N -qubit states as points on the Bloch sphere

N-qubit states as points on the Bloch sphere

Background and motivation

Mäkelä and Messina compare two ways to represent an arbitrary pure $N$-qubit state as a configuration of $2^N-1$ points on one Bloch sphere. The problem is how to make the structure and transformations of a multiqubit state geometrically accessible, especially its separability and response to operations on individual qubits. The points collectively encode the state, up to normalization and global phase; they are not the ordinary Bloch vectors of the $N$ physical qubits.

The work builds on Majorana's 1932 representation of a pure spin-$S$ state using a symmetrized product of $2S$ spin-$\tfrac12$ states. This construction gives a unique configuration of $2S$ points on the sphere, and a spin-$S$ rotation acts as a rigid rotation of that configuration. The introduction situates this property in earlier work on spinor Bose–Einstein condensates, reference-frame alignment, anticoherent states, and geometrical descriptions of multilevel systems. The authors also build on their own earlier polynomial representation of pure multiqubit states, which provided a criterion for separability. The present paper develops its geometrical interpretation and compares it with Majorana's construction.

Geometric state representation is therefore an established problem, but a representation suited to a single spin need not expose the tensor-product structure relevant to quantum computing. An $N$-qubit Hilbert space has the same dimension as that of a spin $S=(2^N-1)/2$, so the Majorana construction applies after choosing a basis correspondence. However, rotations of this equivalent spin differ from rotations of the individual physical qubits. The paper asks which representation better supports these local operations and makes separability visible.

Mapping a multiqubit state to a constellation

The paper first derives the Majorana polynomial from the symmetrized spin construction and explains how its roots determine spherical coordinates. For the multiqubit application, it writes a pure state in a computational basis indexed by decimal integers,

$$\phi=\sum_{i=0}^{2^N-1}C_i|i\rangle_d, \qquad i=\sum_{j=0}^{N-1}i_j2^j.$$

The qubit indexed by $j=0$ is the least significant bit in this convention. A unitary matrix can specify the correspondence between this basis and the spin basis; the examples choose the identity correspondence, $|i\rangle_d\leftrightarrow|S,i-S\rangle$. This produces the Majorana polynomial

$$P_{\mathrm{Majo}}(\phi;x)=\sum_{i=0}^{2^N-1}\sqrt{\binom{2^N-1}{i}}\,C_i x^i.$$

Each root $x_k$ is placed on the sphere using

$$x_k=\tan\frac{\theta_k}{2}\,e^{i\varphi_k}.$$

The modulus of a root determines its polar angle and its complex argument determines its azimuth. A zero root is at the north pole, while a missing highest-order degree contributes a point at the south pole, corresponding to a root at infinity. Counting these points and repeated roots gives $2^N-1$ points for every state. Changing normalization or global phase rescales the polynomial without changing its roots.

The alternative representation uses the same root-to-sphere mapping but omits the binomial weights:

$$P(\phi;x)=\sum_{i=0}^{2^N-1}C_i x^i.$$

For $N>1$, the authors show that no unitary basis correspondence between the qubit and equivalent-spin spaces makes this polynomial identical to the Majorana polynomial for all states. Removing the binomial weights would require a nonunitary diagonal rescaling. Consequently, the alternative is a different geometric encoding, with different transformation properties, rather than a unitary change of basis within the Majorana construction.

Separability as a structured point configuration

The advantage of the alternative polynomial appears when the state is fully separable:

$$\phi_s=\bigotimes_{j=0}^{N-1}(a_j|0\rangle_j+b_j|1\rangle_j).$$

Its polynomial factorizes as

$$P(\phi_s;x)=\prod_{j=0}^{N-1}(a_j+b_jx^{2^j}).$$

The paper invokes the authors' earlier result that a pure state is separable if and only if its polynomial has this form. For $b_j\ne0$, the factor for qubit $j$ contributes $2^j$ roots,

$$x_{jm}=\left(-\frac{a_j}{b_j}\right)^{1/2^j} \exp\left(\frac{2\pi i m}{2^j}\right), \qquad m=0,\ldots,2^j-1.$$

These roots lie at a common latitude with equally spaced azimuths; if $b_j=0$, the corresponding points lie at the south pole. Thus the constellation of a separable two-qubit state consists of one unrestricted point and a pair opposite each other around the $z$-axis at the same latitude. These two points need not be antipodal on the sphere. For three qubits, four further points form a square in a plane perpendicular to the $z$-axis; for four qubits, another eight form an octagon. Degenerate cases allow points to coincide at a pole. This gives a geometric interpretation of the factorization criterion, although the paper qualifies visual detection of separability in general as possible “at least in principle.”

