43 research outputs found
Fortran code for generating random probability vectors, unitaries, and quantum states
The usefulness of generating random configurations is recognized in many
areas of knowledge. Fortran was born for scientific computing and has been one
of the main programming languages in this area since then. And several ongoing
projects targeting towards its betterment indicate that it will keep this
status in the decades to come. In this article, we describe Fortran codes
produced, or organized, for the generation of the following random objects:
numbers, probability vectors, unitary matrices, and quantum state vectors and
density matrices. Some matrix functions are also included and may be of
independent interest.Comment: are most welcome. To obtain the code, download the Source in Others
format
Random Sampling of Quantum States: A Survey of Methods
The numerical generation of random quantum states (RQS) is an important
procedure for investigations in quantum information science. Here we review
some methods that may be used for performing that task. We start by presenting
a simple procedure for generating random state vectors, for which the main tool
is the random sampling of unbiased discrete probability distributions (DPD).
Afterwards the creation of random density matrices is addressed. In this
context we first present the standard method, which consists in using the
spectral decomposition of a quantum state for getting RQS from random DPDs and
random unitary matrices. In the sequence the Bloch vector parametrization
method is described. This approach, despite being useful in several instances,
is not in general convenient for RQS generation. In the last part of the
article we regard the overparametrized method (OPM) and the related Ginibre and
Bures techniques. The OPM can be used to create random positive semidefinite
matrices with unit trace from randomly produced general complex matrices in a
simple way that is friendly for numerical implementations. We consider a
physically relevant issue related to the possible domains that may be used for
the real and imaginary parts of the elements of such general complex matrices.
Subsequently a too fast concentration of measure in the quantum state space
that appears in this parametrization is noticed.Comment: One reference adde
Generating pseudo-random discrete probability distributions
The generation of pseudo-random discrete probability distributions is of
paramount importance for a wide range of stochastic simulations spanning from
Monte Carlo methods to the random sampling of quantum states for investigations
in quantum information science. In spite of its significance, a thorough
exposition of such a procedure is lacking in the literature. In this article we
present relevant details concerning the numerical implementation and
applicability of what we call the iid, normalization, and trigonometric methods
for generating an unbiased probability vector
. An immediate application of these results
regarding the generation of pseudo-random pure quantum states is also
described
Computing partial traces and reduced density matrices
Taking partial traces for computing reduced density matrices, or related
functions, is a ubiquitous procedure in the quantum mechanics of composite
systems. In this article, we present a thorough description of this function
and analyze the number of elementary operations (ops) needed, under some
possible alternative implementations, to compute it on a classical computer. As
we notice, it is worthwhile doing some analytical developments in order to
avoid making null multiplications and sums, what can considerably reduce the
ops. For instance, for a bipartite system
with dimensions
and and for
, while a direct use of partial trace definition applied to
requires ops, its optimized
implementation entails ops. In the sequence, we
regard the computation of partial traces for general multipartite systems and
describe Fortran code provided to implement it numerically. We also consider
the calculation of reduced density matrices via Bloch's parametrization with
generalized Gell Mann's matrices.Comment: The code can be accessed in https://github.com/jonasmaziero/LibFor
Non-monotonicity of trace distance under tensor products
The trace distance (TD) possesses several of the good properties required for
a faithful distance measure in the quantum state space. Despite its importance
and ubiquitous use in quantum information science, one of its questionable
features, its possible non-monotonicity under taking tensor products of its
arguments (NMuTP), has been hitherto unexplored. In this article we advance
analytical and numerical investigations of this issue considering different
classes of states living in a discrete and finite dimensional Hilbert space.
Our results reveal that although this property of TD does not shows up for pure
states and for some particular classes of mixed states, it is present in a
non-negligible fraction of the regarded density operators. Hence, even though
the percentage of quartets of states leading to the NMuTP drawback of TD and
its strength decrease as the system's dimension grows, this property of TD must
be taken into account before using it as a figure of merit for distinguishing
mixed quantum states
The Maccone-Pati uncertainty relation
The existence of incompatible observables constitutes one of the most
prominent characteristics of quantum mechanics (QM) and can be revealed and
formalized through uncertainty relations. The
Heisenberg-Robertson-Schr\"odinger uncertainty relation (HRSUR) was proved at
the dawn of quantum formalism and is ever-present in the teaching and research
on QM. Notwithstanding, the HRSUR possess the so called triviality problem.
That is to say, the HRSUR yields no information about the possible
incompatibility between two observables if the system was prepared in a state
which is an eigenvector of one of them. After about 85 years of existence of
the HRSUR, this problem was solved recently by Lorenzo Maccone and Arun K.
Pati. In this article, we start doing a brief discussion of general aspects of
the uncertainty principle in QM and recapitulating the proof of HRSUR.
Afterwards we present in simple terms the proof of the Maccone-Pati uncertainty
relation, which can be obtained basically via the application of the
parallelogram law and Cauchy-Schwarz inequality.Comment: Didactic text, English versio
Genuine multipartite system-environment correlations in decoherent dynamics
We propose relative entropy-based quantifiers for genuine multipartite total,
quantum, and classical correlations. These correlation measures are applied to
investigate the generation of genuine multiparticle correlations in decoherent
dynamics induced by the interaction of two qubits with local- independent
environments. We consider amplitude- and phase-damping channels and compare
their capabilities to spread information through the creation of many-body
correlations. We identify changes in behavior for the genuine four- and
three-partite total correlations and show that, contrary to amplitude
environments, phase-noise channels transform the bipartite correlation
initially shared between the qubits into genuine multiparticle
system-environment correlations.Comment: 9 pages, 7 figures, published version, one reference update
Preparing tunable Bell-diagonal states on a quantum computer
The class of two-qubit Bell-diagonal states has been the workhorse for
developing understanding about the geometry, dynamics, and applications of
quantum resources. In this article, we present a quantum circuit for preparing
Bell-diagonal states on a quantum computer in a tunable way. We implement this
quantum circuit using the IBM Q 5 Yorktown quantum computer and, as an
application example, we measure the non-local, steering, entanglement, and
discord quantum correlations and non-local quantum coherence of Werner states
Environment-induced quantum coherence spreading of a qubit
We make a thorough study of the spreading of quantum coherence (QC), as
quantified by the -norm QC, when a qubit (a two-level quantum system) is
subjected to noise quantum channels commonly appearing in quantum information
science. We notice that QC is generally not conserved and that even incoherent
initial states can lead to transitory system-environment QC. We show that for
the amplitude damping channel the evolved total QC can be written as the sum of
local and non-local parts, with the last one being equal to entanglement. On
the other hand, for the phase damping channel (PDC) entanglement does not
account for all non-local QC, with the gap between them depending on time and
also on the qubit's initial state. Besides these issues, the possibility and
conditions for time invariance of QC are regarded in the case of bit, phase,
and bit-phase flip channels. Here we reveal the qualitative dynamical
inequivalence between these channels and the PDC and show that the creation of
system-environment entanglement does not necessarily imply in the destruction
of the qubit's QC. We also investigate the resources needed for non-local QC
creation, showing that while the PDC requires initial coherence of the qubit,
for some other channels non-zero population of the excited state (i.e., energy)
is sufficient. Related to that, considering the depolarizing channel we notice
the qubit's ability to act as a catalyst for the creation of joint QC and
entanglement, without need for nonzero initial QC or excited state population
Understanding von Neumann's entropy
We review the postulates of quantum mechanics that are needed to discuss the
von Neumann's entropy. We introduce it as a generalization of Shannon's entropy
and propose a simple game that makes easier understanding its physical meaning.Comment: Didactic text, in Portugues