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arXiv:2508.17140 · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Quantum steering based on partial state information

Shounak Datta, Archan S. Majumdar

Published Aug 26, 2026• 1 citation
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Abstract

Abstract Exploiting partial knowledge of a qubit in terms of its basis-dependent measure of imaginarity, we formulate a steering criterion for bipartite qubit systems, which requires two dichotomic measurements at the untrusted side and two mutually unbiased bases at the trusted side. We show that quantum correlations embodied through our proposed imaginarity steering inequality (ISI) can be witnessed by suitably constructed Hermitian operators that depend on fewer state parameters and require measurement of fewer observables than for witnessing other forms of quantum non-local correlations. The monogamy of such correlations is also demonstrated. We further underscore the steering of imaginarity as an efficient non-local resource in the presence of white noise and under unsharp measurements, showing the robustness of imaginarity steering compared with certain other steering criteria.

DOI
10.1098/rspa.2025.0841

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Abstract Exploiting partial knowledge of a qubit in terms of its basis-dependent measure of imaginarity, we formulate a steering criterion for bipartite qubit systems, which requires two dichotomic measurements at the untrusted side and two mutually unbiased bases at the trusted side. We show that quantum correlations embodied through our proposed imaginarity steering inequality (ISI) can be witnessed by suitably constructed Hermitian operators that depend on fewer state parameters and require measurement of fewer observables than for witnessing other forms of quantum non-local correlations. The monogamy of such correlations is also demonstrated. We further underscore the steering of imaginarity as an efficient non-local resource in the presence of white noise and under unsharp measurements, showing the robustness of imaginarity steering compared with certain other steering criteria.
The bipartite state is said to be steerable iff it does not have single system description, i.e., the bipartite state can not be explained by local hidden state model [Phys. Rev. Lett. {\bf 98}, 140402 (2007)]. Several steering inequalities have been derived using different local uncertainty relations to verify the control on the state of one subsystem by other. Here, we derive complementarity kind of relations between coherences measured on mutually unbiased bases using various coherence measures like the l1l_1l1​-norm, relative entropy and skew information. Using these relations, we formulate steering inequalities to check whether one subsystem has control to the coherence of the other entangled subsystem. According to our results, all pure entangled states are maximally steerable but for mixed entangled states, coherence steerability needs stronger non-local correlation than the existing state steering inequalities. We also show that our steering criteria is monogamous.
Recently, a different form of quantum steering, i.e., certification of quantum steering in a one-sided semi-device-independent way, has been formulated [C. Jebarathinam et al., Phys. Rev. A 108, 042211 (2023)]. In this work, we use this phenomenon to provide operational simultaneous correlations in mutually unbiased bases as quantified by the measures in [S. Wu et al., Sci. Rep. 4, 4036 (2014)]. First, we show that for any bipartite state, such measure of simultaneous correlations in two mutually unbiased bases can be operationally identified as exhibiting one-sided semi-device-independent steering in a two-setting scenario. Next, we demonstrate that for two-qubit Bell-diagonal states, quantifying one-sided semi-device-independent steerability provides an operational quantification of information-theoretic quantification of simultaneous correlations in mutually unbiased bases. Then, we provide a different classification of two-qubit separable states with the above-mentioned information-theoretic quantification of simultaneous correlations in mutually unbiased bases and the quantification of one-sided semi-device-independent steerability. Finally, we invoke quantum steering ellipsoid formalism to shed intuitions on the operational characterization of simultaneous correlations in complementary bases of two-qubit states via one-sided semi-device-independent steerability. This provides us with a geometric visualization of the results.
The present work is motivated by the question what aspect of correlation entailed by the two-qubit state serves as the appropriate quantitative resource for steering. To this end, considering Bell-diagonal states, suitable measures of simultaneous correlations in two and three complementary (mutually unbiased) bases are identified as the relevant resources for quantum steering. Quantitative relations between appropriate measures of quantum steering and corresponding measures of simultaneous correlations in complementary bases are demonstrated which ensure that for two-qubit steerable Bell-diagonal states, a higher value of simultaneous correlations in mutually unbiased bases necessarily implies a higher degree of quantum steering, both for two- and three-setting steering scenarios.
In the present work we study tripartite quantum steering of quantum correlations arising from two local dichotomic measurements on each side in the two types of partially device-independent scenarios: the one-sided device-independent scenario where one of the parties performs untrusted measurements while the other two parties perform trusted measurements and the two-sided device-independent scenario where one of the parties performs trusted measurements while the other two parties perform untrusted measurements. We demonstrate that tripartite steering in the two-sided device-independent scenario is weaker than tripartite steering in the one-sided device-independent scenario by using two families of quantum correlations. That is, these two families of quantum correlations in the two-sided device-independent framework detect tripartite entanglement through tripartite steering for a larger region than that in the one-sided device-independent framework. It is shown that tripartite steering in the two-sided device-independent scenario implies the presence of genuine tripartite entanglement of $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2$ quantum system, even if the correlation does not exhibit genuine nonlocality or genuine steering.
Quantum steering, as a cornerstone of quantum information, is usually used to witness the quantum correlation of bipartite and multi-partite states. Here, we experimentally demonstrate the quantum steering inequality of two-qubit mixed states based on the fine-grained uncertainty relation. Our experimental results show that the steering inequality has potent sensitivity to Werner states and Bell diagonal states. The steering strategy exhibits a strong ability to identify that Werner states are steerable when the decoherence coefficient a>12. Compared to the steering inequality obtained by another stratagem, the steering witness criteria of mixed states based on the fine-grained uncertainty relation demonstrated in our experiment has better precision and accuracy. Moreover, the detection efficiency in our measurement setup is only required to be 50% to close the detection loophole, which means our approach needs less detector efficiency to certificate the steerability of mixed states.
We investigate whether paradigmatic measurements for quantum state tomography, namely mutually unbiased bases and symmetric informationally complete measurements, can be employed to certify quantum correlations. For this purpose, we identify a simple and noise-robust correlation witness for entanglement detection, steering, and nonlocality that can be evaluated based on the outcome statistics obtained in the tomography experiment. This allows us to perform state tomography on entangled qutrits, a test of Einstein-Podolsky-Rosen steering and a Bell inequality test, all within a single experiment. We also investigate the trade-off between quantum correlations and subsets of tomographically complete measurements as well as the quantification of entanglement in the different scenarios. Finally, we perform a photonics experiment in which we demonstrate quantum correlations under these flexible assumptions, namely with both parties trusted, one party untrusted and both parties untrusted.