scholarly journals An operational construction of the sum of two non-commuting observables in quantum theory and related constructions

2020 ◽  
Vol 110 (12) ◽  
pp. 3197-3242
Author(s):  
Nicolò Drago ◽  
Sonia Mazzucchi ◽  
Valter Moretti
Keyword(s):  
Quantum Theory ◽  
Space Structure ◽  
Measuring Instruments ◽  
Spectral Measures ◽  
Feynman Path ◽  
Operational Interpretation ◽  
Real Linear ◽  
Adjoint Operators ◽  
Class Of Functions ◽  
Incompatible Observables

AbstractThe existence of a real linear space structure on the set of observables of a quantum system—i.e., the requirement that the linear combination of two generally non-commuting observables A, B is an observable as well—is a fundamental postulate of the quantum theory yet before introducing any structure of algebra. However, it is by no means clear how to choose the measuring instrument of a general observable of the form $$aA+bB$$ a A + b B ($$a,b\in {{\mathbb {R}}}$$ a , b ∈ R ) if such measuring instruments are given for the addends observables A and B when they are incompatible observables. A mathematical version of this dilemma is how to construct the spectral measure of $$f(aA+bB)$$ f ( a A + b B ) out of the spectral measures of A and B. We present such a construction with a formula which is valid for general unbounded self-adjoint operators A and B, whose spectral measures may not commute, and a wide class of functions $$f: {{\mathbb {R}}}\rightarrow {{\mathbb {C}}}$$ f : R → C . In the bounded case, we prove that the Jordan product of A and B (and suitably symmetrized polynomials of A and B) can be constructed with the same procedure out of the spectral measures of A and B. The formula turns out to have an interesting operational interpretation and, in particular cases, a nice interplay with the theory of Feynman path integration and the Feynman–Kac formula.

scholarly journals A characterization of singular packing subspaces with an application to limit-periodic operators

2017 ◽  
Vol 29 (1) ◽  
Author(s):  
Silas L. Carvalho ◽  
César R. de Oliveira
Keyword(s):  
Packing Dimension ◽  
Spectral Measures ◽  
Adjoint Operators ◽  
Periodic Operators

AbstractA new characterization of the singular packing subspaces of general bounded self-adjoint operators is presented, which is used to show that the set of operators whose spectral measures have upper packing dimension equal to one is a

scholarly journals Bounds on the power of proofs and advice in general physical theories

2016 ◽  
Vol 472 (2190) ◽  
pp. 20160076 ◽  
Author(s):  
Ciarán M. Lee ◽  
Matty J. Hoban
Keyword(s):  
Quantum Theory ◽  
Quantum Structure ◽  
Interactive Proof ◽  
Interactive Proof Systems ◽  
Proof Systems ◽  
Local Measurements ◽  
General Physical ◽  
Adjoint Operators ◽  
Physical Features ◽  
Physical Principles

Quantum theory presents us with the tools for computational and communication advantages over classical theory. One approach to uncovering the source of these advantages is to determine how computation and communication power vary as quantum theory is replaced by other operationally defined theories from a broad framework of such theories. Such investigations may reveal some of the key physical features required for powerful computation and communication. In this paper, we investigate how simple physical principles bound the power of two different computational paradigms which combine computation and communication in a non-trivial fashion: computation with advice and interactive proof systems. We show that the existence of non-trivial dynamics in a theory implies a bound on the power of computation with advice. Moreover, we provide an explicit example of a theory with no non-trivial dynamics in which the power of computation with advice is unbounded. Finally, we show that the power of simple interactive proof systems in theories where local measurements suffice for tomography is non-trivially bounded. This result provides a proof that Q M A is contained in P P , which does not make use of any uniquely quantum structure—such as the fact that observables correspond to self-adjoint operators—and thus may be of independent interest.

Non-self-adjoint operators as observables in quantum theory and nuclear physics

2010 ◽  
Vol 41 (4) ◽  
pp. 508-530 ◽  
Author(s):  
V. S. Olkhovsky ◽  
S. P. Maydanyuk ◽  
E. Recami
Keyword(s):  
Quantum Theory ◽  
Nuclear Physics ◽  
Adjoint Operators

scholarly journals On spectra and Brown's spectral measures of elements in free products of matrix algebras

2008 ◽  
Vol 103 (1) ◽  
pp. 77
Author(s):  
Junsheng Fang ◽  
Don Hadwin ◽  
Xiujuan Ma
Keyword(s):  
Free Product ◽  
Von Neumann Algebra ◽  
Single Point ◽  
Diagonal Operator ◽  
Spectral Measures ◽  
Free Products ◽  
Hyperinvariant Subspace ◽  
Von Neumann ◽  
Matrix Algebras ◽  
Adjoint Operators

