22 pages, 1 figure, new version of paper arXiv:0706.1743 [quant-ph] containing just the Bloch vector part but enlarged with an additional section on experimental application of entanglement witnesses in 3x3 dimensions
We present three different matrix bases that can be used to decompose density matrices of $d$--dimensional quantum systems, so-called qudits: the \emph{generalized Gell-Mann matrix basis}, the \emph{polarization operator basis}, and the \emph{Weyl operator basis}. Such a decomposition can be identified with a vector --the Bloch vector, i.e. a generalization of the well known qubit case-- and is a convenient expression for comparison with measurable quantities and for explicit calculations avoiding the handling of large matrices. We present a new method to decompose density matrices via so--called standard matrices, consider the important case of an isotropic two--qudit state and decompose it according to each basis. In case of qutrits we show a representation of an entanglement witness in terms of expectation values of spin 1 measurements, which is appropriate for an experimental realization.
We investigate the norms of the Bloch vectors for any quantum state with subsystems less than or equal to four. Tight upper bounds of the norms are obtained, which can be used to derive tight upper bounds for entanglement measure defined by the norms of Bloch vectors. By using these bounds a trade-off relation of the norms of Bloch vectors is discussed. Theses upper bounds are then applied on separability. Necessary conditions are presented for different kinds of separable states in four-partite quantum systems. We further present a complete classification of quantum states for four qudits qua…
We present three different matrix bases that can be used to decompose density matrices of d--dimensional quantum systems, so-called qudits: the generalized Gell-Mann matrix basis, the polarization operator basis, and the Weyl operator basis. Such a decomposition can be identified with a vector --the Bloch vector, i.e. a generalization of the well known qubit case-- and is a convenient expression for comparison with measurable quantities and for explicit calculations avoiding the handling of large matrices. We consider the important case of an isotropic two--qudit state and decompose it accordi…
A closed system of equations for the local Bloch vectors and spin correlation functions of three magnetic qudits, which are in an arbitrary, time-dependent, external magnetic field, is obtained using decomplexification of the Liouville-von Neumann equation. The algorithm of the derivation of the dynamic equations is presented. In the basis convenient for the important physical applications structure constants of algebra su(2S+1) are calculated.
We present a multipartite entanglement measure for $N$-qudit pure states, using the norm of the correlation tensor which occurs in the Bloch representation of the state. We compute this measure for important class of $N$-qutrit pure states, namely general GHZ states. We prove that this measure possesses almost all the properties expected of a good entanglement measure, including monotonicity. Finally, we extend this measure to $N$-qudit mixed states via convex roof construction and establish its various properties, including its monotonicity.
We derive a basis for the vector space of bounded operators acting on a $d$-dimensional system Hilbert space $C^d$. In the context of quantum computation the basis elements are identified as the generalised Pauli matrices - the error generators. As an application, we show how such matrices are used in the teleportation a single qudit.
Using the tomographic probability representation of qudit states and the inverse spin-portrait method, we suggest a bijective map of the qudit density operator onto a single probability distribution. Within the framework of the approach proposed, any quantum spin-j state is associated with the (2j+1)(4j+1)-dimensional probability vector whose components are labeled by spin projections and points on the sphere. Such a vector has a clear physical meaning and can be relatively easily measured. Quantum states form a convex subset of the 2j(4j+3) simplex, with the boundary being illustrated for qub…
We determine the set of the Bloch vectors for N-level systems, generalizing the familiar Bloch ball in 2-level systems. An origin of the structural difference from the Bloch ball in 2-level systems is clarified.
