scholarly journals Programmability of covariant quantum channels

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 488
Author(s):  
Martina Gschwendtner ◽  
Andreas Bluhm ◽  
Andreas Winter
Keyword(s):  
Hilbert Space ◽  
Lower Bounds ◽  
Group Action ◽  
Space Dimension ◽  
Upper And Lower Bounds ◽  
Quantum Channels ◽  
Covariant Quantum ◽  
Finite Dimensional ◽  
Channel Input ◽  
Quantum Processor

A programmable quantum processor uses the states of a program register to specify one element of a set of quantum channels which is applied to an input register. It is well-known that such a device is impossible with a finite-dimensional program register for any set that contains infinitely many unitary quantum channels (Nielsen and Chuang's No-Programming Theorem), meaning that a universal programmable quantum processor does not exist. The situation changes if the system has symmetries. Indeed, here we consider group-covariant channels. If the group acts irreducibly on the channel input, these channels can be implemented exactly by a programmable quantum processor with finite program dimension (via teleportation simulation, which uses the Choi-Jamiolkowski state of the channel as a program). Moreover, by leveraging the representation theory of the symmetry group action, we show how to remove redundancy in the program and prove that the resulting program register has minimum Hilbert space dimension. Furthermore, we provide upper and lower bounds on the program register dimension of a processor implementing all group-covariant channels approximately.

scholarly journals Computing the exponent of a Lebesgue space

2020 ◽  
Vol 12 ◽  
Author(s):  
Timothy McNicholl
Keyword(s):  
Lower Bounds ◽  
Lebesgue Space ◽  
Random Variables ◽  
Upper And Lower Bounds ◽  
Effective Theory ◽  
Finite Dimensional ◽  
Stable Random Variables ◽  
Uniform Solution

We consider the question as to whether the exponent of a computably presentable Lebesgue space whose dimension is at least 2 must be computable.  We show this very natural conjecture is true when the exponent is at least 2 or when the space is finite-dimensional.  However, we also show there is no uniform solution even when given upper and lower bounds on the exponent.  The proof of this result leads to some basic results on the effective theory of stable random variables.  

scholarly journals Informationally restricted quantum correlations

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 332 ◽  
Author(s):  
Armin Tavakoli ◽  
Emmanuel Zambrini Cruzeiro ◽  
Jonatan Bohr Brask ◽  
Nicolas Gisin ◽  
Nicolas Brunner
Keyword(s):  
Hilbert Space ◽  
Lower Bounds ◽  
Quantum Communication ◽  
Space Dimension ◽  
Quantum Correlations ◽  
Strong Correlations ◽  
Classical Communication ◽  
Accessible Information ◽  
Classical Correlations ◽  
The One

Quantum communication leads to strong correlations, that can outperform classical ones. Complementary to previous works in this area, we investigate correlations in prepare-and-measure scenarios assuming a bound on the information content of the quantum communication, rather than on its Hilbert-space dimension. Specifically, we explore the extent of classical and quantum correlations given an upper bound on the one-shot accessible information. We provide a characterisation of the set of classical correlations and show that quantum correlations are stronger than classical ones. We also show that limiting information rather than dimension leads to stronger quantum correlations. Moreover, we present device-independent tests for placing lower bounds on the information given observed correlations. Finally, we show that quantum communication carrying log⁡d bits of information is at least as strong a resource as d-dimensional classical communication assisted by pre-shared entanglement.

scholarly journals The Minimum Numbers for Certain Positive Operators

2020 ◽  
Vol 12 (5) ◽  
pp. 15
Author(s):  
Ching-Yun Suen
Keyword(s):  
Hilbert Space ◽  
Lower Bounds ◽  
Positive Operators ◽  
Upper And Lower Bounds ◽  
Numerical Radius ◽  
Minimum Numbers

In this paper we give upper and lower bounds of the infimum of k  such that kI+2ReT⊗Sm  is positive, where Sm  is the m×m  matrix whose entries are all 0’s except on the superdiagonal where they are all 1’s and T∈BH  for some Hilbert space H. When T  is self-adjoint, we have the minimum of k. When m=3  and T∈B(H)  , we obtain the minimum of k  and an inequality Involving the numerical radius w(T) .

