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arXiv:quant-ph/0601013 · Journal of Physics A Mathematical and General

Generalized Bloch spheres form-qubit states

Klaus Dietz

Published Jan 24, 2006• 21 citations
arXivHTMLPDF

Abstract

m -qubit states are embedded in Clifford algebras. Their probability spectrum then depends on O (2 m )- or O (2 m + 1)-invariants, respectively. Parameter domains for O (2 m (+1))-vector and -tensor configurations, generalizing the notion of a Bloch sphere, are derived.
DOI
10.1088/0305-4470/39/6/016

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We represent a two-qubit density matrix in the basis of Pauli matrix tensor products, with the coefficients constituting a Bloch matrix, analogous to the single qubit Bloch vector. We find the quantum state positivity requirements on the Bloch matrix components, leading to three important inequalities, allowing us to parametrize and visualize the two-qubit state space. Applying the singular value decomposition naturally separates the degrees of freedom to local and nonlocal, and simplifies the positivity inequalities. It also allows us to geometrically represent a state as two entangled Bloch spheres with superimposed correlation axes. It is shown that unitary transformations, local or nonlocal, have simple interpretations as axis rotations or mixing of certain degrees of freedom. The nonlocal unitary invariants of the state are then derived in terms of local unitary invariants. The positive partial transpose criterion for entanglement is generalized, and interpreted as a reflection, or a change of a single sign. The formalism is used to characterize maximally entangled states, and generalize two qubit isotropic and Werner states.
While the Bloch sphere provides a canonical geometric representation for qubits, its generalization to pure qudit states (d>2d>2d>2) is not unique. This representation can be extended following two different approaches: increasing the number of points, as in the Majorana representation, or using a single point but in a space of larger dimension, as in the generalized Bloch representation. In this work, we compare these representations of pure qudit states by finding explicit analytical relations between them. In particular, a general formula of the Bloch vector in terms of the Majorana points of a given state is provided. We also analyze some suitable extensions of these representations in the case of pure permutation-symmetric two- and three-qubit states, deriving the corresponding relation between them. These analytical results allow us to study, from a geometric point of view, quantum correlations of special classes of two-qubit states.
The Bloch sphere is an elegant tool for representing single-qubit states. However, a widely accepted generalization for multi-qubit systems with entanglement remains absent. We propose a novel geometric model extending the Bloch sphere representation to arbitrary NNN-qubit systems using 2N−12^N-12N−1 spheres. We demonstrate that any pure 2-qubit state is uniquely described by three spheres: two for individual qubits and a third encapsulating bipartite entanglement. Generalizing this, we establish an NNN-qubit parameterization through the hierarchical application of controlled rotation gates along the ZZZ and YYY axes. We formally prove a strict bijection between the standard state vector representation and our model's angular parameters. This framework provides an intuitive visualization of multiple entanglement, offering potential computational advantages for quantum simulators and new analytical perspectives on quantum gates.
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 a\mathbf{a}a and a metric tensor $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}$ which satisfy $\mathbf{a}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{a}\ensuremath{\le}1$. This resembles the well known condition for qubit Bloch vectors in which case $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}=\mathbb{1}$. In our case the vector a\mathbf{a}a corresponds to spin-1 polarization, whereas the tensor $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}$ is a function of polarization uncertainties. Alternatively, a\mathbf{a}a is a local Bloch vector of a symmetric two-qubit state and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Gamma}}}$ is a function of the corresponding correlation tensor.
Abstract We study an analogous Bloch sphere representation of higher-level quantum systems using the Heisenberg-Weyl operator basis. We introduce a parametrization method that will allow us to identify a real-valued Bloch vector for an arbitrary density operator. Before going into arbitrary d-level (d ≥ 3) quantum systems (qudits), we start our analysis with three-level ones (qutrits). It is well known that we need at least eight real parameters in the Bloch vector to describe arbitrary three-level quantum systems (qutrits). However, using our method we can divide these parameters into four weight, and four angular parameters, and find that the weight parameters are inducing a unit sphere in four-dimension. And, the four angular parameters determine whether a Bloch vector is physical. Therefore, unlike its qubit counterpart, the qutrit Bloch sphere does not exhibit a solid structure. Importantly, this construction allows us to define different properties of qutrits in terms of Bloch vector components. We also examine the two and three-dimensional sections of the sphere, which reveal a non-convex yet closed structure for physical qutrit states. Further, we apply our representation to derive mutually unbiased bases (MUBs), characterize unital maps for qutrits, and assess ensembles using the Hilbert-Schmidt and Bures metrics. Moreover, we extend this construction to qudits, showcasing its potential applicability beyond the qutrit scenario.