An ability to describe quantum states directly by average values of measurement outcomes is provided by the Bloch vector. For an informationally complete set of measurements one can construct unique Bloch vector for any quantum state. However, not every Bloch vector corresponds to a quantum state. It seems that only for two dimensional quantum systems it is easy to distinguish proper Bloch vectors from improper ones, i.e. the ones corresponding to quantum states from the other ones. I propose an alternative approach to the problem in which more than one vector is used. In particular, I show th…
Quantum computing is among the most far-reaching technologies of the 21st century, tackling challenges at the cutting edge of physics. This new paradigm in computer science harnesses quantum entanglement, one striking non-intuitive feature of quantum mechanics and a cornerstone of quantum information, to provide computation with a quantum speed-up over the best-known classical algorithms and to enable encrypted data communication against eavesdropping. The bulk of this article is focused on providing a deep and abiding understanding of non-relativistic matrix mechanics by demonstrating the fun…
Too much noise kills entanglement. This is the main problem in its production and transmission. We use a handy approach to indicate noise resistance of entanglement of a bi-partite system described by $d\times d$ Hilbert space. Our analysis uses a geometric approach based on the fact that if a scalar product of a vector $\vec{s}$ with a vector $\vec {e}$ is less than the square of the norm of $\vec{e}$, then $\vec{s}\neq\vec{e}$. We use such concepts for correlation tensors of separable and entangled states. As a general form correlation tensors for pairs of qudits, for $d>2$, is very difficul…
Using the Bloch representation of an N-dimensional quantum system and immediate results from quantum stochastic calculus, we establish a closed formula for the Bloch vector, hence also for the density operator, of a quantum system following a Lindblad evolution with selfadjoint Lindblad operators.
We obtain a new inequality for arbitrary Hermitian matrices. We describe particular linear maps called the matrix portrait of arbitrary NxN matrices. The maps are obtained as analogs of partial tracing of density matrices of multipartite qudit systems. The structure of the maps is inspired by "portrait" map of the probability vectors corresponding to the action on the vectors by stochastic matrices containing either unity or zero matrix elements. We obtain new entropic inequalities for arbitrary qudit states including a single qudit and an discuss entangled single qudit state. We consider in d…
We describe and characterize an equal-fidelity surface in Bloch space targeted for a qubit state by means of equal-distance concept. The distance is generalized and defined as the Euclidean distance between extended Bloch vectors for arbitrary dimensional states. The distance is a genuine distance according to the definition and is related to other distances between quantum states and super-fidelity.
So far there is no closed formula for relative entropy of entanglement of arbitrary two-qubit states. In this paper we present a method, which guarantees the derivation of the relative entropy of entanglement for most states that have $z$-directional Bloch vectors. It is shown that the closest separable states for those states also have $z$-directional Bloch vectors though there are few exceptions.
Bloch-vector spaces for $N$-level systems are investigated from the spherical-coordinate point of view in order to understand their geometrical aspects. We show that the maximum radius in each direction, which is due to the construction of the Bloch-vector space, is determined by the minimum eigenvalue of the corresponding observable (orthogonal generator of SU(N)). From this fact, we reveal the dual property of the structure of the Bloch-vector space; if in some direction the space reachs the large sphere (pure state), then in the opposite direction the space can only get to the small sphere,…
We present a method for preparing arbitrary pure states of spatial qudits, namely, D-dimensional (D > 2) quantum systems carrying information in the transverse momentum and position of single photons. For this purpose, a set of D slits with complex transmission are displayed on a spatial light modulator (SLM). In a recent work we have shown a method that requires a single phase-only SLM to control independently the complex coefficients which define the quantum state of dimension D. The amplitude information was codified by introducing phase gratings inside each slit and the phase value of the …
We have considered a system of two superconducting charge qubits capacitively coupled to a microwave resonator. The dynamics of the Bloch vectors are investigated for different regimes. By means of the Bloch vectors and cross dyadic we quantify the degree of entanglement contained in the generated entangled state. We consider different values of the system parameters to discuss the dynamics of the channel capacity between the qubits. We show that there is an important role played by initial state settings, coupling constant and the mean photon number on generating entangled state with high deg…
In this paper we present a simple algorithm for representation of statistical data of any origin by complex probability amplitudes. Numerical simulation with Mathematica-6 is performed. The Bloch's sphere is used for visualization of results of numerical simulation. On the one hand, creation of such a quantum-like (QL) representation and its numerical approval is an important step in clarification of extremely complicated interrelation between classical and quantum randomness. On the other hand, it opens new possibilities for application the mathematical formalism of QM in other domains of sci…
We introduce a Hermitian generalization of Pauli matrices to higher dimensions which is based on Heisenberg-Weyl operators. The complete set of Heisenberg-Weyl observables allows us to identify a real-valued Bloch vector for an arbitrary density operator in discrete phase space, with a smooth transition to infinite dimensions. Furthermore, we derive bounds on the sum of expectation values of any set of anti-commuting observables. Such bounds can be used in entanglement detection and we show that Heisenberg-Weyl observables provide a first non-trivial example beyond the dichotomic case.
