From Majorana to Bloch: two ways to generalize the Bloch sphere representation

arXiv:2610.11868 · quant-ph · Submitted 2026-10-08 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai, and with me are Mira and Lev, guest researcher.

Mira: Today's paper: "From Majorana to Bloch".

Kai: The gist The generalization of the Bloch sphere representation to pure qudit states involves comparing two distinct geometric approaches: increasing the number of points, as in the Majorana representation,

Mira: First, who's behind it and why it matters.

Paper summary: Mira: So, to wrap up this paper "From Majorana to Bloch: two ways to generalize the Bloch sphere representation", we established that there are distinct geometric pathways for representing pure qudit states beyond qubits, namely the generalized Bloch representation and the Majorana constellation.

Kai: And what we found was that linking these two representations through explicit analytical relations is a powerful way to study quantum correlations geometrically, especially when looking at special classes of states like permutation-symmetric two-qubit states.

Lev: For me, the most practical part is how they connect geometric discord to one-anticoherence for those two-qubit states, which gives us a quantifiable relationship between those correlation measures <ref:2610.11868#pg1>.

Mira: Exactly. And because they showed how these relations help reduce the number of parameters for rotationally invariant functions, it gives us a way to simplify optimization tasks in quantum information theory.

Kai: It opens up questions about how general tensors might be expressed using the Majorana constellation for an arbitrary number of parties, which is a big area for future work.

Lev: The paper's limitation is that they focus on pure states and symmetric cases, so applying these exact relations to more complex mixed states or non-symmetric setups would require new mathematical machinery.

Mira: Right. But overall, the work shows the utility of the Majorana representation in analyzing rotationally invariant properties of quantum states, providing a clear comparison between geometric tools for different dimensions.

Conclusion: Kai: So, we've looked at how these two ways of mapping pure qudit states—the Majorana constellation and the generalized Bloch vector—actually relate to each other.

Mira: Right, so basically, they're just different lenses you can use to look at the same quantum state; one gives you a set of points on a sphere, the other gives you one central point in a higher-dimensional space.

Kai: I think that distinction is what makes this paper really interesting because it shows how these two geometric ideas aren't just academic exercises, but tools we can actually use to measure quantum correlations.

Mira: Exactly. It’s not just about drawing pretty pictures on the Bloch sphere; it’s about finding a mathematical bridge between them so we can quantify things like discord.

Kai: And what the authors did was find that direct link for pure states, which means we can start using these Majorana points to calculate things like correlation measures without having to deal with all that extra complexity upfront.

Mira: It simplifies the math significantly, especially when they move on to those specific two-qubit and three-qubit symmetric states where they actually provide formulas for the reduced states and correlation matrices.

Kai: So, we've seen that these representations are comparable, useful for optimization, and give us new ways to measure things like geometric discord.

Mira: But the real weight of this work is showing how this geometric approach connects something called quantum discord directly to one-anticoherence for those specific states.

Kai: Yeah, and it suggests that these two measures are linked in a very concrete way when you look at the spin vector in that context.

Mira: It opens up questions about applying this general framework to even more complex systems, like multipartite ones, which is where the real future work lies.

Tikhonov Moscow Institute of Electronics and Mathematics, HSE University

quant-ph

Submitted: 2026-10-08

Updated: 2026-10-08

Comments: 20 pages

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 72/100

The gist: The gist The generalization of the Bloch sphere representation to pure qudit states involves comparing two distinct geometric approaches: increasing the number of points, as in the Majorana

Key concepts

Bloch Vector
This is a vector used in the generalized Bloch representation to associate every qudit state with a single point in R^(d-1). It serves as a geometric descriptor of the state, allowing researchers to study quantum states using standard vector space geometry.
Majorana Representation
This approach generalizes the Bloch sphere by associating each pure qudit state with d-1 specific points on the unit sphere (called stars). The collection of these points forms a 'Majorana constellation,' providing a richer geometric structure for higher dimensions.
Generalized Bloch Representation
This method represents every pure qudit state as a single point within a higher-dimensional space, specifically R^(d-1). While simpler in terms of representation (one point), it requires analyzing the geometry within this larger space to understand the state's properties.
Geometric Quantum Discord
This is a measure used to quantify quantum correlations based on the geometric distance between states. The paper shows that for certain symmetric states, this discord can be related directly to spin 1-anticoherence, offering a new way to link correlation measures.

Terminology

Summary

The gist The generalization of the Bloch sphere representation to pure qudit states involves comparing two distinct geometric approaches: increasing the number of points, as in the Majorana representation, or using a single point in a larger space, as in the generalized Bloch representation.

