We discuss the question of entanglement versus separability of pure quantum states in direct product Hilbert spaces and the relevance of this issue to physics. Different types of separability may be possible, depending on the particular factorization or split of the Hilbert space. A given orthonormal basis set for a Hilbert space is defined to be of type (p,q) if p elements of the basis are entangled and q are separable, relative to a given bi-partite factorization of that space. We conjecture that not all basis types exist for a given Hilbert space.
We present a simple and general factorization law for quantum systems shared by two parties, which describes the time evolution of entanglement upon passage of either component through an arbitrary noisy channel. The robustness of entanglement-based quantum information processing protocols is thus easily and fully characterized by a single quantity.
We propose two algorithms to factor numbers using Gauss sums and entanglement: (i) in a Shor-like algorithm we encode the standard Gauss sum in one of two entangled states and (ii) in an interference algorithm we create a superposition of Gauss sums in the probability amplitudes of two entangled states.These schemes are rather efficient provided that there exists a fast algorithm that can detect a period of a function hidden in its zeros.
For a bi-partite quantum system defined in a finite dimensional Hilbert space we investigate in what sense entanglement change and interactions imply each other. For this purpose we introduce an entanglement operator, which is then shown to represent a non-conserved property for any bi-partite system and any type of interaction. This general relation does not exclude the existence of special initial product states, for which the entanglement remains small over some period of time, despite interactions. For this case we derive an approximation to the full Schroedinger equation, which allows the…
There exist zero-temperature states in quantum many-body systems that are fully factorized, thereby possessing vanishing entanglement, and hence being of no use as resource in quantum information processing tasks. Such states can become useful for quantum protocols when the temperature of the system is increased, and when the system is allowed to evolve under either the influence of an external environment, or a closed unitary evolution driven by its own Hamiltonian due to a sudden change in the system parameters. Using the one-dimensional anisotropic XY model in a uniform and an alternating t…
We show that any quantum density matrix can be represented by a Bayesian network (a directed acyclic graph), and also by a Markov network (an undirected graph). We show that any Bayesian or Markov net that represents a density matrix, is logically equivalent to a set of conditional independencies (symmetries) satisfied by the density matrix. We show that the d-separation theorems of classical Bayesian and Markov networks generalize in a simple and natural way to quantum physics. The quantum d-separation theorems are shown to be closely connected to quantum entanglement. We show that the graphi…
We determine the conditions for the existence of a pair of degenerate parity breaking separable eigenstates in general arrays of arbitrary spins connected through $XYZ$ couplings of arbitrary range and placed in a transverse field, not necessarily uniform. Sufficient conditions under which they are ground states are also provided. It is then shown that in finite chains, the associated definite parity states, which represent the actual ground state in the immediate vicinity of separability, can exhibit entanglement between any two spins regardless of the coupling range or separation, with the r…
The dynamics of entanglement between two photons with one of them passing through noisy quantum channels is characterized. It is described by a simple factorization law which was first theoretically proposed by Konrad {\it et al.} [Nature Phys., 4, 99 (2008)]. Quantum state tomography process is employed to reconstruct the reduced density matrixes of the final states and the corresponding concurrences are calculated. Good fittings between experimental results and theoretical predictions are found, which imply the validity of the general factorization law in the characterization of entanglement…
In this work, we consider the question of local Hilbert space factorization in 2D conformal field theory. Generalizing previous work on entanglement and open-closed TQFT, we interpret the factorization of CFT states in terms of path integral processes that split and join the Hilbert spaces of circles and intervals. More abstractly, these processes are cobordisms of an extended CFT which are defined purely in terms of the OPE data. In addition to the usual sewing axioms, we impose an entanglement boundary condition that is satisfied by the vacuum Ishibashi state. This choice of entanglement bou…