We extend the black hole/qudit correspondence by identifying five and six-dimensional 1/2-BPS black string and hole charge vectors in N=8 and N=2 magic supergravities with qubits and qutrits over composition algebras. In D=6, this is accomplished via Hopf fibrations, which map qubits over composition algebras to rank one elements of Jordan algebras of degree two. An analogous procedure maps qutrits over composition algebras to D=5 charge vectors, which are rank one elements of Jordan algebras of degree three. In both cases, the U-duality groups are interpreted as qudit SLOCC transformation gro…
Quantum manipulation protocols for quantum sensors and quantum computation often require many single qubit rotations. However, the impact of phase noise in the field that performs the qubit rotations is often neglected or treated only for special cases. We present a general framework for calculating the impact of phase noise on the state of a qubit, as described by its equivalent Bloch vector. The analysis applies to any Bloch vector orientation, and any rotation axis azimuthal angle for both a single pulse, and pulse sequences. Experimental examples are presented for several special cases. We…
In this paper, we investigate the dynamics of the Bloch vector of a single two-level atom which interacts with a single quantized electromagnetic field mode according to the Jaynes-Cummings model, where the field is initially prepared in a thermal state. The time evolution of the Bloch vector S(t) seems to be in complete disorder because of the thermal distribution of the initial state of the field. Both the norm and the direction of S(t) oscillate hard and their periods seem infinite. We observe that the trajectory of the time evolution of S(t) in the two- or three-dimensional space does not …
A relation is established in the present paper between Dicke states in a d-dimensional space and vectors in the representation space of a generalized Weyl-Heisenberg algebra of finite dimension d. This provides a natural way to deal with the separable and entangled states of a system of N = d-1 symmetric qubit states. Using the decomposition property of Dicke states, it is shown that the separable states coincide with the Perelomov coherent states associated with the generalized Weyl-Heisenberg algebra considered in this paper. In the so-called Majorana scheme, the qudit (d-level) states are r…
As is well known, when an SU(2) operation acts on a two-level system, its Bloch vector rotates without change of magnitude. Considering a system composed of two two-level systems, it is proven that for a class of nonlocal interactions of the two subsystems including σ_i\otimesσ_j (with i,j \in {x,y,z}) and the Heisenberg interaction, the geometric description of the motion is particularly simple: each of the two Bloch vectors follows an elliptical orbit within the Bloch sphere. The utility of this result is demonstrated in two applications, the first of which bears on quantum control via quant…
Geometric algebra is a mathematical structure that is inherent in any metric vector space, and defined by the requirement that the metric tensor is given by the scalar part of the product of vectors. It provides a natural framework in which to represent the classical groups as subgroups of rotation groups, and similarly their Lie algebras. In this article we show how the geometric algebra of a six-dimensional real Euclidean vector space naturally allows one to construct the special unitary group on a two-qubit (quantum bit) Hilbert space, in a fashion similar to that used in the well-establish…
Physical constraints such as positivity endow the set of quantum states with a rich geometry if the system dimension is greater than two. To shed some light on the complicated structure of the set of quantum states, we consider a stratification with strata given by unitary orbit manifolds, which can be identified with flag manifolds. The results are applied to study the geometry of the coherence vector for n-level quantum systems. It is shown that the unitary orbits can be naturally identified with spheres in R^{n^2-1} only for n=2. In higher dimensions the coherence vector only defines a non-…
We show that the Bloch vectors lying on any great circle is the largest set S(L) for which the parallel states |n,n> can always be transformed into the anti-parallel states |n,-n>. Thus more information about the Bloch vector is not extractable from |n,-n> than from |n,n> by any measuring strategy, for the Bloch vector belonging to S(L). Surprisingly, the largest set of Bloch vectors for which the corresponding qubits can be flipped is again S(L). We then show that probabilistic exact parallel to anti-parallel transformation is not possible if the corresponding anti-parallel spins span the who…
For one qubit systems, we present a short, elementary argument characterizing unital quantum operators in terms of their action on Bloch vectors. We then show how our approach generalizes to multi-qubit systems, obtaining inequalities that govern when a ``diagonal'' superoperator on the Bloch region is a quantum operator. These inequalities are the n-qubit analogue of the Algoet-Fujiwara conditions. Our work is facilitated by an analysis of operator-sum decompositions in which negative summands are allowed.