General Representations

The paper compares two primary ways to generalize the Bloch sphere for pure qudit states of dimension d > 2: increasing the number of points via the Majorana representation or using a single point in a higher-dimensional space through the generalized Bloch representation. The generalized Bloch representation associates every qudit state, pure or mixed, with a point in R d 2−1 referred to as its Bloch vector. In contrast, the Majorana representation associates every pure qudit state with a set of d − 1 points on the unit sphere S2⊂R3 that are usually referred to as stars, and the whole set as a (Majorana) constellation.

Relation Between Representations

The core of the work is finding explicit analytical relations between these two representations for pure qudit states. Specifically, a general formula of the Bloch vector in terms of the Majorana points of a given state is provided. Theorem 1 provides an exact analytical expression for the Bloch vector in terms of the Majorana constellation. This relation allows for studying quantum correlations from a geometric point of view.

Application to Symmetric States

The analytical results are extended to pure permutation-symmetric two- and three-qubit states. For two-qubit states, Theorem 2 expresses the Bloch vector of the reduced state rρ2 = sρ2 and the correlation matrix Tρ2 in terms of the Majorana stars. For three-qubit states, Theorem 3 provides expressions for the correlation parameters rρ3, Tρ3, and Mρ3 in terms of its Majorana stars.

Quantum Correlations

The geometric approach allows for the quantification of quantum correlations through measures like the geometric quantum discord. For permutation-symmetric two-qubit states, this is connected to a measure of 1-anticoherence based on total variance. Theorem 4 establishes that the geometric quantum discord with respect to the Hilbert-Schmidt distance coincides with half of the value of the 1-anticoherence of its corresponding spin vector under a specific correspondence. This confirms a quantitative relation between spin 1-anticoherence and quantum correlations in this context.

Optimization Potential

These analytical relations are useful for reducing the number of parameters in optimization tasks involving rotationally invariant functions. For two- and three-qubit cases, this reduction is achieved by expressing parameters in terms of the Majorana stars, allowing optimization to be carried out in the shape space.

Conclusion

The work provides explicit analytical relations between the Bloch vector and the Majorana constellation for pure qudit states. These relations are applicable to multipartite pure symmetric states of n qubits, allowing analysis of quantum correlations via geometric measures. The findings offer a path to reduce parameters in optimization problems concerning rotationally invariant functions on pure symmetric quantum states. The paper also opens questions regarding the expression of general tensors in terms of the Majorana constellation for arbitrary number of parties. The results establish a connection between geometric quantum discord and 1-anticoherence for pure-symmetric two-qubit states. The study demonstrates the utility of the Majorana representation in analyzing rotationally invariant properties of quantum states. The comparison between the two representations is essential for selecting the appropriate tool for specific tasks. The paper concludes by showing how geometric discord and 1-anticoherence are constant along SU(2)-fibers for this class of states. The results can be used to analyze quantum correlations in terms of the corresponding tensors of different ranks. The geometric approach is particularly useful for problems involving the optimization of rotationally invariant functions on quantum states. The paper concludes by showing how geometric discord and 1-anticoherence are constant along SU(2)-fibers for this class of states. The results can be used to analyze quantum correlations in terms of the corresponding tensors of different ranks. The geometric approach is particularly useful for problems involving the optimization of rotationally invariant functions on quantum states. The paper concludes by showing how geometric discord and 1-anticoherence are constant along SU(2)-fibers for this class of states.

Improvements for AI systems

  1. Improve state representation for pure qudit states by using a single Bloch vector in R d2−1, as described in The generalized Bloch representation, hereafter Bloch representation, associates every qudit state, pure or mixed, with a point in R d2−1, which is suitable for experimental contexts [1].

  2. Develop analytical tools to find the general formula of the Bloch vector in terms of the Majorana points of a given state, allowing for geometric analysis by finding an explicit analytical relation between them.

  3. Enable rotationally invariant function optimization by reducing parameters, as suggested by the paper's conclusion: "these results could also be useful when solving optimization problems of rotationally invariant functions on the set of pure symmetric quantum states in the two- and three-qubit cases, by reducing the number of required parameters to 1 and 3 parameters, respectively."

  4. Quantify quantum correlations for permutation-symmetric two-qubit states using geometric discord, as shown in Section 5, allowing for a geometric approach that is invariant under local rotations.

  5. Establish a direct link between the geometric quantum discord and the measure of 1-anticoherence for spin systems, as demonstrated by Theorem 4: The geometric quantum discord with respect to the Hilbert-Schmidt distance of a pure-symmetric state of two-qubits coincides with half of the value of the 1-anticoherence.

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