In this paper it has been described how to use the unitary dynamics of quantum mechanics to solve the prime factorization problem on a spin ensemble without any quantum entanglement. The ensemble quantum computation for the prime factorization is based on the basic principle that both a closed quantum system and its ensemble obey the same unitary dynamics of quantum mechanics if there is not any decoherence effect in both the quantum system and its ensemble. It uses the NMR multiple-quantum measurement techniques to output the quantum computational results that are the inphase multiple-quantum…
We investigate the scaling of the entanglement spectrum and of the Rényi block entropies and determine its universal aspects in the ground state of critical and noncritical one-dimensional quantum spin models. In all cases, the scaling exhibits an oscillatory behavior that terminates at the factorization point and whose frequency is universal. Parity effects in the scaling of the Rényi entropies for gapless models at zero field are thus shown to be a particular case of such universal behavior. Likewise, the absence of oscillations for the Ising chain in transverse field is due to the vanishing…
Distribution of quantum entanglement is investigated for an anisotropic quantum XY model with variable range interactions and in the presence of a uniform transverse magnetic field. We report the possibility of \emph{qualitative} growth in the entanglement between distant sites with an increase in the range of interactions that vary either exponentially or polynomially as the distance between the sites increases. Interestingly, we find that such entanglement enhancement is not ubiquitous and is dependent on the factorization points, a specific set of system parameters where the zero-temperatur…
We describe a decomposition of the Lie group of unitary evolutions for a bipartite quantum system of arbitrary dimensions. The decomposition is based on a recursive procedure which systematically uses the Cartan classification of the symmetric spaces of the Lie group SO(n). The resulting factorization of unitary evolutions clearly displays the local and entangling character of each factor.
We discuss the concept of how entanglement changes with respect to different factorizations of the total algebra which describes the quantum states. Depending on the considered factorization a quantum state appears either entangled or separable. For pure states we always can switch unitarily between separability and entanglement, however, for mixed states a minimal amount of mixedness is needed. We discuss our general statements in detail for the familiar case of qubits, the GHZ states, Werner states and Gisin states, emphasizing their geometric features. As theorists we use and play with this…
We study quantum entanglements induced on product states by the action of 8-vertex braid matrices, rendered unitary with purely imaginary spectral parameters (rapidity). The unitarity is displayed via the "canonical factorization" of the coefficients of the projectors spanning the basis. This adds one more new facet to the famous and fascinating features of the 8-vertex model. The double periodicity and the analytic properties of the elliptic functions involved lead to a rich structure of the 3-tangle quantifying the entanglement. We thus explore the complex relationship between topological an…
Quantum information theory and quantum computing are theoritical basis of quantum computers. Thanks to entanglement, quantum mechanical systems are provisioned to realize many information processing problems faster than classical counterparts. For example, Shor factorization algorithm, Grover search algorithm, quantum Fourrier transformation, etc. Entanglement, is the theoretical basis providing the expected speedups. It can be view in bipartite or multipartite forms. In order to quantify entanglement, some measures are defined. On the other hand, a general and accepted criterion, which can me…
We consider quantum computing with pseudo-pure states. This framework arises in certain implementations of quantum computing using NMR. We analyze quantum computational protocols which aim to solve exponential classical problems with polynomial resources and ask whether or not entanglement of the pseudo-pure states is needed to achieve this aim. We show that for a large class of such protocols, including Shor's factorization, entanglement is necessary. We also show that achieving entanglement is not sufficient: if the noise in the state is sufficiently large, exponential resources are needed e…
We discover a simple factorization law describing how multipartite entanglement of a composite quantum system evolves when one of the subsystems undergoes an arbitrary physical process. This multipartite entanglement decay is determined uniquely by a single factor we call the entanglement resilient factor (ERF). Since the ERF is a function of the quantum channel alone, we find that multipartite entanglement evolves in exactly the same way as bipartite (two qudits) entanglement. For the two qubits case, our factorization law reduces to the main result of Nature Physics 4, 99 (2008). In addition…