Entanglement, a manifestation of quantumness of correlations between the observables of the subsystems of a composite system, and the quantumness of their mutual information are widely studied characteristics of a system of spin-1/2 particles. The concept of quantumness of correlations between the observables of a system is based on incommensurability of the correlations with the predictions of some local hidden variable (LHV) theory. However, the concept of quantumness of mutual information does not invoke the LHV theory explicitly. In this paper, by invoking explicitly the local hidden varia…
A two-sphere ("Bloch" or "Poincare") is familiar for describing the dynamics of a spin-1/2 particle or light polarization. Analogous objects are derived for unitary groups larger than SU(2) through an iterative procedure that constructs evolution operators for higher-dimensional SU in terms of lower-dimensional ones. We focus, in particular, on the SU(4) of two qubits which describes all possible logic gates in quantum computation. For a general Hamiltonian of SU(4) with 15 parameters, and for Hamiltonians of its various sub-groups so that fewer parameters suffice, we derive Bloch-like rotatio…
The $d^2$ Pauli operators attached to a composite qudit in dimension $d$ may be mapped to the vectors of the symplectic module $\mathcal{Z}_d^{2}$ ($\mathcal{Z}_d$ the modular ring). As a result, perpendicular vectors correspond to commuting operators, a free cyclic submodule to a maximal commuting set, and disjoint such sets to mutually unbiased bases. For dimensions $d=6,~10,~15,~12$, and 18, the fine structure and the incidence between maximal commuting sets is found to reproduce the projective line over the rings $\mathcal{Z}_{6}$, $\mathcal{Z}_{10}$, $\mathcal{Z}_{15}$, $\mathcal{Z}_6 \ti…
In this paper, we show that the time evolution of the Bloch vector governed by the thermal Jaynes-Cummings model is equivalent to a compressible inviscid flow with zero vorticity. Because of its quasiperiodicity, the dynamics of the Bloch vector includes countably infinite angular momenta as integrals of motion. Moreover, to derive the Bloch vector, we trace out the Hilbert space of the cavity field and remove entanglement between the single atom and the cavity mode. These facts indicate that the dynamics of the Bloch vector can be described with a hidden-variable model that has local determin…
The Bloch vector's path of a two-level system exposed to a monochromatic field exhibits, in the regime of strong coupling, complex corkscrew trajectories. By considering the infinitesimal evolution of the two-level system when the field is treated as a classical object, we show that the Bloch vector's rotation speed oscillates between zero and twice the rotation speed predicted by the rotating wave approximation. Cusps appear when the rotation speed vanishes. We prove analytically that in correspondence to cusps the curvature of the Bloch vector's path diverges. On the other hand, numerical da…
We present a practical scheme for the decomposition of a bipartite mixed state into a sum of direct products of local density matrices, using the technique developed in Li and Qiao (Sci. Rep. 8: 1442, 2018). In the scheme, the correlation matrix which characterizes the bipartite entanglement is first decomposed into two matrices composed of the Bloch vectors of local states. Then we show that the symmetries of Bloch vectors are consistent with that of the correlation matrix, and the magnitudes of the local Bloch vectors are lower bounded by the correlation matrix. Concrete examples for the sep…
We use polarization operators known from quantum theory of angular momentum to expand the $N \times N$ dimensional density operators. Thereby, we construct generalized Bloch vectors representing density matrices. We study their properties and derive positivity conditions for any $N$. We also apply the procedure to study Bloch vector space for a qubit and a qutrit.
