---
title: "BEADS: a canonical visualization of quantum states for applications in quantum information processing"
authors:
  - D Huber
  - S J Glaser
abstract: We introduce a generalized phase-space representation of qubit systems called the BEADS representation which makes it possible to visualize arbitrary quantum states in an intuitive and an easy to grasp way. Our representation is exact, bijective, and general. It bridges the gap between the highly abstract mathematical description of quantum mechanical phenomena and the mission to convey them to non-specialists in terms of meaningful pictures and tangible models. Several levels of simplifications can be chosen, e.g. when using the BEADS representation in the communication of quantum mechanics to the general public. In particular, this visualization has predictive power in contrast to simple metaphors such as Schrödinger’s cat.
summaryType: survey
sourceStatus: null
sources:
  - https://doi.org/10.1088/1367-2630/ae0514
---

# BEADS: a canonical visualization of quantum states for applications in quantum information processing

[Original paper (PDF)](https://doi.org/10.1088/1367-2630/ae0514)

## Background and motivation

BEADS addresses the established problem of making multiqubit quantum states understandable without abandoning their mathematical content.
A Bloch vector gives a complete picture of an individual qubit, but a collection of individual Bloch vectors omits the correlations required to characterize a multiqubit state.
State-vector displays and density-matrix cityscape plots can preserve the relevant information, yet identifying entanglement, spatial symmetries, or the effects of collective rotations from them can require substantial calculation.
The authors aim to represent these properties in pictures that also support quantitative predictions of measurement statistics, with applications in quantum-information education, communication, and analysis of small quantum systems.

The work builds on continuous phase-space representations, including Wigner and Husimi functions, and particularly on the authors’ earlier DROPS representation.
DROPS decomposes operators using the LISA tensor basis, which organizes components by how many qubits they involve, which qubits they involve, and auxiliary symmetry properties.
This preserves individually addressable qubits, a useful distinction from multipole representations organized primarily by angular momentum.
However, the values of DROPS functions correspond to expectation values of tensor operators that may combine several Pauli products, so interpreting them as familiar measurement quantities is difficult.
BEADS extends this construction through new scaling factors, explicit separation of symmetry components, and a representation of connected and compound correlation functions.
Its name refers to the beadlike appearance of its colored spheres; it is not the expansion of an acronym.

## Mathematical construction and completeness

BEADS maps the density operator onto a collection of real functions defined on spheres.
An operator is first expanded in the LISA basis, with each component assigned to a labeled set of tensor operators.
BEADS further separates these sets according to even or odd point-inversion symmetry and scales spherical harmonics of different ranks independently.
In schematic form, the mapping is

$$
A^{(\ell')}=\sum_{j,m}c_{j,m}^{(\ell')}T_{j,m}^{(\ell')}
\quad\longleftrightarrow\quad
b_A^{(\ell')}(\theta,\phi)=\sum_{j,m}s_j^{(\ell')}(N,g)c_{j,m}^{(\ell')}Y_{j,m}(\theta,\phi),
$$

where $N$ is the number of qubits, $g$ is the number involved in the component, and $\ell'$ identifies its subsystem and symmetries.
The nonzero scaling factors can be inverted to recover the original operator coefficients, establishing completeness of the full representation.
For fully permutation-symmetric components, the scaling makes function values equal to expectation values of Pauli products measured along a common direction.
Other symmetry components use scaling based on global unitary bounds to provide a bounded interpretation of their values.
The resulting representation is related to a generalized Husimi function rather than retaining all the normalization conditions of a Wigner representation.

A bead’s even or odd label describes the behavior of its function at antipodal surface points.
Even functions have equal values there, while odd functions have opposite values.
Separating these components makes the hidden hemisphere inferable from a visible hemisphere, which is useful in static top views and circuit diagrams.
These labels must be distinguished from the parity of measurement outcomes and from the permutation symmetries of the density operator.
The latter are also distinct from the exchange symmetry of a state vector, because a global sign change does not change its density operator.