We introduce a method for analyzing ground state properties of quantum many body systems, based on the characterization of separability and entanglement by single subsystem unitary operations. We apply the method to the study of the ground state structure of several interacting spin-1/2 models, described by Hamiltonians with different degrees of symmetry. We show that the approach based on single qubit unitary operations allows to introduce {\it ``entanglement excitation energies''}, a set of observables that can characterize ground state properties, including the quantification of single-site…
In the traditional quantum theory, one-dimensional quantum spin models possess a factorization surface where the ground states are fully separable having vanishing bipartite as well as multipartite entanglement. We report that in the non-Hermitian counterpart of these models, these factorization surfaces either can predict the exceptional points where the unbroken-to-broken transition occurs or can guarantee the reality of the spectrum, thereby proposing a procedure to reveal the unbroken phase. We first analytically demonstrate it for the nearest-neighbour rotation-time RT-symmetric XY model …
We report a quantum-classical hybrid scheme for factorization of bi-prime numbers (which are odd and square-free) using IBM's quantum processors. The hybrid scheme proposed here involves both classical optimization techniques and adiabatic quantum optimization techniques, and is build by extending a previous scheme of hybrid factorization [Pal et al., Pramana 92, 26 (2019) and Xu et al., Phys. Rev. Lett. 108, 130501 (2012)]. The quantum part of the scheme is very general in the sense that it can be implemented using any quantum computing architecture. Here, as an example, we experimentally imp…
Entanglement in high energy and and nuclear reactions is receiving great attention. A proper description of these reactions uses density matrices, and the express of entanglement in terms of {\it separability}. Quantum tomography bypasses field-theoretic formalism to determine density matrices in terms of experimental observables. We review recent work applying quantum tomography to practical experimental data analysis. We discuss the relation between separability, as defined in quantum information science, and factorization, as defined in high energy physics. When factorization applies, it co…
We study the factorization conditions of a wave function made up of states of two, three and four qubits and propose and analytical expression which can characterize entangled states in terms of the coefficients of the wave function and density matrix elements.
This paper investigates the physical effects of Yang-Baxter equation (YBE) to quantum entanglements through the 3-body S-matrix in entangling parameter space. The explicit form of 3-body S-matrix $\breve{R}_{123}(θ,φ)$ based on the 2-body S-matrices is given due to the factorization condition of YBE. The corresponding chain Hamiltonian has been obtained and diagonalized, also the Berry phase for 3-body system is given. It turns out that by choosing different spectral parameters the $\breve{R}(θ,φ)$-matrix gives GHZ and W state respectively. The extended 1-D Kitaev toy model has been derived. E…
We analyze exact ground state (GS) separability in general $N$ particle systems with two-site couplings. General necessary and sufficient conditions for full separability, in the form of one and two-site eigenvalue equations, are first derived. The formalism is then applied to a class of $SU(n)$-type interacting systems, where each constituent has access to $n$ local levels, and where the total number parity of each level is preserved. Explicit factorization conditions for parity-breaking GS's are obtained, which generalize those for $XYZ$ spin systems and correspond to a fundamental GS multil…
We elaborate on the extended Hilbert space factorization of Chern Simons theory and show how this arises naturally from a proper regularization of the entangling surface in the Euclidean path integral. The regularization amounts to stretching the entangling surface into a co-dimension one surface which hosts edge modes of the Chern Simons theory when quantized on a spatial subregion. The factorized state is a regularized Ishibashi state and reproduces the well known topological entanglement entropies. We illustrate how the same factorization arises from the glueing of two spatial subregions vi…
We find a connection between the existence of a factorization of a quantum channel and the existence of low-rank solutions to certain linear matrix equations. Using this, we show that if a quantum channel is factorized by a direct integral of factors, it must lie in the convex hull of quantum channels which are factorized respectively by the factors in the direct integral. We use this to characterize some non-trivial extreme points in the set of factorizable quantum channels and give an example.
New exactly solvable problems have already been studied by using a modification of the factorization method introduced by Mielnik. We review this method and its connection with the traditional factorization method. The survey includes the discussion on a generalization of the factorization energies used in the traditional Infeld and Hull method.
