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Physical Review A

Characterizing the set of quantum correlations in prepare-and-measure quantum multichain-shaped networks

Ya Jia, Fen-Zhuo Guo, Yukun Wang, Haifeng Dong, Fei Gao

Published Feb 24, 2025• 3 citations
Publisher

Abstract

We introduce a hierarchy of tests satisfied by any probability distribution $P$ that represents the quantum correlations generated in prepare-and-measure (P) quantum multichain-shaped networks, assuming only the inner-product information of prepared states. The P quantum multichain-shaped networks involve multiple measurement parties, with each measurement party potentially having multiple sequential receivers. We adapt the original Navascu\'es, Pironio, and Ac\'{\i}n (NPA) hierarchy by incorporating a finite number of linear and positive semi-definite constraints to characterize the quantum correlations in P quantum multichain-shaped networks. These constraints in each hierarchy are derived from sequential measurements and the inner-product matrix of prepared states. The adapted NPA hierarchy is further applied to tackle some quantum information tasks, including sequential quantum random access codes (QRACs) and semi-device-independent randomness certification. First, we derive the optimal trade-off between the two sequential receivers in the $2\ensuremath{\rightarrow}1$ sequential QRACs, and investigate randomness certification in its double violation region. Second, considering the presence of an eavesdropper in actual communication, we show how much local and global randomness can be certified using the optimal trade-off of $2\ensuremath{\rightarrow}1$ sequential QRACs. We also quantify the amount of local and global randomness that can be certified from the complete set of probabilities generated by the two sequential receivers. Our conclusion is that utilizing the full set of probabilities certifies more randomness than relying solely on the optimal trade-off relationship.

