Exploring Imaginary Coordinates: Disparity in the Shape of Quantum State Space in Even and Odd Dimensions

Simon Morelli; Santiago Llorens; Jens Siewert

doi:10.22331/q-2026-07-08-2153

Exploring Imaginary Coordinates: Disparity in the Shape of Quantum State Space in Even and Odd Dimensions

Simon Morelli^1,2, Santiago Llorens^3,4, and Jens Siewert^5,6

¹Technische Universität Wien, Atominstitut, Stadionallee 2, 1020 Vienna, Austria
²BCAM - Basque Center for Applied Mathematics, Mazarredo 14, E48009 Bilbao, Basque Country - Spain
³Division of Quantum Information, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, Wita Stwosza 57, 80-308 Gdańsk, Poland
⁴Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain
⁵Department of Physical Chemistry and EHU Quantum Center, University of the Basque Country UPV/EHU, E-48080 Bilbao, Spain
⁶IKERBASQUE Basque Foundation for Science, E-48009 Bilbao, Spain

Published:	2026-07-08, volume 10, page 2153
Editor:	Leon Loveridge
Eprint:	arXiv:2404.15179v2
Doi:	https://doi.org/10.22331/q-2026-07-08-2153
Citation:	Quantum 10, 2153 (2026).

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Abstract

The state of a finite-dimensional quantum system is described by a density matrix that can be decomposed into a real diagonal, a real off-diagonal and and an imaginary off-diagonal part. The latter plays a peculiar role. While it is intuitively clear that some of the imaginary coordinates cannot have the same extension as their real counterparts the precise relation is not obvious. We give a complete characterization of the constraints in terms of tight inequalities for real and imaginary Bloch-type coordinates. Our description entails a three-dimensional Bloch ball-type model for the state space. We uncover a surprising qualitative difference for the state-space boundaries in even and odd dimensions.

Featured image: The figure shows the restrictions of the real ($S_\text{X}$ and $S_\text{D}$) and imaginary ($S_\text{I}$) components of a quantum system in dimension 5. While the real components (diagonal and off-diagonal) are only bounded by the total Bloch length, the imaginary component obeys a stricter constraint.

Popular summary

One of the distinguishing features of quantum theory is that a system can exist in a superposition of two distinct states. Such a superposition is itself a valid quantum state, making the quantum state space much larger than its classical counterpart. This can be illustrated in the three-dimensional Bloch sphere, which represents the set of pure states of the simplest two-level quantum system, the qubit. While there are only two "classical" states, they are connected by a continuous family of superposition states. The position of a state on the Bloch sphere is determined by the relative weights of the two basis states their complex phase difference.

A similar picture emerges in higher dimensions: the set of pure states consists of all possible superpositions of the classical basis states. However, an important difference arises. Although pure states still lie on a high-dimensional sphere, they no longer occupy it completely. As a result, the state space is not fully symmetric, and its coordinates are subject to nontrivial constraints. Moreover, describing the full quantum state space, the convex hull of all pure states, becomes increasingly challenging as the dimension grows. It is therefore important to identify general constraints that characterize this structure.

In this work, we investigate restrictions relating the real and imaginary components of quantum states. We derive new constraints on the imaginary coordinates of arbitrary finite-dimensional quantum systems, revealing an asymmetry between real and imaginary coordinates that emerges beyond dimension two. Remarkably, the form of these bounds depends on the parity of the dimension, and their generic form for odd dimensions appears only from dimension five onward.

► BibTeX data

@article{Morelli2026exploringimaginary,
  doi = {10.22331/q-2026-07-08-2153},
  url = {https://doi.org/10.22331/q-2026-07-08-2153},
  title = {Exploring {I}maginary {C}oordinates: {D}isparity in the {S}hape of {Q}uantum {S}tate {S}pace in {E}ven and {O}dd {D}imensions},
  author = {Morelli, Simon and Llorens, Santiago and Siewert, Jens},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2153},
  month = jul,
  year = {2026}
}

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