The block canonical coset decomposition is developed as a universal algorithmic tool to synthesize n-qubit quantum gates out of experimentally realizable NMR elements. The two-, three-, and four-qubit quantum Fourier transformations are worked out as examples. The proposed decomposition bridges the state of the art numerical analysis with NMR quantum gate synthesis.
It is unusual to find QCD factorization explained in the language of quantum information science. However, we will discuss how the issue of factorization and its breaking in high-energy QCD processes relates to phenomena like decoherence and entanglement. We will elaborate with several examples and explain them in terms familiar from basic quantum mechanics and quantum information science.
In this paper we address the relativist-perspectival nature of the orthodox definition of quantum entanglement in terms of preferred factorizations. We also consider this aspect aspect within the generalized definition of entanglement proposed by Barnum et al. [6, 7] in terms of preferred observables. More specifically, we will discuss the non-separable relativism implied by the orthodox definition of entanglement, the contextual relativism implied by its generalization as well as some other serious problems presently discussed within the specialized literature. In the second part of this work…
We demonstrate that, in the case of Shor's algorithm for factoring, highly mixed states will allow efficient quantum computation, indeed factorization can be achieved efficiently with just one initial pure qubit and a supply of initally maximally mixed qubits (S. Parker and M. B. Plenio, Phys. Rev. Lett. 85, 3049 (2000)) . This leads us to ask how this affects the entanglement in the algorithm. We thus investigate the behaviour of entanglement in Shor's algorithm for small numbers of qubits by classical computer simulation of the quantum computer at different stages of the algorithm. We find t…
We propose an adiabatic quantum algorithm capable of factorizing numbers, using fewer qubits than Shor's algorithm. We implement the algorithm in an NMR quantum information processor and experimentally factorize the number 21. Numerical simulations indicate that the running time grows only quadratically with the number of qubits.
Quantum processors are potentially superior to their classical counterparts for many computational tasks including factorization. Circuit methods as well as adiabatic methods have already been proposed and implemented for finding the factors of a given composite number. The main challenge in scaling it to larger numbers is the unavailability of large number of qubits. Here we propose a hybrid scheme that involves both classical and quantum computation, which reduces the number of qubits required for factorization. The classical part involves setting up and partially simplifying a set of bit-wi…
It was shown in [Augusiak et al.,\;Phys. Rev. A \textbf{77}, 030301(R) (2008)] that discrimination between entanglement and separability in a two qubit state can be achieved by a measurement of a single observable on four copies of it. Moreover, a pseudo entanglement monotone $π$ was proposed to quantify entanglement in such states. The main goal of the present paper is to show that close relationship between $π$ and concurrence reported there is a result of sharing the same underlying construction of a spin flipped matrix. We also show that monogamy of entanglement can be rephrased in terms o…
Quantum algorithms could be much faster than classical ones in solving the factoring problem. Adiabatic quantum computation for this is an alternative approach other than Shor's algorithm. Here we report an improved adiabatic factoring algorithm and its experimental realization to factor the number 143 on a liquid crystal NMR quantum processor with dipole-dipole couplings. We believe this to be the largest number factored in quantum-computation realizations, which shows the practical importance of adiabatic quantum algorithms.