9.—Bivariational Bounds associated with Non-self-adjoint Linear Operators

1976 ◽  
Vol 75 (2) ◽  
pp. 109-118 ◽  
Author(s):  
Michael F. Barnsley ◽  
Peter D. Robinson
Keyword(s):  
Hilbert Space ◽  
Lower Bounds ◽  
Positive Constant ◽  
Real Hilbert Space ◽  
Linear Operators ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Inner Product ◽  
Upper And Lower Bounds ◽  
Fredholm Integral Equations

SYNOPSISLet A be a closed linear transformation from a real Hilbert space ℋ, with symmetric inner product 〈, 〉, into itself; and let f ∈ ℋ be given such that the problem Aø = f has a solution ø ∈ D(A), the domain of A. Then bivariational upper and lower bounds on 〈g, ø〉 for any g ∈ ℋ are exhibited when there exists a positive constant a such that 〈AΦ, AΦ⊖ ≧ a2〈Φ, Φ〉 for all Φ ∈ D(A). The applicability of the theory both to Fredholm integral equations and also to time-dependent diffusion equations is demonstrated.

The Ranks of the Homotopy Groups of a Finite Dimensional Complex

2013 ◽  
Vol 65 (1) ◽  
pp. 82-119
Author(s):  
Yves Félix ◽  
Steve Halperin ◽  
Jean-Claude Thomas
Keyword(s):  
Error Bound ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Homotopy Groups ◽  
Finite Dimensional ◽  
Simply Connected

AbstractLet X be an n-dimensional, finite, simply connected CWcomplex and setWhen 0 < αX < ∞, we give upper and lower bounds for for k sufficiently large. We also show for any r that αX can be estimated from the integers rk πi (X), i ≤ nr with an error bound depending explicitly on r.

Complementary bivariational principles for linear problems involving non-self-adjoint operators

1980 ◽  
Vol 86 (1-2) ◽  
pp. 115-128 ◽  
Author(s):  
R. J. Cole
Keyword(s):  
Hilbert Space ◽  
Integral Equation ◽  
Linear Operator ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Fredholm Type ◽  
Linear Problems ◽  
Adjoint Operators

SynopsisBivariational principles are constructed that yield upper and lower bounds to the quantity 〈g0, f〉, where f is the solution of the equation f0−Tf = 0, g0 is a given function and T is a non-self-adjoint linear operator from a Hilbert space into itself. The theory is illustrated by an integral equation of Fredholm type.

Upper and lower bounds for solutions of linear operator problems with unilateral constraints

1977 ◽  
Vol 76 (2) ◽  
pp. 95-105 ◽  
Author(s):  
W. D. Collins
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Lower Bounds ◽  
Original Problem ◽  
Adjoint Operator ◽  
Decomposition Theorem ◽  
Positive Definite ◽  
Upper And Lower Bounds ◽  
Unilateral Constraints ◽  
Space Decomposition

SynopsisDual extremum principles characterising the solutions of problems for a positive-definite self-adjoint operator on a Hilbert space which involve unilateral constraints are formulated using a Hilbert space decomposition theorem due to Moreau. Various upper and lower bounds to these solutions are then obtained, these bounds involving the solutions to subsidiary problems with less restrictive conditions than the solution to the original problem.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

A note on normal operators

1963 ◽  
Vol 59 (4) ◽  
pp. 727-729 ◽  
Author(s):  
S. J. Bernau ◽  
F. Smithies
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Adjoint Operator ◽  
Spectral Theorem ◽  
Unitary Operators ◽  
Unitary Space ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Normal Operators ◽  
Finite Dimensional Case

We recall that a bounded linear operator T in a Hilbert space or finite-dimensional unitary space is said to be normal if T commutes with its adjoint operator T*, i.e. TT* = T*T. Most of the proofs given in the literature for the spectral theorem for normal operators, even in the finite-dimensional case, appeal to the corresponding results for Hermitian or unitary operators.

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