Randomness Certification from Multipartite Quantum Steering for Arbitrary Dimensional Systems

Question

1. Is multipartite steering necessary for randomness certification?

2. What is multipartite steering necessary for in the tripartite scenario?

3. How does the see-saw algorithm help in calculating the upper bound of min-entropy?

4. What is the SDP optimization process in generating POVMs?

Accepted Answer

In the case of m B = m C = 2, multipartite steering is necessary and sufficient for certifying randomness. This perfect equivalence ceases to hold with more measurement settings. The argument is valid for both multipartite and bipartite scenarios. In the bipartite scenario, if problem Eq. (A1) is feasible, then problem Eq. (A2) gives P g (y *) = 1 for any measurement settings m A. If there is no randomness on Bob's outputs, then {s obs b|y} b,y is unsteerable. Therefore, multipartite steering is necessary for randomness certification.

Accepted Answer

Multipartite steering is necessary for certifying randomness on Bob and Charlie in the tripartite scenario. It ensures that the observed assemblage is unsteerable, meaning that the measurements cannot be manipulated to reveal any information about the randomness. In the case of m B = m C = 2, multipartite steering is sufficient for randomness. However, in scenarios where m B >= 3 or m C >= 3, additional measurements can express steerability without being involved in generating randomness. This highlights the importance of multipartite steering in ensuring the security and integrity of the tripartite scenario.

Accepted Answer

The see-saw algorithm is used to convert the problem Eq. (B1) into three Semidefinite Programming (SDP) problems. By fixing the dimensions of each subsystem (dA, dB, dC), the algorithm helps in calculating the upper bound of min-entropy. It achieves this by setting POVMs {Mb|y}b,y and ensemble {s ee' ABC} ee' randomly. To obtain matrices M(0)b|y >= 0, the algorithm takes nBmBd2. This process allows for the calculation of the upper bound of min-entropy in a tripartite scenario, even though the problem is neither an SDP nor a linear problem.

Accepted Answer

The SDP optimization process involves maximizing the equation EQUATION, where s ee ' (0) BC = Tr A [s ee ' (0) ABC]. Initially, POVMs {M (0) b|y } b,y and states {s ee ' (0) ABC } e,e ' are set randomly. An optimal solution {M (1) c|z } c,z is found easily. Then, the optimal solution {M (1) b|y } b,y is obtained by changing variables in Eq. (B4). In the third step, POVMs M (1) b|y b,y and M (1) c|z c,z are fixed to find an optimal set of states s ee ' (0) ABC e,e ' that maximizes Eq. (B4). Iterating these steps until max{l bc|yz } decreases to about 1e-9 and the optimal guessing probability reaches its convergency, an upper bound H Dim min can be calculated. Adding random POVMs and states (with tiny weight) helps in finding a larger P g. The solution found by the see-saw algorithm may not be a global optimal solution, but it provides an upper bound of actual randomness.

Accepted Answer

In a multipartite scenario, the guessing probability P B g is given by max where s e B = Tr AC [s e ABC]. This probability represents Eve's ability to guess the measurement results of Bob or Charlie solely. The main difference between Eq. (B5) and Eq. (B1) lies in the objective function, which certifies the randomness on Bob in a real tripartite scenario. However, solving the optimization problem directly is not feasible. The lower bounds of Eq. (B5) can be calculated using the no-signaling principle, which ensures that information cannot be transmitted faster than the speed of light.

Accepted Answer

To certify randomness in a continuous-variable system, an example can be provided using the pure state |Ps>CV generated from a setup. The squeezed operator S(x) = e^(1/2)x*a^2 - x(a+)^2 is used, where x = re^i. The homodyne measurement results on Bob and Charlie are binned using a coarse-graining scheme. By analyzing the assemblage on Alice, upper and lower bounds of H min can be calculated for Bob's and Charlie's measurement results. The randomness is maximized over binning periods in the range Tx(Bob) [2, 10] and Tx(Charlie) [2, 10]. The method can be extended to randomness on only Bob or Charlie. The actual min-entropy is described by its lower and upper bounds. This approach can be extended to a real tripartite scenario and is similar to the method in Appendix B. The violation of a genuine tripartite steering inequality, such as Eq. (D2), can be used to observe genuine tripartite steering and demonstrate randomness in the system.

Randomness Certification from Multipartite Quantum Steering for Arbitrary Dimensional Systems

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Most frequently asked questions

1. Is multipartite steering necessary for randomness certification?

2. What is multipartite steering necessary for in the tripartite scenario?

3. How does the see-saw algorithm help in calculating the upper bound of min-entropy?

4. What is the SDP optimization process in generating POVMs?

5. What is the guessing probability P B g in a multipartite scenario?

6. How can randomness in a continuous-variable system be certified?

Citations

Tripartite entanglement and tripartite steering in three-qubit pure states induced by vacuum-one-photon superpositions

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