For a separable state, the effect of single-qubit rotations can be calculated by rotating each pair of amplitudes $(a_j,b_j)$ and substituting the resulting ratio into the root formula. The paper gives explicit expressions for identical Euler-angle rotations of all qubits and explains that qubit-dependent angles extend the calculation to different local rotations. This is an explicit way to compute the changing constellation of a product state; it does not mean that the whole constellation rotates rigidly.

Figures and worked comparisons

The two figures compare the same unnormalized states,

$$\phi_{\mathrm{ent}}=\sqrt{3}|00\rangle+|10\rangle+|01\rangle+\sqrt{3}|11\rangle, \qquad \phi_{\mathrm{sep}}=|00\rangle+|10\rangle+|01\rangle+|11\rangle.$$

Each figure has two rows, one for the entangled state and one for the separable state, and three columns showing the original state, a rotation of the equivalent spin-$\tfrac32$ through $\pi/2$ about the $y$-axis, and simultaneous rotations of both physical qubits through the same angle. The sphere, coordinate axes, equator, and point positions establish the geometric relationships; connecting segments make relative positions easier to compare. The label $\times3$ marks three coincident points at the north pole. These are static illustrations of calculated configurations, not interface screenshots or interactive visualizations.

Figure 1, on PDF page 6, shows the Majorana representation. The equivalent-spin rotation preserves each configuration's relative geometry, as expected from Majorana's rigid-rotation property. Rotating the two physical qubits instead changes the relative positions of the points, with all three points coinciding at the north pole for the separable example. The distinction follows from which degrees of freedom are rotated: the three auxiliary spin-$\tfrac12$ constituents of the Majorana construction are not the two physical qubits.

Figure 2, on PDF page 8, shows the alternative representation. Its separable-state configurations expose the pair symmetry around the $z$-axis described above, and local rotations preserve separability even when they deform the constellation. An equivalent-spin rotation can change that property: in this example it takes $\phi_{\mathrm{ent}}$ to a separable state. The figure also illustrates that the alternative representation loses the rigid-rotation property under equivalent-spin rotations. Finally, the unrotated alternative constellation for $\phi_{\mathrm{sep}}$ equals the unrotated Majorana constellation for $\phi_{\mathrm{ent}}$, because their polynomials agree up to an overall factor. A geometric pattern therefore cannot be interpreted as evidence of entanglement without specifying the representation.

Contributions, evidence, and limitations

The contribution is a mathematical comparison supported by derivations and worked examples. The paper explains the Majorana construction, applies it to arbitrary pure multiqubit states through the equivalent-spin mapping, and develops the geometry of the authors' earlier coefficient polynomial. It establishes why the two encodings are not related by a unitary basis correspondence and shows how the alternative connects product-state factorization to recognizable point configurations and calculable local-rotation behavior. The authors favor Majorana for spin-$S$ systems and their alternative for multiqubit systems because these settings emphasize different operations. There is no user study, performance benchmark, hardware experiment, or implemented interactive visualization system reported.

The treatment is restricted to pure states, and the authors explicitly state that a mixed-state generalization of their alternative description was not known at the time. Their simple local-rotation calculation also applies only to separable states; they do not provide a general rule of this form for the motion of the points of an entangled multiqubit state. The examples show particular entangled-state transformations but do not resolve that general problem. Both encodings require $2^N-1$ points, so the construction itself does not reduce the exponential size of a general pure-state description; the paper does not evaluate visual readability or computational cost as $N$ grows. Its forward-looking suggestion is that the alternative geometry may help visualize transformations and identify special classes of multiqubit states, by analogy with the role of Majorana geometry in spinor-condensate research. That possibility remains a proposed use rather than a demonstrated discovery or validated analysis workflow.