Spectral-angular parametrization of open qudit dynamics: gap coordinates, Cartan structure, and GKLS decoupling
Université Paris Cité, CNRS（巴黎西岱大学，法国国家科学研究中心）；Faculty of Mathematics, University of Białystok（比亚韦斯托克大学数学系）；ESMaR, Faculty of Sciences at Univ. Mohammed V - Rabat（穆罕默德五世大学理学院ESMaR）
We study a quasiperiodic structure in the time evolution of the Bloch vector, whose dynamics is governed by the thermal Jaynes-Cummings model (JCM). Putting the two-level atom into a certain pure state and the cavity field into a mixed state in thermal equilibrium at initial time, we let the whole system evolve according to the JCM Hamiltonian. During this time evolution, motion of the Bloch vector seems to be in disorder. Because of the thermal photon distribution, both a norm and a direction of the Bloch vector change hard at random. In this paper, taking a different viewpoint compared with …
By use of a generalized Bloch vector construction, we study the decoherence of a system composed of two interacting qubits in a general noisy environment. In particular, we investigate the effects of correlations in the noise acting on distinct qubits. Our treatment of the two-qubit system by use of the generalized Bloch vector leads to tractable analytic equations for the dynamics of the 4-level Bloch vector and allows for the application of geometrical concepts from the well known 2-level Bloch sphere. We find that in the presence of correlated or anticorrelated noise, the rate of decoherenc…
We present a surprisingly simple three-dimensional Bloch sphere representation of a qutrit, i.e., a single three-level quantum system. We start with a symmetric state of a two-qubit system and relate it to the spin-1 representation. Using this representation we associate each qutrit state with a three-dimensional vector $\mathbf{a}$ and a metric tensor $\mathbf{\hatΓ}$ which satisfy $\mathbf{a}\cdot \mathbf{\hatΓ} \cdot \mathbf{a}\leq 1$. This resembles the well known condition for qubit Bloch vectors in which case $\mathbf{\hatΓ}=\hat{I}$. In our case the vector $\mathbf{a}$ corresponds to sp…
In the Bloch sphere picture, one finds the coefficients for expanding a single-qubit density operator in terms of the identity and Pauli matrices. A generalization to $n$ qubits via tensor products represents a density operator by a real vector of length $4^n$, conceptually similar to a statevector. Here, we study this approach for the purpose of quantum circuit simulation, including noise processes. The tensor structure leads to computationally efficient algorithms for applying circuit gates and performing few-qubit quantum operations. In view of variational circuit optimization, we study ``b…
A topological formulation of the eigenspace anholonomy, where eigenspaces are interchanged by adiabatic cycles, is introduced. The anholonomy in two-level systems is identified with a disclination of the director (headless vector) of a Bloch vector, which characterizes eigenprojectors. The covering map structure behind the exotic quantum holonomy and the role of the homotopy classification of adiabatic cycles are elucidated. The extensions of this formulation to nonadiabatic cycles and N-level systems are outlined.
A visualization scheme for dynamics of a qudit polarization vector in a time-dependent magnetic field is presented by solving equations for a density matrix in Hermitian basis. This is realized by means of mapping solution for the polarization vector on the three-dimensional spherical curve (vector hodograph). The obtained results obviously display the interference of precessional and nutational effects on the polarization vector in a magnetic resonance. The study can find the practical applications in a magnetic resonance and 3D visualization of computational data.
We consider the problem of carrying an initial Bloch vector to a final Bloch vector in a specified amount of time under the action of three control fields (a vector control field). We show that this control problem is solvable and therefore it is possible to optimize the control. We choose the physically motivated criteria of minimum energy spent in the control, minimum magnitude of the rate of change of the control and a combination of both. We find exact analytical solutions.
This article gives a brief overview of some recent progress in the characterization and parametrization of density matrices of finite dimensional systems. We discuss in some detail the Bloch-vector and Jarlskog parametrizations and mention briefly the coset parametrization. As applications of the Bloch parametrization we discuss the trace invariants for the case of time dependent Hamiltonians and in some detail the dynamics of three-level systems. Furthermore, the Bloch vector of two-qubit systems as well as the use of the polarization operator basis is indicated. As the main application of th…
