n​⟨βj​∣A∣βj​⟩ where β j for j = 1,..., n is any orthonormal basis in HThus tr(P)=1. The eigenvalues of a density operator are greater than zero. By the spectral theorem every density operator can be represented as a mixture of pure states ρ=∑j=1nPj∣⟨aj⟨aj∣\rho = \sum\limits_{j = 1}^n {Pj|} \left\langle {aj} \right.\langle aj|ρ=j=1∑n​Pj∣⟨aj⟨aj∣ where α j for j = 1,..., n are the orthonormal eigenvectors of ρ (which form a basis in H and pj∈R,pj0,∑j=1npj=1pj \in R,{p_j}0,\sum\limits_{j = 1}^n {{p_j}} = 1pj∈R,pj​0,j=1∑n​pj​=1
We analyse possibility to extend a quantum operation (sub-unital normal completely positive linear map on the algebra B(H)\mathfrak{B}(\mathcal{H})B(H) of bounded operators on a separable Hilbert space H\mathcal{H}H ) to the space of all operators on H\mathcal{H}H relatively bounded w.r.t. a given positive unbounded operator. We show that a quantum operation Φ\PhiΦ can be uniquely extended to a bounded linear operator on the Banach space of all G\sqrt{G}G​ -bounded operators on H\mathcal{H}H provided that the operation Φ\PhiΦ is GGG -limited: the predual operation Φ∗\Phi_{*}Φ∗​ maps the set of positive trace class operators ρ\rhoρ with finite value of TrρG\textrm{Tr}\rho GTrρG into itself. Assuming that GGG has discrete spectrum of finite multiplicity we prove that for a wide class of quantum operations the existence of the above extension implies the GGG -limited property. Applications to the theory of Bosonic Gaussian channels are considered.
Given a physical quantum system described by a Hilbert H, for any bounded quantum observable (a bounded self-adjoint operator) T it is possible to define several ''hidden observable'' functions f:H->R associated to T and for any quantum mixed state (a density matrix) D it is possible to define several ''hidden mixed states'' (probability measures) m on H associated to D in such a way that the following equality is verified: Trace[ b(T). D] =integral[b(f(psi)).dm(psi) whatever is the continuous function b:R->R. This formula gives a general way to express any expectation value computable in a quantum theory as a classical statistical mean value.
Josza’s definition of fidelity [R. Jozsa, J. Mod. Opt. 41(12), 2315–2323 (1994)] for a pair of (mixed) quantum states is studied in the context of two types of operator algebras. The first setting is mainly algebraic in that it involves unital C∗-algebras A that possess a faithful trace functional τ. In this context, the role of quantum states (that is, density operators) in the classical quantum-mechanical framework is assumed by positive elements ρ ∈ A for which τ(ρ) = 1. The second setting is more operator theoretic: by fixing a faithful normal semifinite trace τ on a semifinite von Neumann algebra M, we define and consider the fidelity of pairs of positive operators in M of unit trace. The main results of this paper address monotonicity and preservation of fidelity under the action of certain trace-preserving positive linear maps of A or of the predual M∗. Our results in the von Neumann algebra setting are novel in that we focus on the Schrödinger picture rather than the Heisenberg picture, and they also yield a new proof of a theorem of Molnár [Rep. Math. Phys. 48(3), 299–303 (2001)] on the structure of fidelity-preserving quantum channels on the trace-class operators.