## Visual encoding of qubits and correlation functions

The standard rendering uses spheres of fixed radius and encodes each function’s value through surface color, rather than using distance from the center to encode magnitude.
Q-Beads represent individual reduced density operators and contain the same information as local Bloch vectors.
Their red pole points along the Bloch vector, while decreasing color intensity indicates a shorter vector.
Red, black, and green correspond to expectation values $+1$, $0$, and $-1$, respectively, or probabilities $0$, $1/2$, and $1$ of the corresponding single-qubit outcome labeled 1.
For a globally pure state, reduced Q-Bead brightness indicates entanglement of that qubit with the rest of the system; an entirely black Q-Bead represents a maximally mixed reduced state.
For a general mixed global state, black Q-Beads alone do not establish entanglement.



Figure 1 connects red and green classical-bit patches to the north-pole colors of Q-Beads and compares top and oblique views with Bloch vectors.
Figure 2 gives a schematic explanation of the color scale through simulated measurement-outcome grids, rather than reporting a hardware experiment or a usability result.
The probability associated with a surface point concerns a measurement in the direction from the bead’s center to that point.
Thus the sphere simultaneously displays directional measurement statistics, rather than merely marking a state’s orientation.

Correlation beads add information that Q-Beads cannot contain.
T-Beads visualize total correlation functions, which are expectation values of products of local observables.
For pure states, the authors separate these into connected contributions, shown by E-Beads, and compound contributions, shown by C-Beads.
For two qubits with local observables $O_1$ and $O_2$, this decomposition is

$$
T_{12}=\langle O_1O_2\rangle=C_{12}+E_{12},
\qquad
C_{12}=\langle O_1\rangle\langle O_2\rangle,
\qquad
E_{12}=\langle O_1O_2\rangle-\langle O_1\rangle\langle O_2\rangle.
$$

Higher-order connected functions are obtained through Ursell-function relations that subtract products of lower-order contributions.
Compound components are redundant given the lower-order information; they should not be interpreted as a separate category of classical correlation in a pure state.
E-Beads use yellow and blue for positive and negative connected coefficients, with black at zero.
All-black correlation beads are normally omitted to reduce clutter, while black Q-Beads remain visible because they represent physical qubits.
Lines connect relevant qubits, and their thickness can provide an auxiliary indication of the norm of an entanglement-related component.



Figure 5 shows $|\psi_\theta\rangle=\cos(\theta/2)|00\rangle+\sin(\theta/2)|11\rangle$ as $\theta$ increases from $0$ to $\pi/2$.
The Q-Beads become darker while the connected-correlation bead becomes brighter, distinguishing the product-state endpoint from the Bell-state endpoint.
Figure 6 then separates the compound and connected components and shows their combination in T-Beads.
The extended T-Bead color scheme uses brightness for the magnitude of the total coefficient and hue for the relative connected and compound contributions.
For example, the Schmidt-family $zz$ total correlation remains 1 while its two contributing components change, so the total-correlation color changes hue without implying a change in that total coefficient.

Measurement prediction requires this distinction between E-Beads and T-Beads.
For symmetric measurements along a common direction $\mathbf r$, a fully permutation-symmetric T-Bead gives the expectation value of $O=\bigotimes_{k\in G}(\mathbf r\cdot\boldsymbol\sigma_k)$ and hence the probability of an odd number of outcome-1 results:

$$
p_{\mathrm{odd}}=\frac{1-\langle O\rangle}{2}.
$$

Connected coefficients alone are generally insufficient for this prediction when compound terms are nonzero.
For the Bell-state examples, the total and connected functions coincide, allowing their E-Beads to support the direct probability interpretation.
Beads for other permutation symmetries generally encode linear combinations of measurement quantities and do not support the same direct reading as a single projective-measurement probability.
Measurements in different directions on different qubits require a fuller analysis of both symmetric and antisymmetric components.

## Circuit views, dynamics, and software

BEADS-augmented circuits place Q-Beads on their corresponding qubit wires after operations and correlation beads on additional lighter lines.
The latter lines organize correlation components; they do not denote extra physical qubits.
Figure 7 uses this arrangement to show the creation of GHZ and W states, together with oblique views of their final representations.
The GHZ example has black single-qubit beads and a threefold rotational pattern in its three-qubit correlation bead, whereas the W example retains nonzero single-qubit components and has axial symmetry around the $z$ axis.
These pictures expose different correlation structures without requiring readers to infer them from amplitudes alone.

