scholarly journals The Prime state and its quantum relatives

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 371
Author(s):  
D. García-Martín ◽  
E. Ribas ◽  
S. Carrazza ◽  
J.I. Latorre ◽  
G. Sierra
Keyword(s):  
Entanglement Entropy ◽  
Prime Numbers ◽  
Quantum States ◽  
Direct Access ◽  
Distribution Of Primes ◽  
Quantum Fourier Transform ◽  
Arithmetic Properties ◽  
Primes In Arithmetic Progressions ◽  
Arbitrary Precision ◽  
Computational Basis

The Prime state of n qubits, |Pn⟩, is defined as the uniform superposition of all the computational-basis states corresponding to prime numbers smaller than 2n. This state encodes, quantum mechanically, arithmetic properties of the primes. We first show that the Quantum Fourier Transform of the Prime state provides a direct access to Chebyshev-like biases in the distribution of prime numbers. We next study the entanglement entropy of |Pn⟩ up to n=30 qubits, and find a relation between its scaling and the Shannon entropy of the density of square-free integers. This relation also holds when the Prime state is constructed using a qudit basis, showing that this property is intrinsic to the distribution of primes. The same feature is found when considering states built from the superposition of primes in arithmetic progressions. Finally, we explore the properties of other number-theoretical quantum states, such as those defined from odd composite numbers, square-free integers and starry primes. For this study, we have developed an open-source library that diagonalizes matrices using floats of arbitrary precision.

scholarly journals Probing Rényi entanglement entropy via randomized measurements

Science ◽  
2019 ◽  
Vol 364 (6437) ◽  
pp. 260-263 ◽  
Author(s):  
Tiff Brydges ◽  
Andreas Elben ◽  
Petar Jurcevic ◽  
Benoît Vermersch ◽  
Christine Maier ◽  
...  
Keyword(s):  
Quantum System ◽  
Entanglement Entropy ◽  
Quantum Systems ◽  
Quantum States ◽  
Many Body ◽  
Trapped Ion ◽  
Statistical Correlations ◽  
Quantum Simulator ◽  
Entanglement Structure ◽  
Arbitrary Quantum

Entanglement is a key feature of many-body quantum systems. Measuring the entropy of different partitions of a quantum system provides a way to probe its entanglement structure. Here, we present and experimentally demonstrate a protocol for measuring the second-order Rényi entropy based on statistical correlations between randomized measurements. Our experiments, carried out with a trapped-ion quantum simulator with partition sizes of up to 10 qubits, prove the overall coherent character of the system dynamics and reveal the growth of entanglement between its parts, in both the absence and presence of disorder. Our protocol represents a universal tool for probing and characterizing engineered quantum systems in the laboratory, which is applicable to arbitrary quantum states of up to several tens of qubits.

scholarly journals Photon Enhanced Interaction and Entanglement in Semiconductor Position-Based Qubits

2019 ◽  
Vol 9 (21) ◽  
pp. 4534 ◽  
Author(s):  
Panagiotis Giounanlis ◽  
Elena Blokhina ◽  
Dirk Leipold ◽  
Robert Staszewski
Keyword(s):  
Potential Energy ◽  
Entanglement Entropy ◽  
Electrostatic Actuation ◽  
Quantum States ◽  
Quantum Trajectory ◽  
Energy Profiles ◽  
Si Quantum Dot ◽  
And Control ◽  
Potential Energy Profiles ◽  
External Driving

CMOS technologies facilitate the possibility of implementing quantum logic in silicon. In this work, we discuss a minimalistic modelling of entangled photon communication in semiconductor qubits. We demonstrate that electrostatic actuation is sufficient to construct and control desired potential energy profiles along a Si quantum dot (QD) structure allowing the formation of position-based qubits. We further discuss a basic mathematical formalism to define the position-based qubits and their evolution under the presence of external driving fields. Then, based on Jaynes–Cummings–Hubbard formalism, we expand the model to include the description of the position-based qubits involving four energy states coupled with a cavity. We proceed with showing an anti-correlation between the various quantum states. Moreover, we simulate an example of a quantum trajectory as a result of transitions between the quantum states and we plot the emitted/absorbed photos in the system with time. Lastly, we examine the system of two coupled position-based qubits via a waveguide. We demonstrate a mechanism to achieve a dynamic interchange of information between these qubits over larger distances, exploiting both an electrostatic actuation/control of qubits and their photon communication. We define the entanglement entropy between two qubits and we find that their quantum states are in principle entangled.