DOI
10.1103/physreva.111.022439

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We introduce a hierarchy of tests satisfied by any probability distribution PPP that represents the quantum correlations generated in prepare-and-measure (P) quantum multichain-shaped networks, assuming only the inner-product information of prepared states. The P quantum multichain-shaped networks involve multiple measurement parties, with each measurement party potentially having multiple sequential receivers. We adapt the original Navascu'es, Pironio, and Ac'{\i}n (NPA) hierarchy by incorporating a finite number of linear and positive semi-definite constraints to characterize the quantum correlations in P quantum multichain-shaped networks. These constraints in each hierarchy are derived from sequential measurements and the inner-product matrix of prepared states. The adapted NPA hierarchy is further applied to tackle some quantum information tasks, including sequential quantum random access codes (QRACs) and semi-device-independent randomness certification. First, we derive the optimal trade-off between the two sequential receivers in the 2\ensuremath→12\ensuremath{\rightarrow}12\ensuremath→1 sequential QRACs, and investigate randomness certification in its double violation region. Second, considering the presence of an eavesdropper in actual communication, we show how much local and global randomness can be certified using the optimal trade-off of 2\ensuremath→12\ensuremath{\rightarrow}12\ensuremath→1 sequential QRACs. We also quantify the amount of local and global randomness that can be certified from the complete set of probabilities generated by the two sequential receivers. Our conclusion is that utilizing the full set of probabilities certifies more randomness than relying solely on the optimal trade-off relationship.
We introduce a hierarchy of tests satisfied by any probability distribution PPP that represents the quantum correlations generated in prepare-and-measure (P\&M) quantum chain-shaped networks, assuming only the inner-product information of the non-orthogonal quantum states. The P\&M quantum chain-shaped networks involve multiple measurement parties, each measurement party potentially having multiple sequential receivers. Specifically, we adapt the original NPA-hierarchy by incorporating a finite number of linear and positive semi-definite constraints to characterize the quantum correlations in P\&M quantum chain-shaped networks. These constraints in each hierarchy are derived from sequential measurement operators and the inner-product matrix of the non-orthogonal quantum states. We apply the adapted NPA-hierarchy to tackle some quantum information tasks, including sequential quantum random access codes (QRACs) and randomness certification. First, we derive the optimal trade-off between the two sequential receivers in the 2→12 \to 12→1 sequential QRACs. Furthermore, we have investigated semi-device-independent randomness certification in the double violation region of 2→12 \to 12→1 sequential QRACs. Second, considering the presence of eavesdropper (Eve) in actual communication, we show how much global and local randomness can be certified using the optimal trade-off of 2→12 \to 12→1 sequential QRACs. Additionally, we quantify the amount of local and global randomness that can be certified from the complete probabilities generated by the two sequential receivers. We conclude that utilizing the complete set of probabilities certifies more local and global randomness than relying solely on the optimal trade-off relationship.
An important problem in quantum information theory is that of bounding sets of correlations that arise from making local measurements on entangled states of arbitrary dimension. Currently, the best-known method to tackle this problem is the NPA hierarchy; an infinite sequence of semidefinite programs that provides increasingly tighter outer approximations to the desired set of correlations. In this work we consider a more general scenario in which one performs sequences of local measurements on an entangled state of arbitrary dimension. We show that a simple adaptation of the original NPA hierarchy provides an analogous hierarchy for this scenario, with comparable resource requirements and convergence properties. We then use the method to tackle some problems in device-independent quantum information. First, we show how one can robustly certify over 2.3 bits of device-independent local randomness from a two-quibt state using a sequence of measurements, going beyond the theoretical maximum of two bits that can be achieved with non-sequential measurements. Finally, we show tight upper bounds to two previously defined tasks in sequential Bell test scenarios.
Quantum Random Access Codes (QRACs) are key tools for a variety of protocols in quantum information theory. These are commonly studied in prepare-and-measure scenarios in which a sender prepares states and a receiver measures them. Here, we consider a three-party prepare-transform-measure scenario in which the simplest QRAC is implemented twice in sequence based on the same physical system. We derive optimal trade-off relations between the two QRACs. We apply our results to construct semi-device independent self-tests of quantum instruments, i.e. measurement channels with both a classical and quantum output. Finally, we show how sequential QRACs enable inference of upper and lower bounds on the sharpness parameter of a quantum instrument.
We investigate the prepare-and-broadcast scenario, a multipartite extension of dimension-constrained prepare-and-measure experiments in which a quantum system is distributed to multiple receivers. We derive fundamental trade-offs between prepare-and-measure witnesses, Bell nonlocality, and quantum random access code performance. We further develop a semi-device-independent randomness certification framework based on prepare-and-broadcast witnesses, showing that the maximal quantum violation certifies two bits of joint randomness, exceeding the limit achievable from the CHSH inequality while remaining robust to noise. Finally, we show that the prepare-and-broadcast scenario naturally accommodates stronger adversarial models in which the eavesdropper retains a quantum system correlated with the measurement device, providing a natural framework for semi-device-independent randomness certification against quantum side information.
The prepare-and-measure framework based on energy constraints offers a practical middle ground between fully device-dependent and device-independent quantum cryptography. The only assumption on an otherwise uncharacterized prepare-and-measure device is that the energy of the prepared states is bounded. Existing security analyses of this framework assume that the preparation and measurement devices share at most classical correlations, and under this assumption certified lower bounds on the extractable randomness have been established. Recent work has shown that an adversary who pre-distributes entanglement between the devices can mount attacks that are strictly more powerful than those available when the devices share only classical correlations, reducing the extractable randomness below the previously certified rates. This leaves open the fundamental question of whether randomness can be certified at all in this scenario. We address this open question by constructing semidefinite programming relaxations of the guessing probability by adapting the Navascués-Pironio-Acín hierarchy to the energy-constrained setting where shared entanglement between the devices is permitted. Our relaxations yield certified lower bounds on the extractable randomness without enforcing any restrictions on the dimensions of the quantum state shared between the preparation and measurement devices. We show that these certified lower bounds are strictly positive for a range of energy values, thereby answering the open question affirmatively: certified randomness generation is theoretically possible in the energy-constrained SDI framework even in the presence of a fully quantum adversary.
Abstract Unsharp measurements play an increasingly important role in quantum information theory. In this paper, we study a three-party prepare-transform-measure experiment with unsharp measurements based on 3 → 1 sequential random access codes (RACs). We derive optimal trade-off between the two correlation witnesses in 3 → 1 sequential quantum random access codes (QRACs), and use the result to complete the self-testing of quantum preparations, instruments and measurements for three sequential parties. We also give the upper and lower bounds of the sharpness parameter to complete the robustness analysis of the self-testing scheme. In addition, we find that classical correlation witness violation based on 3 → 1 sequential RACs cannot be obtained by both correlation witnesses simultaneously. This means that if the second party uses strong unsharp measurements to overcome the classical upper bound, the third party cannot do so even with sharp measurements. Finally, we give the analysis and comparison of the random number generation efficiency under different sharpness parameters based on the determinant value, 2 → 1 and 3 → 1 QRACs separately. This letter sheds new light on generating random numbers among multi-party in semi-device independent framework.