Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0812.1572 · Physical Review A

Bounding the dimension of bipartite quantum systems

Tamás Vértesi, Károly F. Pál

Published Apr 27, 2009• 72 citations
arXivHTMLPDF

Abstract

Let us consider the set of joint quantum correlations arising from two-outcome local measurements on a bipartite quantum system. We prove that no finite dimension is sufficient to generate all these sets. We approach the problem in two different ways by constructing explicit examples for every dimension $d$, which demonstrates that there exist bipartite correlations that necessitate $d$-dimensional local quantum systems in order to generate them. We also show that at least ten two-outcome measurements must be carried out by the two parties altogether so as to generate bipartite joint correlations not achievable by two-dimensional local systems. The smallest explicit example we found involves 11 settings.

DOI
10.1103/physreva.79.042106

Related papers

Upper Bounding Hilbert Space Dimensions which can Realize all the Quantum Correlations

Yasamin Panahi, Maria Ciudad Alañón, Daniel Centeno +4 more

We introduce novel upper bounds on the Hilbert space dimensions required to realize quantum correlations in Bell scenarios. We start by considering bipartite cases wherein one of the two parties has two settings and two outcomes. Regardless of the number of measurements and outcomes of the other party, the Hilbert space dimension of the first party can be limited to two while still achieving all convexly extremal quantum correlations. We then leverage Schmidt decomposition to show that the remaining party can losslessly also be restricted to a qubit Hilbert space. We then extend this idea to multipartite scenarios. We also adapt our results to provide upper bounds of local Hilbert space dimensions to achieve any quantum correlation, including convexly non-extremal correlations, by utilizing Caratheodory's theorem. Finally, we generalize our results to nonstandard Bell scenarios with communication. Taken together, our results fill in several previously unresolved aspects of the problem of determining sufficient Hilbert space dimensionality, expanding the collection of scenarios for which finite-dimensional quantum systems are known to be sufficient to reproduce any quantum correlation.

arXiv:2505.20519•May 26, 2025•0 citations

Minimum Dimension of a Hilbert Space Needed to Generate a Quantum Correlation

Jamie Sikora, Antonios Varvitsiotis, Zhaohui Wei

Consider a two-party correlation that can be generated by performing local measurements on a bipartite quantum system. A question of fundamental importance is to understand how many resources, which we quantify by the dimension of the underlying quantum system, are needed to reproduce this correlation. In this Letter, we identify an easy-to-compute lower bound on the smallest Hilbert space dimension needed to generate a given two-party quantum correlation. We show that our bound is tight on many well-known correlations and discuss how it can rule out correlations of having a finite-dimensional quantum representation. We show that our bound is multiplicative under product correlations and also that it can witness the nonconvexity of certain restricted-dimensional quantum correlations.

arXiv:1507.00213•Physical Review Letters•Aug 4, 2016•44 citations

Separation of quantum, spatial quantum, and approximate quantum correlations

Salman Beigi

Quantum nonlocal correlations are generated by implementation of local quantum measurements on spatially separated quantum subsystems. Depending on the underlying mathematical model, various notions of sets of quantum correlations can be defined. In this paper we prove separations of such sets of quantum correlations. In particular, we show that the set of bipartite quantum correlations with four binary measurements per party becomes strictly smaller once we restrict the local Hilbert spaces to be finite dimensional, i.e. , Cq(4,4,2,2)≠Cqs(4,4,2,2) . We also prove non-closure of the set of bipartite quantum correlations with four ternary measurements per party, i.e. , Cqs(4,4,3,3)≠Cqa(4,4,3,3) .

arXiv:2004.11103•Quantum•Jan 28, 2021•4 citations

Efficiency of higher-dimensional Hilbert spaces for the violation of Bell inequalities

