Geometric Visualizations of Quantum Mixed States and Density Matrices

TL;DR

The paper develops a geometric framework in which every quantum state, pure or mixed, corresponds to a point in an $n$-dimensional Euclidean state space with $n=d^2-1$, placing the maximally mixed state at the center. It generalizes the Bloch-sphere picture from qubits to qudits via regular probability simplices (unit $(d-1)$-simplices) and decoherence leaves, linking density-matrix elements to geometric coordinates. Key contributions include a barycentric center-of-mass rule for mixtures, a basis-diagonalization interpretation of density-matrix entries, and explicit expressions for distances and angles in this state space, including the infinite-dimensional limit to $\hat{\rho}_\infty$. These geometric pictures support intuition for measurements, decoherence, and state evolution while offering educational and computational benefits alongside conventional algebraic methods.

Abstract

This paper presents an introduction to geometric representations of quantum states in which each distinct quantum state, pure and mixed, corresponds to a unique point in a Euclidean space. Beginning with a review of some underappreciated properties of the most commonly used geometric representation, the Bloch sphere visualization of qubit states, we show how concepts, algorithms, and spatial relations viewable on this geometric representation can be extended to representations of qudit states of any finite quantum dimension $d$ and on to the infinite-dimensional limit. A primary goal of the work is helping the reader develop a visual intuition of these spaces, which can complement the understanding of the algebraic formalism of quantum mechanics for learners, teachers, and researchers at any level. Particular emphasis is given both to understanding states in a basis-independent way and to understanding how probability amplitudes and density matrix elements used to algebraically represent states in a particular basis correspond to line segments and angles in the geometric representations. In addition to providing visualizations for such concepts as superpositions, mixtures, decoherence, and measurement, we demonstrate how the representations can be used to substitute simple geometrical calculations for more cumbersome linear algebra ones, which may be of particular use in introducing mixed states and density matrices to beginning quantum students at an early stage. The work concludes with the geometrical interpretation of some commonly used metrics such as the purity of states and their relation to real, Euclidean vectors in the infinite-dimensional limit of the space, which contains all lower-dimensional qudit spaces as subspaces.

Figures (10)

• Figure 1: (a) A selection of pure qubit statepoints on a Bloch circle cross section of a Bloch sphere corresponding to different linear light polarization states (left) and different spin-1/2 states (right). (b) The statepoint representing a 70% chance of an up state and a 30% chance of a down state is found by dividing the line connecting these two points in segments corresponding to the probabilities. (c) In general, statepoints can be found using a center-of-mass algorithm with probabilities replacing masses, such as is shown for the state with equal likelihood of being an up, down, or right state. (d) By this algorithm, it is clear that the statepoint for the state with equal likelihood of all pure states is the center of the circle, corresponding to the maximally mixed qubit state.
• Figure 2: (a) Geometrically, the probabilities of a projective quantum measurement on a qubit involve choosing a Bloch sphere diameter that terminates in two pure statepoints and drawing a perpendicular line from the statepoint being measured to this diameter. The perpendicular projection cuts the diameter into two sections, equal in length to the probabilities of the measurement resulting in one of the two pure states. (b) A single indeterministic, irreversible process of measurement conventionally taught when only pure states are referenced: a measured pure state evolves to one of two pure states determined by the choice of measurement diameter. (c) The same measurement as a two-step process: an, in principle, deterministic decoherence process of the pure state evolving in the decoherence leaf disk to a mixed state along the measurement diameter, followed by an indeterministic outcome of the measured mixed state.
• Figure 3: (a) Representations of the state of Fig. \ref{['figBlochCircle']}(c) in two different bases. The matrix elements of the representations are equal to the lengths of the perpendicular from the statepoint to the basis diameter (off-diagonal) and the lengths of the segments of the diameter (diagonal elements). (b) A section of the full Bloch sphere made by rotating a semicircle about a possible measurement axis. For a fixed set of measurement probabilities, the location of a statepoint within the "decoherence leaf" disk can be specified by a single complex number $r_ce^{i\phi}$ encoding its distance $r_c$ from and angle $\phi$ about the measurement diameter. (c) For any semicircular section of the Bloch sphere the $r_ce^{i\phi}$ decoherence leaf coordinate and its complex conjugate serve as the off-diagonal elements of the density matrix. For pure statepoints, an alternate representation is a column vector, where the magnitudes of the entries $a$ and $b$ are equal to the lengths of the lines connecting the statepoint to the basis statepoints.
• Figure 4: (a) A regular, unit 1-simplex diameter of a partially-shown Bloch Sphere terminating at two zero-entropy pure states and bisected by the $d=2$ maximally mixed state with entropy $S=k_B\log{W}=k_B\log{2}$. All qubit statepoints lie along one such diameter. (b) A regular, unit 2-simplex equilateral triangle with sides formed by qubit 1-simplices with the maximally mixed $W=3$ statepoint at its center. The three Bloch Spheres corresponding to these 1-simplices are embeded in the 8-dimensional qutrit statespace where they intersect 2-simplices only at their edges and touch each other only at a single pure statepoint. (c) The regular, unit 3-simplex cross-section of the $d=4$ statespace with the $W=4$ maximally mixed statepoint at its center. The tetrahedron's sides are distinct qutrit 2-simplices and its edges the qubit 1-simplices. (d) All qutrit statepoints lie in one of the 2-simplices; drawing lines through this statepoint parallel to the edges cuts the unit edges in lengths equal to the diagonalized density matrix measurement probabilities. An interactive version of this diagram is found https://quantum.bard.edu/singleAnimation/qutritsimplex.html. (e) All $d=4$ statepoints lie in one of the 3-simplices; drawing planes through this statepoint parallel to the sides cuts the unit edges in lengths equal to the diagonalized density-matrix measurement probabilities and their sums. An interactive version of this diagram is found https://quantum.bard.edu/singleAnimation/2qubitsimplex.html.
• Figure 5: For any state $\hat{\rho}$ and any choice of basis, the line $r_c$ in the decoherence leaf of the basis containing the $\hat{\rho}$ statepoint and its projection $\hat{\rho}_P$ statepoint on the basis simplex is perpendicular to the simplex. Connecting these two statepoints to the maximally mixed statepoint $\hat{\rho}_{M,d}$ at the center of the simplex forms a right triangle as shown in the (a) $d=2$ qubit case where the leaf is 2-dimensional, and (b) $d=3$ qutrit case where the leaf is 6-dimensional. (c) The length of line $r_{B\Psi}$ between any two pure states $\ket{B}$ and $\ket{\Psi}$ on the extremal boundary of a statespace is equal in length to a probability amplitude for the state $\ket{\Psi}$ expanded in a qubit basis containing $\ket{B}$. Lines from the basis states to the statepoint form an alternate geometric parametrization of its location used in column- and row-vector representations of pure states.