Measurement-Device-Independent Approach to Entanglement Measures

Abstract

Within the context of semiquantum nonlocal games, the trust can be removed from the measurement devices in an entanglement-detection procedure. Here, we show that a similar approach can be taken to quantify the amount of entanglement. To be specific, first, we show that in this context, a small subset of semiquantum nonlocal games is necessary and sufficient for entanglement detection in the local operations and classical communication paradigm. Second, we prove that the maximum payoff for these games is a universal measure of entanglement which is convex and continuous. Third, we show that for the quantification of negative-partial-transpose entanglement, this subset can be further reduced down to a single arbitrary element. Importantly, our measure is measurement device independent by construction and operationally accessible. Finally, our approach straightforwardly extends to quantify the entanglement within any partitioning of multipartite quantum states. DOI: 10.1103/PhysRevLett.118.150505 Introduction.-Entanglement is a valuable resource for practical as well as fundamental applications of quantum theory, ranging from quantum computation and communication to metrology [1-3]. There are two major challenges in understanding entanglement that stimulates this research. First, it is extremely difficult to specify all the nonentangled bipartite or multipartite quantum states. In fact, the problem is known to be NP-hard [4,5]. Second, not surprisingly, the characterization of entangled states, i.e., the quantification of entanglement within quantum states, is an equally difficult task. The answer to the second challenge is practically very important because it tells us how well our protocols will perform using a given state [6][7][8][9].Focusing on the second challenge above, a first level of hardness is that the quantification of entanglement using almost any entanglement measure, e.g., entanglement of formation [10], negativity [11,12], or random robustness [13], requires estimating a large number of density matrix elements, a task which is difficult to perform on bipartite and multipartite quantum states. While this difficulty can be partially circumvented by making use of entanglement witnesses (EWs) when lower bounds on the entanglement are desired [14][15][16][17][18][19], errors and misalignments of the measurement devices can still lead to incorrect estimations of the quantities and thus, erroneous conclusions. A measurementdevice-independent approach is therefore desirable.Recent work by Buscemi [20] has introduced a new way to think about entanglement detection [21][22][23][24]. The idea is to map the problem onto a modified class of nonlocal games, called semiquantum nonlocal games (SQNLGs). In any such game, two players (Alice and Bob) share a possibly entangled state. A referee (Charlie) starts by asking them

Experimental Measurement-Device-Independent Quantum Steering and Randomness Generation Beyond Qubits

“…It immediately leads to the fact that the input states are given by the eigenstate vectors as per Eqs. (22) and (23). In this experiment, instead of preparing these 3-level states directly, we generate a set of 2-level state vectors {τ T k } which are much easier to be prepared with high fidelity.…”

“…For the 3-dimensional system, these quantum measurements are chosen as the same as Bob's measurements, i.e., three measurement outcomes of the setting x = 1 are given by Eq. (22) while the setting x = 2 has a description in Eq. (23).…”

“…even when Charlie does not trust Alice and Bob. This was further explored in [18][19][20][21][22] and extended by Cavalcanti et al [18] to the case of EPR steering: With access to a quantum channel to Bob and a classical channel to Alice, Charlie can certify entanglement for all EPR-steerable (from Alice to Bob) states. An experimental MDI verification of steering for qubits was reported in [23], together with a method to construct quantum-refereed steering (QRS) witnesses from a given steering inequality.…”

Measurement-Device-Independent Verification of Quantum Channels

“…MDI schemes were first proposed in the context of entanglement verification [12] for spatially separated subsystems, within the framework of semi-quantum games [13], with several advantages over Bell tests [14]. These methods have since been extended to quantum steering [15] and to the analysis of entanglement structure [16] and its quantification [17][18][19], which has been demonstrated experimentally [20][21][22]. Moving beyond the classic scenario of two spacelike separated parties, the MDI approach and semi-quantum games can be adapted to situations where one party wants to test another party's ability to maintain quantum coherence over time [23], such as in a quantum memory.…”

“…Fidelity, storage time and recall properties could be tested by changing the timing between the probe photons. On the theory side, the protocol could be extended to quantify the quantum nature of the channel instead of verifying it as has been done for the MDI quantification of entanglement [17][18][19]; in this direction, one would need to use a capacity that quantifies specifically the quantum part of the channel: the negativity of the channel Choi state, or the quantum relative entropies [31] could be used for that purpose.…”