Károly F. Pál, Tamás Vértesi

We have determined numerically the maximum quantum violation of over 100 tight bipartite Bell inequalities with two-outcome measurements by each party on systems of up to four-dimensional Hilbert spaces. We have found several cases, including the ones where each party has only four measurement choices, where two-dimensional systems, i.e., qubits, are not sufficient to achieve maximum violation. In a significant proportion of those cases when qubits are sufficient, one or both parties have to make trivial, degenerate ``measurements'' in order to achieve maximum violation. The quantum state corresponding to the maximum violation in most cases is not the maximally entangled one. We also obtain the result that bipartite quantum correlations can always be reproduced by measurements and states which require only real numbers if there is no restriction on the size of the local Hilbert spaces. Therefore in order to achieve maximum quantum violation on any bipartite Bell inequality (with any number of settings and outcomes), there is no need to consider complex Hilbert spaces.

arXiv:0712.4320•Physical Review A•Apr 9, 2008•55 citations

Characterization of Quantum Correlations with Local Dimension Constraints and Its Device-Independent Applications

Miguel Navascués, de la, Torre Gonzalo, Tamás Vértesi

The future progress of semi-device independent quantum information science depends crucially on our ability to bound the strength of the nonlocal correlations achievable with finite dimensional quantum resources. In this work, we characterize quantum nonlocality under local dimension constraints via a complete hierarchy of semidefinite programming relaxations. In the bipartite case, we find that the first level of the hierarchy returns non-trivial bounds in all cases considered, allowing to study nonlocality scenarios with four measurement settings on one side and twelve (12) on the other in a normal desktop. In the tripartite case, we apply the hierarchy to derive a Bell-type inequality that can only be violated when each of the three parties has local dimension greater than two, hence certifying three-dimensional tripartite entanglement in a device independent way. Finally, we show how the new method can be trivially modified to detect non-separable measurements in two-qubit scenarios.

•Jan 1, 2014•37 citations

Certifying bipartite pure quantum states efficiently using untrusted devices

Lijinzhi Lin, Zhenyu Chen, Xiaodie Lin +1 more

It has been known that all bipartite pure quantum states can be certified by quantum self-testing, i.e., any such states can be pinned down completely based on the statistics produced by local quantum measurements. A notable feature of quantum self-testing is that the conclusions remain reliable even when the quantum measurements involved are untrusted, where quantum nonlocality is crucial. This necessitates that each party conducts at least two different quantum measurements to produce the desired correlation. Here, we prove that when the underlying Hilbert space dimension is known beforehand, which is very common in quantum experiments, an arbitrary $d\times d$ bipartite pure state can be certified completely (up to local unitaries) by a certain correlation generated with a single measurement setting on each party, where each measurement yields only $2d$ or even $d+1$ outcomes. We also prove the robustness of our protocols to quantum noises and experimental imperfections. Compared with quantum self-testing, our protocols do not hinge on quantum nonlocality and are much more efficient, yet they maintain the essential feature of not requiring additional assumptions about the quantum devices involved. This advancement could offer significant convenience when certifying bipartite quantum states using untrusted quantum devices in future quantum industries.

