Exploring Imaginary Coordinates: Disparity in the Shape of Quantum State Space in Even and Odd Dimensions
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AbstractThe 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 summaryOne 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} }► References [1] J. v. Neumann, Mathematische Begründung der Quantenmechanik, Nachr. Ges. Wiss. Gött., Math.-Phys. Kl. , 1 (1927). http://eudml.org/doc/59215 [2] P. A. M. Dirac, The Principles of Quantum Mechanics (Clarendon Press, Oxford, 1930). [3] J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed. (Cambridge University Press, 2020). [4] K. F. Pál and T. Vértesi, Efficiency of higher-dimensional Hilbert spaces for the violation of Bell inequalities, Phys. Rev. A 77 (2008). https://doi.org/10.1103/PhysRevA.77.042105 [5] M. McKague, M. Mosca, and N. Gisin, Simulating quantum systems using real hilbert spaces, Phys. Rev. Lett. 102 (2009). https://doi.org/10.1103/PhysRevLett.102.020505 [6] M.-O. Renou, D. Trillo, M. Weilenmann, T. P. Le, A. Tavakoli, N. Gisin, A. Acín, and M. Navascués, Quantum theory based on real numbers can be experimentally falsified, Nature 600, 625 (2021). https://doi.org/10.1038/s41586-021-04160-4 [7] C. Caves, C. Fuchs, and P. Rungta, Entanglement of Formation of an Arbitrary State of Two Rebits, Found. Phys. Lett. 114, 199 (2001). https://doi.org/10.1023/A:1012215309321 [8] W. Wootters, Entanglement Sharing in Real-Vector-Space Quantum Theory, Found. Phys. 42, 19 (2012). https://doi.org/10.1007/s10701-010-9488-1 [9] Q. Chen, T. Gao, and F. Yan, Measures of imaginarity and quantum state order, Sci. China-Phys. Mech. Astron. 66 (2023). https://doi.org/10.1007/s11433-023-2126-9 [10] J. Xu, Imaginarity of Gaussian states, Phys. Rev. A 108 (2023). https://doi.org/10.1103/PhysRevA.108.062203 [11] C. Fernandes, R. Wagner, L. Novo, and E. F. Galvão, Unitary-Invariant Witnesses of Quantum Imaginarity, Phys. Rev. Lett. 133 (2024). https://doi.org/10.1103/PhysRevLett.133.190201 [12] P. B. Hita, A. Trushechkin, H. Kampermann, M. Epping, and D. Bruß, Quantum mechanics based on real numbers: A consistent description, Phys. Rev. Lett. (2026). https://doi.org/10.1103/4k13-sdjh [13] T. Hoffreumon and M. Woods, Quantum theory based on real numbers cannot be experimentally falsified, (2026). arXiv:2603.19208 [14] A. Hickey and G. Gour, Quantifying the imaginarity of quantum mechanics, J. Phys. A 51, 414009 (2018). https://doi.org/10.1088/1751-8121/aabe9c [15] K.-D. Wu, T. V. Kondra, S. Rana, C. M. Scandolo, G.-Y. Xiang, C.-F. Li, G.-C. Guo, and A. Streltsov, Operational resource theory of imaginarity, Phys. Rev. Lett. 126 (2021a). https://doi.org/10.1103/PhysRevLett.126.090401 [16] K.-D. Wu, T. V. Kondra, S. Rana, C. M. Scandolo, G.-Y. Xiang, C.-F. Li, G.-C. Guo, and A. Streltsov, Resource theory of imaginarity: Quantification and state conversion, Phys. Rev. A 103 (2021b). https://doi.org/10.1103/PhysRevA.103.032401 [17] I. Bengtsson and K. Życzkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement, 2nd edition (Cambridge University Press, 2017). https://doi.org/10.1017/9781139207010 [18] I. Bengtsson, S. Weis, and K. Życzkowski, Geometry of the Set of Mixed Quantum States: An Apophatic Approach (Springer Basel, 2012) p. 175. https://doi.org/10.1007/978-3-0348-0448-6_15 [19] R. A. Bertlmann and P. Krammer, Bloch vectors for qudits, J. Phys. A 41, 235303 (2008). https://doi.org/10.1088/1751-8113/41/23/235303 [20] J. Siewert, On orthogonal bases in the Hilbert-Schmidt space of matrices, J. Phys. Comm. 6 (2022). https://doi.org/10.1088/2399-6528/ac6f43 [21] G. Kimura, The bloch vector for n-level systems, Phys. Lett. A 314, 339 (2003). https://doi.org/10.1016/s0375-9601(03)00941-1 [22] C. Eltschka, M. Huber, S. Morelli, and J. Siewert, The shape of higher-dimensional state space: Bloch-ball analog for a qutrit, Quantum 5, 485 (2021). https://doi.org/10.22331/q-2021-06-29-485 [23] A. Khvedelidze, D. Mladenov, and A. Torosyan, Parameterizing qudit states, Discrete Contin. Models Appl. Comput. Sci. 29, 361 (2021). https://journals.rudn.ru/miph/article/view/29429 [24] M. H. Levitt and C. Bengs, Hyperpolarization and the physical boundary of Liouville space, Magn. Reson. 2, 395 (2021). https://doi.org/10.5194/mr-2-395-2021 [25] A. R. P. Rau, Symmetries and Geometries of Qubits, and Their Uses, Symmetry 13 (2021). https://doi.org/10.3390/sym13091732 [26] S. Jarov and M. V. Raamsdonk, Mapping the space of quantum expectation values, (2023), arXiv:2310.13111. arXiv:2310.13111 [27] G. Sharma, S. Ghosh, and S. Sazim, Bloch sphere analog of qudits using Heisenberg-Weyl Operators, Phys. Scr. 99, 045105 (2024). https://doi.org/10.1088/1402-4896/ad2ccf [28] S. Morelli, C. Eltschka, M. Huber, and J. Siewert, Correlation constraints and the Bloch geometry of two qubits, Phys. Rev. A 109 (2024). https://doi.org/10.1103/PhysRevA.109.012423 [29] S. Shravan, S. Morelli, O. Gühne, and S. Imai, Geometry of two-body correlations in three-qubit states, Phys. Rev. A 110 (2024). https://doi.org/10.1103/PhysRevA.110.062419 [30] P. Kurzyński, A. Kołodziejski, W. Laskowski, and M. Markiewicz, Three-dimensional visualization of a qutrit, Phys. Rev. A 93, 062126 (2016). https://doi.org/10.1103/physreva.93.062126 [31] N. Wyderka and O. Gühne, Characterizing quantum states via sector lengths, J. Phys. A 53, 345302 (2020). https://doi.org/10.1088/1751-8121/ab7f0a [32] M. Ozawa, Entanglement measures and the Hilbert-Schmidt distance, Phys. Lett. A 268, 158 (2000). https://doi.org/10.1016/s0375-9601(00)00171-7 [33] M. B. Ruskai, Beyond strong superadditivity? Improved bounds on the contraction of generalized relative entropy, Rev. Math. Phys. 06, 1147 (1994). https://doi.org/10.1142/S0129055X94000407 [34] R. Feynman, F. Vernon, and R. Hellwarth, Description of states in quantum mechanics by density matrix and operator techniques, J. Appl. Phys. 28, 49 (1957). https://doi.org/10.1063/1.1722572 [35] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2010) p. 702, 1011.1669v3. https://doi.org/10.1017/CBO9780511976667 arXiv:1011.1669v3 [36] C. Cafaro, L. Rossetti, and P. M. Alsing, Complexity of quantum-mechanical evolutions from probability amplitudes, Nucl. Phys. B 1010, 116755 (2025). https://doi.org/10.1016/j.nuclphysb.2024.116755 [37] T. Durt, B.-G. Englert, I. Bengtsson, and K. Życzkowski, On mutually unbiased bases, Int. J. Quantum Inf. 08, 535 (2010). https://doi.org/10.1142/s0219749910006502 [38] A. J. Scott and M. Grassl, Symmetric informationally complete positive-operator-valued measures: A new computer study, J. Math. Phys. 51 (2010). https://doi.org/10.1063/1.3374022Cited byCould not fetch Crossref cited-by data during last attempt 2026-07-08 18:56:45: Could not fetch cited-by data for 10.22331/q-2026-07-08-2153 from Crossref. This is normal if the DOI was registered recently. On SAO/NASA ADS no data on citing works was found (last attempt 2026-07-08 18:56:45).This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractThe 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 summaryOne 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} }► References [1] J. v. Neumann, Mathematische Begründung der Quantenmechanik, Nachr. Ges. Wiss. Gött., Math.