Probing Rényi entanglement entropy via randomized measurements

Abstract

Entanglement is the key feature of many-body quantum systems, and the development of new tools to probe it in the laboratory is an outstanding challenge. Measuring the entropy of different partitions of a quantum system provides a way to probe its entanglement structure. Here, we present and experimentally demonstrate a new protocol for measuring entropy, based on statistical correlations between randomized measurements. Our experiments, carried out with a trapped-ion quantum simulator, prove the overall coherent character of the system dynamics and reveal the growth of entanglement between its parts -both in the absence and presence of disorder. Our protocol represents a universal tool for probing and characterizing engineered quantum systems in the laboratory, applicable to arbitrary quantum states of up to several tens of qubits.Engineered quantum systems, consisting of tens of individually-controllable interacting quantum particles, are currently being developed using a number of different physical platforms; including atoms in optical arrays (1-3), ions in radio-frequency traps (4, 5), and superconducting circuits (6-9). These systems offer the possibility of generating and 1 arXiv:1806.05747v2 [quant-ph] 14 Jan 2019 probing complex quantum states and dynamics particle by particle -finding application in the near-term as quantum simulators, and in the longer-term as quantum computers. As these systems are developed, new protocols are required to characterize them -to verify that they are performing as desired and to measure quantum phenomena of interest.A key property to measure in engineered quantum systems is entanglement. For example, in order for quantum simulators and computers to provide an advantage over their classical analogues, they must generate large amounts of entanglement between their parts (10). Furthermore, when using these devices to tackle open questions in physics, the dynamics of entanglement provides signatures of the phenomena of interest, such as thermalization (11) and many-body localization (12,13).Entanglement can be probed by measuring entanglement entropies. In particular, consider the second-order Rényi entropywith ρ A the reduced density matrix for a part A of the total system described by ρ. If the entropy of part A is greater than the entropy of the total system; i.e S (2) (ρ A ) > S (2) (ρ), bipartite entanglement exists between A and the rest of the system (14). Thus, a measurement of the entropy of the whole system, as well as of its subsystems, provides information about the entanglement contained within the system. Additionally, a measurement of the entropy of the total state ρ provides the opportunity to verify the overall coherence of the system, as for pure quantum states S (2) (ρ) = 0.Recently, a protocol to directly measure the second-order Rényi entropy, S (2) , has been demonstrated, requiring collective measurements to be made on two identical copies ρ of a quantum system (15)(16)(17)(18). In (17), that protocol was used to study entanglement growth and thermali...

Measuring von Neumann entanglement entropies without wave functions

“…The second class of imperfections we discuss is the presence of inhomogeneities in the BW-EH couplings. This is motivated by potential experimental realizations of the EH: indeed, the generic approach described above can be extended to formulate protocols to measure the von Neumann entropy in experiments (complementing previous approaches based on Renyi entropies [59][60][61][62][63][64][65] and entanglement spectra [15,66]) as follows.…”

Rényi entanglement entropies for the compactified massless boson with open boundary conditions

“…It has been understood that different phases of manybody systems can be characterized in terms of their entanglement properties [5,6], which also cover an important role in the efficiency of tensor networks algorithms [8][9][10][11][12][13]. Furthermore, recent times have witnessed a direct engineering of entanglement measurements in cold atom experiments [14][15][16][17][18][19][20]. Given a quantum system initialized in a pure state |ψ and bipartited in two halves A ∪ B, arguably the most widespread measurement of the entanglement between A and B is given by the Von Neumann entropy…”

Statistical localization: From strong fragmentation to strong edge modes

“…This has been motivated by experimental advances in preparing and manipulating quantum systems that are isolated from their environments to a high precision. An interesting question to ask is how such systems approach thermal equilibrium under their unitary dynamics [1][2][3][4][5][6][7][8][9]. The eigenstate thermalization hypothesis (ETH) has emerged as a sufficient condition for thermalization, and has been subsequently demonstrated to hold in a variety of interacting quantum systems [7,[10][11][12][13].…”