arXiv:2306.07755•Jun 13, 2023•0 citations
Let us consider the set of joint quantum correlations arising from two-outcome local measurements on a bipartite quantum system. We prove that no finite dimension is sufficient to generate all these sets. We approach the problem in two different ways by constructing explicit examples for every dimension ddd, which demonstrates that there exist bipartite correlations that necessitate ddd-dimensional local quantum systems in order to generate them. We also show that at least ten two-outcome measurements must be carried out by the two parties altogether so as to generate bipartite joint correlations not achievable by two-dimensional local systems. The smallest explicit example we found involves 11 settings.
We introduce novel upper bounds on the Hilbert space dimensions required to realize quantum correlations in Bell scenarios. We start by considering bipartite cases wherein one of the two parties has two settings and two outcomes. Regardless of the number of measurements and outcomes of the other party, the Hilbert space dimension of the first party can be limited to two while still achieving all convexly extremal quantum correlations. We then leverage Schmidt decomposition to show that the remaining party can losslessly also be restricted to a qubit Hilbert space. We then extend this idea to multipartite scenarios. We also adapt our results to provide upper bounds of local Hilbert space dimensions to achieve any quantum correlation, including convexly non-extremal correlations, by utilizing Caratheodory's theorem. Finally, we generalize our results to nonstandard Bell scenarios with communication. Taken together, our results fill in several previously unresolved aspects of the problem of determining sufficient Hilbert space dimensionality, expanding the collection of scenarios for which finite-dimensional quantum systems are known to be sufficient to reproduce any quantum correlation.
Consider a two-party correlation that can be generated by performing local measurements on a bipartite quantum system. A question of fundamental importance is to understand how many resources, which we quantify by the dimension of the underlying quantum system, are needed to reproduce this correlation. In this Letter, we identify an easy-to-compute lower bound on the smallest Hilbert space dimension needed to generate a given two-party quantum correlation. We show that our bound is tight on many well-known correlations and discuss how it can rule out correlations of having a finite-dimensional quantum representation. We show that our bound is multiplicative under product correlations and also that it can witness the nonconvexity of certain restricted-dimensional quantum correlations.
Quantum nonlocal correlations are generated by implementation of local quantum measurements on spatially separated quantum subsystems. Depending on the underlying mathematical model, various notions of sets of quantum correlations can be defined. In this paper we prove separations of such sets of quantum correlations. In particular, we show that the set of bipartite quantum correlations with four binary measurements per party becomes strictly smaller once we restrict the local Hilbert spaces to be finite dimensional, i.e. , Cq(4,4,2,2)≠Cqs(4,4,2,2) . We also prove non-closure of the set of bipartite quantum correlations with four ternary measurements per party, i.e. , Cqs(4,4,3,3)≠Cqa(4,4,3,3) .
We have determined numerically the maximum quantum violation of over 100 tight bipartite Bell inequalities with two-outcome measurements by each party on systems of up to four-dimensional Hilbert spaces. We have found several cases, including the ones where each party has only four measurement choices, where two-dimensional systems, i.e., qubits, are not sufficient to achieve maximum violation. In a significant proportion of those cases when qubits are sufficient, one or both parties have to make trivial, degenerate ``measurements'' in order to achieve maximum violation. The quantum state corresponding to the maximum violation in most cases is not the maximally entangled one. We also obtain the result that bipartite quantum correlations can always be reproduced by measurements and states which require only real numbers if there is no restriction on the size of the local Hilbert spaces. Therefore in order to achieve maximum quantum violation on any bipartite Bell inequality (with any number of settings and outcomes), there is no need to consider complex Hilbert spaces.
The future progress of semi-device independent quantum information science depends crucially on our ability to bound the strength of the nonlocal correlations achievable with finite dimensional quantum resources. In this work, we characterize quantum nonlocality under local dimension constraints via a complete hierarchy of semidefinite programming relaxations. In the bipartite case, we find that the first level of the hierarchy returns non-trivial bounds in all cases considered, allowing to study nonlocality scenarios with four measurement settings on one side and twelve (12) on the other in a normal desktop. In the tripartite case, we apply the hierarchy to derive a Bell-type inequality that can only be violated when each of the three parties has local dimension greater than two, hence certifying three-dimensional tripartite entanglement in a device independent way. Finally, we show how the new method can be trivially modified to detect non-separable measurements in two-qubit scenarios.
It has been known that all bipartite pure quantum states can be certified by quantum self-testing, i.e., any such states can be pinned down completely based on the statistics produced by local quantum measurements. A notable feature of quantum self-testing is that the conclusions remain reliable even when the quantum measurements involved are untrusted, where quantum nonlocality is crucial. This necessitates that each party conducts at least two different quantum measurements to produce the desired correlation. Here, we prove that when the underlying Hilbert space dimension is known beforehand, which is very common in quantum experiments, an arbitrary d×dd\times dd×d bipartite pure state can be certified completely (up to local unitaries) by a certain correlation generated with a single measurement setting on each party, where each measurement yields only 2d2d2d or even d+1d+1d+1 outcomes. We also prove the robustness of our protocols to quantum noises and experimental imperfections. Compared with quantum self-testing, our protocols do not hinge on quantum nonlocality and are much more efficient, yet they maintain the essential feature of not requiring additional assumptions about the quantum devices involved. This advancement could offer significant convenience when certifying bipartite quantum states using untrusted quantum devices in future quantum industries.