scholarly journals Generic Entanglement Entropy for Quantum States with Symmetry

Entropy ◽  
2020 ◽  
Vol 22 (6) ◽  
pp. 684
Author(s):  
Yoshifumi Nakata ◽  
Mio Murao
Keyword(s):  
Phase Transition ◽  
Information Science ◽  
Entanglement Entropy ◽  
Quantum States ◽  
Permutation Symmetry ◽  
Phase Transition Behavior ◽  
Bipartite Entanglement ◽  
Translation Symmetry ◽  
The One ◽  
Random States

When a quantum pure state is drawn uniformly at random from a Hilbert space, the state is typically highly entangled. This property of a random state is known as generic entanglement of quantum states and has been long investigated from many perspectives, ranging from the black hole science to quantum information science. In this paper, we address the question of how symmetry of quantum states changes the properties of generic entanglement. More specifically, we study bipartite entanglement entropy of a quantum state that is drawn uniformly at random from an invariant subspace of a given symmetry. We first extend the well-known concentration formula to the one applicable to any subspace and then show that 1. quantum states in the subspaces associated with an axial symmetry are still highly entangled, though it is less than that of the quantum states without symmetry, 2. quantum states associated with the permutation symmetry are significantly less entangled, and 3. quantum states with translation symmetry are as entangled as the generic one. We also numerically investigate the phase-transition behavior of the distribution of generic entanglement, which indicates that the phase transition seems to still exist even when random states have symmetry.

EFFECTIVE LOWER BOUND FOR THE VARIANCE OF DISTRIBUTION OF PRIMES IN ARITHMETIC PROGRESSIONS

2008 ◽  
Vol 04 (01) ◽  
pp. 45-56 ◽  
Author(s):  
EMMANUEL KNAFO
Keyword(s):  
Lower Bound ◽  
Arithmetic Progressions ◽  
Distribution Of Primes ◽  
Primes In Arithmetic Progressions ◽  
Siegel Zeros

Through a refinement for the estimation of the effect of Siegel zeros, we show how to avoid the use of Siegel's theorem in order to obtain the first effective lower bound for the variance of distribution of primes in arithmetic progressions.

Almost-primes in arithmetic progressions and short intervals

1978 ◽  
Vol 83 (3) ◽  
pp. 357-375 ◽  
Author(s):  
D. R. Heath-Brown
Keyword(s):  
Prime Numbers ◽  
Arithmetic Progressions ◽  
Short Intervals ◽  
Almost Primes ◽  
Primes In Arithmetic Progressions ◽  
Image Position

In this paper we shall investigate the occurrence of almost-primes in arithmetic progressions and in short intervals. These problems correspond to two well-known conjectures concerning prime numbers. The first conjecture is that, if (l, k) = 1, there exists a prime p satisfying

scholarly journals Elliptic curves with a given number of points over finite fields

2012 ◽  
Vol 149 (2) ◽  
pp. 175-203 ◽  
Author(s):  
Chantal David ◽  
Ethan Smith
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Short Interval ◽  
Distribution Of Primes ◽  
Primes In Arithmetic Progressions ◽  
Interval Distribution ◽  
Better Than

AbstractGiven an elliptic curve E and a positive integer N, we consider the problem of counting the number of primes p for which the reduction of E modulo p possesses exactly N points over 𝔽p. On average (over a family of elliptic curves), we show bounds that are significantly better than what is trivially obtained by the Hasse bound. Under some additional hypotheses, including a conjecture concerning the short-interval distribution of primes in arithmetic progressions, we obtain an asymptotic formula for the average.

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