We compute spectra and Brown measures of some non self-adjoint operators in $(M_2(\mathsf {C}), \frac{1}{2}\mathrm{Tr})*(M_2(\mathsf{C}), \frac{1}{2}\mathrm{Tr})$, the reduced free product von Neumann algebra of $M_2(\mathsf {C})$ with $M_2(\mathsf {C})$. Examples include $AB$ and $A+B$, where $A$ and $B$ are matrices in $(M_2(\mathsf {C}), \frac{1}{2}\mathrm{Tr})*1$ and $1*(M_2(\mathsf {C}), \frac{1}{2}\mathrm{Tr})$, respectively. We prove that $AB$ is an R-diagonal operator (in the sense of Nica and Speicher [12]) if and only if $\mathrm{Tr}(A)=\mathrm{Tr}(B)=0$. We show that if $X=AB$ or $X=A+B$ and $A,B$ are not scalar matrices, then the Brown measure of $X$ is not concentrated on a single point. By a theorem of Haagerup and Schultz [9], we obtain that if $X=AB$ or $X=A+B$ and $X\neq \lambda 1$, then $X$ has a nontrivial hyperinvariant subspace affiliated with $(M_2(\mathsf{C}), \frac{1}{2}\mathrm{Tr})*(M_2(\mathsf{C}), \frac{1}{2}\mathrm{Tr})$.

scholarly journals Suppressing measurement uncertainty in an inhomogeneous spin star system

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Saeed Haddadi ◽  
Mehrdad Ghominejad ◽  
Ahmad Akhound ◽  
Mohammad Reza Pourkarimi
Keyword(s):  
Quantum Theory ◽  
Lower Bound ◽  
Measurement Uncertainty ◽  
Thermal Regime ◽  
Thermal Evolution ◽  
Star System ◽  
Uncertainty Bound ◽  
Inhomogeneity Parameter ◽  
Inhomogeneous Spin ◽  
Incompatible Observables

AbstractThe uncertainty principle is known as a foundational element of quantum theory, providing a striking lower bound to quantify our prediction for the measured result of two incompatible observables. In this work, we study the thermal evolution of the entropic uncertainty bound in the presence of quantum memory for an inhomogeneous four-qubit spin-star system that is in the thermal regime. Intriguingly, our results show that the entropic uncertainty bound can be controlled and suppressed by adjusting the inhomogeneity parameter of the system.

scholarly journals On time derivatives for and : formal 1D calculations

2013 ◽  
Vol 35 (2) ◽  
Author(s):  
Salvatore De Vincenzo
Keyword(s):  
Fisher Information ◽  
Spatial Derivative ◽  
Mean Values ◽  
Probability Current ◽  
Boundary Terms ◽  
The Mean ◽  
Adjoint Operators ◽  
Quantum Force ◽  
Derivatives Of ◽  
Class Of Functions

We present formal 1D calculations of the time derivatives of the mean values of the position (x) and momentum (p) operators in the coordinate representation. We call these calculations formal because we do not care for the appropriate class of functions on which the involved (self-adjoint) operators and some of its products must act. Throughout the paper, we examine and discuss in detail the conditions under which two pairs of relations involving these derivatives (which have been previously published) can be formally equivalent. We show that the boundary terms present in d{x}/dt and d{x}/dt can be written so that they only depend on the values taken there by the probability density, its spatial derivative, the probability current density and the external potential V= V9 (x) V = V(x). We also show that d(p)/dt is equal to -dv /dx=(FQ) plus a boundary term (Fq = aQ/ax)is the quantum force and Q is the Bohm's quantum potential). We verify that (fq) is simply obtained by evaluating a certain quantity on each end of the interval containing the particle and by subtracting the two results. That quantity is precisely proportional to the integrand of the so-called Fisher information in some particular cases. We have noted that fQ has a significant role in situations in which the particle is confined to a region, even if V is zero inside that region.

scholarly journals BLIND QUANTUM COMPUTATION

2006 ◽  
Vol 04 (05) ◽  
pp. 883-898 ◽  
Author(s):  
PABLO ARRIGHI ◽  
LOUIS SALVAIL
Keyword(s):  
Quantum Theory ◽  
Quantum Computation ◽  
Security Analysis ◽  
Input Output ◽  
Random Input ◽  
Efficient Procedure ◽  
Computational Resources ◽  
Class Of Functions ◽  
Blind Quantum Computation

We investigate the possibility of having someone carry out the work of executing a function for you, but without letting him learn anything about your input. Say Alice wants Bob to compute some known function f upon her input x, but wants to prevent Bob from learning anything about x. The situation arises for instance if client Alice has limited computational resources in comparison with mistrusted server Bob, or if x is an inherently mobile piece of data. Could there be a protocol whereby Bob is forced to compute ,f(x)blindly, i.e. without observing x? We provide such a blind computation protocol for the class of functions which admit an efficient procedure to generate random input–output pairs, e.g. factorization. The cheat-sensitive security achieved relies only upon quantum theory being true. The security analysis carried out assumes the eavesdropper performs individual attacks.

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