Measurements in two bases are sufficient for certifying high-dimensional entanglement

Abstract

High-dimensional encoding of quantum information provides a promising method of transcending current limitations in quantum communication. One of the central challenges in the pursuit of such an approach is the certification of high-dimensional entanglement. In particular, it is desirable to do so without resorting to inefficient full state tomography. Here, we show how carefully constructed measurements in two bases (one of which is not orthonormal) can be used to faithfully and efficiently certify bipartite high-dimensional states and their entanglement for any physical platform. To showcase the practicality of this approach under realistic conditions, we put it to the test for photons entangled in their orbital angular momentum. In our experimental setup, we are able to verify 9dimensional entanglement for a pair of photons on a 11-dimensional subspace each, at present the highest amount certified without any assumptions on the state.Quantum communication offers advantages such as enhanced security in quantum key distribution (QKD) protocols [1] and increased channel capacities [2] with respect to classical means of communication. All of these improvements, ranging from early proposals [3] to recent exciting developments such as fully device-independent QKD [4, 5], rely on one fundamental phenomenon: quantum entanglement. Currently, the workhorse of most implementations is entanglement between qubits, i.e., between two-dimensional quantum systems (e.g., photon polarization). However, it has long been known that higher-dimensional entanglement can be useful in overcoming the limitations of qubit entanglement [6, 7], offering better key rates [8], higher noise resistance [9, 10] and improved security against different attacks [11].Attempting to capitalize on this insight, recent experiments have successfully generated and certified highdimensional entanglement in different degrees of freedom. In particular, the canonical way of generating two-dimensional polarization entanglement in downconversion processes already offers the potential for exploring entanglement in higher dimensions. This can be achieved by exploiting spatial degrees of freedom [12, 13], orbital angular momentum (OAM) [14-16], energy-time based encodings [17][18][19][20], or combinations thereof in hyper-entangled quantum systems [21,22]. High-dimensional quantum systems have recently also been explored in matter-based systems such as Cesium atoms [23] and superconducting circuits [24]. Thus, high- dimensional quantum systems are not only of fundamental interest but are also becoming more readily available.In this context, the certification and quantification of entanglement in many dimensions is a crucial challenge since full state tomography (FST) for bipartite systems of local dimension d requires measurements in (d + 1) 2 global product bases (i.e., tensor product bases for the global state) [25], which quickly becomes impractical in high dimensions. Due to the complexity of realizing measurements in high-dimensional spaces, previous ex...

Entanglement protection of high-dimensional states by adaptive optics

“…More generally in the context of the open system entanglement theory, and irrespective of the specific resource character of the transmitted state for QKD, the here considered scenario defines an interesting and timely testing ground for different approaches to economically assess entanglement properties of high dimensional, mixed quantum states [60][61][62][63], a topic to be addressed in future work.…”

Entanglement certification from theory to experiment

“…S i for the chosen orthonormal basis {|v i } i . For non-orthonormal bases, this approach can still be used with minor modifications [115]. Crucially, the data corresponding to a d-outcome measurement can also be obtained from d individual singleoutcome measurements.…”

“…Collective observables are (weighted) averages of single-party observables that can be used to witness entanglement via their second moments [ * the second moments of a single observable given by a weighted sum of local observables, i.e., where terms are local but may act nontrivially on more than one subsystem, e.g., for an interaction Hamiltonian, are sufficient to certify entanglement [117,118]]. Local (bipartite or n-partite) measurements in MUBs (or tilted bases [115]) can be used for local tomography and direct fidelity estimation ( * * or for certifying a lower bound with only two product bases [115]). Determining the coefficients of the Bloch decomposition requires the measurement of all d 2 − 1 local Bloch vector elements in every possible combination.…”

High-dimensional quantum gates using full-field spatial modes of photons

“…The existence of such simple and efficient elements is one of the main reasons why this degree of freedom (DOF) has been used in a myriad of quantum experiments both in fundamental studies [2,5,6] as well as in applications [7,8]. However, it is known that high-dimensional systems, so-called qudits, offer access to several advantages such as an increase in channel capacity as well as an improved resistance to noise in communication protocols [9][10][11] with feasible experimental effort [12].…”