The representation also supports continuous simulated dynamics.
Single-qubit gates rotate Q-Beads, and multiqubit operations can change their brightness as well as their orientation.
Correlation beads can rotate, change intensity, or change their surface patterns, a transformation the paper calls morphing.
Applying the same local rotation to every qubit involved in a correlation component produces a simple spatial rotation, whereas selective operations can change its pattern.
Figure 8 illustrates the creation and removal of correlations during two-qubit Grover search and the variation over successive three-qubit Grover iterations.
Figure 9 visualizes teleportation, including four possible measurement branches, their classical corrections, and a final weighted mixture of corrected outcomes.
The displayed summation step represents construction of that mixed-state description, rather than a physical gate.

The implemented QuBeads software supplies a graphical gate palette and modular circuit construction without requiring code.
Its standard mode displays the evolving current state alongside the circuit, while its augmented-circuit mode shows bead representations after individual operations and can display possible measurement outcomes together.
Figures H1 and H2 are screenshots of these two software modes, including the circuit editor and playback controls.
The publication describes a beta implementation supporting systems of up to three qubits and states that its figures were generated with QuBeads simulations.
This implementation limit is separate from the broader mathematical construction.

## Evidence and contributions

The paper’s support consists of mathematical derivations, illustrative state and circuit examples, and qualitative comparisons of representations.
Figure 4 compares BEADS with density-matrix cityscapes and Q-Spheres for selected states, including two states related by a collective rotation.
Table 1 presents the authors’ assessment against criteria such as completeness, measurement interpretation, visibility of entanglement, and educational suitability.
Figure G3 further demonstrates that the scaling changes the functions themselves, not only their rendering: the selected Pauli-product examples have directly interpretable cosine-power functions in BEADS but different combinations in DROPS.
These comparisons explain the design rationale; they are not measured task-performance rankings.

The principal contribution is an invertible multiqubit representation that connects local state information, symmetry components, and directional correlation functions in a common visual notation.
The separation of connected and compound contributions is adapted from prior work rather than introduced as a new theory of correlation.
The new mapping, its scaling, the coordinated bead types, circuit augmentation, and working simulator together make those quantities accessible in static and dynamic views.
The authors also report educational and outreach use and the creation of tangible models, but provide no controlled user study, learning-outcome measurements, or quantitative scalability benchmark.
Claims about ease of understanding therefore rest on the construction and worked examples rather than a demonstrated advantage for a sampled user population.

## Limitations and future directions

The full mapping can represent pure and mixed density operators, but the paper’s entanglement-specific interpretation of connected functions is restricted to pure global states.
In mixed states, connected correlations can reflect classical correlations, quantum discord, or entanglement, and the presented subtraction procedure does not distinguish these sources.
Isolating mixed-state entanglement-related contributions is explicitly identified as an open problem.
The representation also omits global phase and does not directly visualize probability amplitudes; the authors suggest complementing it with state-vector representations such as dimensional circle notation when those quantities matter, particularly for algorithm design.

Computational and visual complexity still grow exponentially with the number of qubits, and calculating the connected-correlation operator can become expensive.
Figure 10 organizes alternatives that trade detail for simplicity.
Omitting compound beads can retain complete information because those terms are recoverable, but showing only selected permutation symmetries or only Q-Beads generally loses information.
Combining even and odd components reduces the number of beads but removes the guarantee that one viewing perspective determines the hidden hemisphere.
Consequently, mathematical completeness applies to the appropriate full representation, not to every simplified display.

Color is another practical constraint.
The default banded scales facilitate rough numerical reading but discretize the displayed values, so an image is not an exact numerical substitute for the underlying functions.
Appendix F provides continuous, grayscale, high-contrast, and alternative paired-color schemes, including alternatives intended for color-vision deficiencies.
These options are proposed and illustrated without a perceptual evaluation.
The authors plan to extend QuBeads beyond three qubits and discuss the representation’s extension to qudit systems, while the paper’s detailed demonstrations remain centered on small qubit systems.