Quantum-Kit is a graphical desktop application for quantum circuit simulations. Its powerful, memory-efficient computational engine enables large-scale simulations on a desktop. The ability to design hybrid circuits, with both quantum and classical bits and controls, is employed to demonstrate Kitaev's approach to Shor's factorization algorithm. For the first time, Shor's factorization of a 24-bit integer is simulated with Quantum-Kit in a mere 26 minutes on a modest desktop with Intel Core i5 7400T, 2.4GHz and12GB RAM. While the largest number factorized so far has been a 20-bit integer, requ…
For a one-dimensional chain of four nuclear spins (1/2) and taking into account first and second neighbor interactions among the spin system, we make the numerical simulation of Shor prime factorization algorithm of the integer number N=4 to study the influence of the second neighbor interaction on the performance of this algorithm. It is shown that the optimum Rabi's frequency to control the non-resonant effects is dominated by the second neighbor interaction coupling parameter ($J'$), and that a good Shor quantum factorization is achieved for a ratio of second to first coupling constant of $…
We consider a probabilistic quantum implementation of a variable of the Pocklington-Lehmer $N-1$ primality test using Shor's algorithm. O($\log^3 N \log\log N \log\log\log N$) elementary q-bit operations are required to determine the primality of a number $N$, making it (asymptotically) the fastest known primality test. Thus, the potential power of quantum mechanical computers is once again revealed.
We present the general ideas on SuperSymmetric Quantum Mechanics (SUSY-QM) using different representations for the operators in question, which are defined by the corresponding bosonic Hamiltonian as part of SUSY Hamiltonian and its supercharges, which are defined as matrix or differential operators. We show that, although most of the SUSY partners of one-dimensional Schrödinger problems have already been found,there are still some unveiled aspects of the factorization procedure which may lead to richer insights of the problem involved.
Determining the prime factors of a given number N is a problem, which requires super-polynomial time for conventional digital computers. A polynomial-time algorithm was invented by P. Shor for quantum computers. However, the realization of quantum computers is associated with significant technological challenges in achieving and preserving quantum entanglement. Prime factorization can be also solved by using classical wave interference without quantum entanglement. In this work, we present experimental data showing prime factorization by utilizing spin wave interference. The prime factorizatio…
Quantum computation is a continuously growing research area which is based on nature and resources of quantum mechanics, as superposition and entanglement. In its quantum circuits version, the use of convenient and appropriate gates is essential. But while those gates adopt convenient forms for computational algorithms, their design depends on specific quantum systems and stu? being used. These gates need manage quantum systems based on physical interactions ruled by quantum Hamiltonians. With this, gates design is restricted to properties and limitations of interactions and physical elements …
In two companion papers it was shown how to separate out from a scattering function in quantum electrodynamics a distinguished part that meets the correspondence-principle and pole-factorization requirements. The integrals that define the terms of the remainder are here shown to have singularities on the pertinent Landau singularity surface that are weaker than those of the distinguished part. These remainder terms therefore vanish, relative to the distinguished term, in the appropriate macroscopic limits. This shows, in each order of the perturbative expansion, that quantum electrodynamics do…
This paper discusses relationships between topological entanglement and quantum entanglement. Specifically, we propose that for this comparison it is fundamental to view topological entanglements such as braids as "entanglement operators" and to associate to them unitary operators that are capable of creating quantum entanglement.
New families of time-dependent potentials related with the stationary singular oscillator are introduced. This is achieved after noticing that a non stationary quantum invariant can be constructed for the singular oscillator. Such invariant depends on coefficients that are related to solutions of an Ermakov equation, the latter becomes essential since it guarantees the regularity of the solutions at each time. In this form, after applying the factorization method to the quantum invariant, rather than the Hamiltonian, one manages to introduce the time parameter into the transformation, leading …
In quantum systems, entanglement corresponds to nonclassical correlation of nonlocal observables. Thus, entanglement (or, to the contrary, separability) of a given quantum state is not uniquely determined by properties of the state, but may depend on the choice of the factorization of the algebra of observables. In the present work, we expose and systematize some recently reported results about the possibility to represent a single quantum state as either entangled or separable. We will distinguish in particular the cases of pure and mixed states. For pure states, it has been shown that observ…
It is shown that Smolin four-qubit bound entangled states [Phys. Rev. A, 63 032306 (2001)] can maximally violate two-setting Bell inequality similar to standard CHSH inequality. Surprisingly this entanglement does not allow for secure key distillation, so neither entanglement nor violation of Bell inequalities implies quantum security. It is also pointed out how that kind of bound entanglement can be useful in reducing communication complexity.