-Phys. Kl. , 1 (1927). http://eudml.org/doc/59215 [2] P. A. M. Dirac, The Principles of Quantum Mechanics (Clarendon Press, Oxford, 1930). [3] J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed. (Cambridge University Press, 2020). [4] K. F. Pál and T. Vértesi, Efficiency of higher-dimensional Hilbert spaces for the violation of Bell inequalities, Phys. Rev. A 77 (2008). https://doi.org/10.1103/PhysRevA.77.042105 [5] M. McKague, M. Mosca, and N. Gisin, Simulating quantum systems using real hilbert spaces, Phys. Rev. Lett. 102 (2009). https://doi.org/10.1103/PhysRevLett.102.020505 [6] M.-O. Renou, D. Trillo, M. Weilenmann, T. P. Le, A. Tavakoli, N. Gisin, A. Acín, and M. Navascués, Quantum theory based on real numbers can be experimentally falsified, Nature 600, 625 (2021). https://doi.org/10.1038/s41586-021-04160-4 [7] C. Caves, C. Fuchs, and P. Rungta, Entanglement of Formation of an Arbitrary State of Two Rebits, Found. Phys. Lett. 114, 199 (2001). https://doi.org/10.1023/A:1012215309321 [8] W. Wootters, Entanglement Sharing in Real-Vector-Space Quantum Theory, Found. Phys. 42, 19 (2012). https://doi.org/10.1007/s10701-010-9488-1 [9] Q. Chen, T. Gao, and F. Yan, Measures of imaginarity and quantum state order, Sci. China-Phys. Mech. Astron. 66 (2023). https://doi.org/10.1007/s11433-023-2126-9 [10] J. Xu, Imaginarity of Gaussian states, Phys. Rev. A 108 (2023). https://doi.org/10.1103/PhysRevA.108.062203 [11] C. Fernandes, R. Wagner, L. Novo, and E. F. Galvão, Unitary-Invariant Witnesses of Quantum Imaginarity, Phys. Rev. Lett. 133 (2024). https://doi.org/10.1103/PhysRevLett.133.190201 [12] P. B. Hita, A. Trushechkin, H. Kampermann, M. Epping, and D. Bruß, Quantum mechanics based on real numbers: A consistent description, Phys. Rev. Lett. (2026). https://doi.org/10.1103/4k13-sdjh [13] T. Hoffreumon and M. Woods, Quantum theory based on real numbers cannot be experimentally falsified, (2026). arXiv:2603.19208 [14] A. Hickey and G. Gour, Quantifying the imaginarity of quantum mechanics, J. Phys. A 51, 414009 (2018). https://doi.org/10.1088/1751-8121/aabe9c [15] K.-D. Wu, T. V. Kondra, S. Rana, C. M. Scandolo, G.-Y. Xiang, C.-F. Li, G.-C. Guo, and A. Streltsov, Operational resource theory of imaginarity, Phys. Rev. Lett. 126 (2021a). https://doi.org/10.1103/PhysRevLett.126.090401 [16] K.-D. Wu, T. V. Kondra, S. Rana, C. M. Scandolo, G.-Y. Xiang, C.-F. Li, G.-C. Guo, and A. Streltsov, Resource theory of imaginarity: Quantification and state conversion, Phys. Rev. A 103 (2021b). https://doi.org/10.1103/PhysRevA.103.032401 [17] I. Bengtsson and K. Życzkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement, 2nd edition (Cambridge University Press, 2017). https://doi.org/10.1017/9781139207010 [18] I. Bengtsson, S. Weis, and K. Życzkowski, Geometry of the Set of Mixed Quantum States: An Apophatic Approach (Springer Basel, 2012) p. 175. https://doi.org/10.1007/978-3-0348-0448-6_15 [19] R. A. Bertlmann and P. Krammer, Bloch vectors for qudits, J. Phys. A 41, 235303 (2008). https://doi.org/10.1088/1751-8113/41/23/235303 [20] J. Siewert, On orthogonal bases in the Hilbert-Schmidt space of matrices, J. Phys. Comm. 6 (2022). https://doi.org/10.1088/2399-6528/ac6f43 [21] G. Kimura, The bloch vector for n-level systems, Phys. Lett. A 314, 339 (2003). https://doi.org/10.1016/s0375-9601(03)00941-1 [22] C. Eltschka, M. Huber, S. Morelli, and J. Siewert, The shape of higher-dimensional state space: Bloch-ball analog for a qutrit, Quantum 5, 485 (2021). https://doi.org/10.22331/q-2021-06-29-485 [23] A. Khvedelidze, D. Mladenov, and A. Torosyan, Parameterizing qudit states, Discrete Contin. Models Appl. Comput. Sci. 29, 361 (2021). https://journals.rudn.ru/miph/article/view/29429 [24] M. H. Levitt and C. Bengs, Hyperpolarization and the physical boundary of Liouville space, Magn. Reson. 2, 395 (2021). https://doi.org/10.5194/mr-2-395-2021 [25] A. R. P. Rau, Symmetries and Geometries of Qubits, and Their Uses, Symmetry 13 (2021). https://doi.org/10.3390/sym13091732 [26] S. Jarov and M. V. Raamsdonk, Mapping the space of quantum expectation values, (2023), arXiv:2310.13111. arXiv:2310.13111 [27] G. Sharma, S. Ghosh, and S. Sazim, Bloch sphere analog of qudits using Heisenberg-Weyl Operators, Phys. Scr. 99, 045105 (2024). https://doi.org/10.1088/1402-4896/ad2ccf [28] S. Morelli, C. Eltschka, M. Huber, and J. Siewert, Correlation constraints and the Bloch geometry of two qubits, Phys. Rev. A 109 (2024). https://doi.org/10.1103/PhysRevA.109.012423 [29] S. Shravan, S. Morelli, O. Gühne, and S. Imai, Geometry of two-body correlations in three-qubit states, Phys. Rev. A 110 (2024). https://doi.org/10.1103/PhysRevA.110.062419 [30] P. Kurzyński, A. Kołodziejski, W. Laskowski, and M. Markiewicz, Three-dimensional visualization of a qutrit, Phys. Rev. A 93, 062126 (2016). https://doi.org/10.1103/physreva.93.062126 [31] N. Wyderka and O. Gühne, Characterizing quantum states via sector lengths, J. Phys. A 53, 345302 (2020). https://doi.org/10.1088/1751-8121/ab7f0a [32] M. Ozawa, Entanglement measures and the Hilbert-Schmidt distance, Phys. Lett. A 268, 158 (2000). https://doi.org/10.1016/s0375-9601(00)00171-7 [33] M. B. Ruskai, Beyond strong superadditivity? Improved bounds on the contraction of generalized relative entropy, Rev. Math. Phys. 06, 1147 (1994). https://doi.org/10.1142/S0129055X94000407 [34] R. Feynman, F. Vernon, and R. Hellwarth, Description of states in quantum mechanics by density matrix and operator techniques, J. Appl. Phys. 28, 49 (1957). https://doi.org/10.1063/1.1722572 [35] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2010) p. 702, 1011.1669v3. https://doi.org/10.1017/CBO9780511976667 arXiv:1011.1669v3 [36] C. Cafaro, L. Rossetti, and P. M. Alsing, Complexity of quantum-mechanical evolutions from probability amplitudes, Nucl. Phys. B 1010, 116755 (2025). https://doi.org/10.1016/j.nuclphysb.2024.116755 [37] T. Durt, B.-G. Englert, I. Bengtsson, and K. Życzkowski, On mutually unbiased bases, Int. J. Quantum Inf. 08, 535 (2010). https://doi.org/10.1142/s0219749910006502 [38] A. J. Scott and M. Grassl, Symmetric informationally complete positive-operator-valued measures: A new computer study, J. Math. Phys. 51 (2010). https://doi.org/10.1063/1.3374022Cited byCould not fetch Crossref cited-by data during last attempt 2026-07-08 18:56:45: Could not fetch cited-by data for 10.22331/q-2026-07-08-2153 from Crossref. This is normal if the DOI was registered recently. On SAO/NASA ADS no data on citing works was found (last attempt 2026-07-08 18:56:45).This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.