Extraction of Product and Higher Moment Weak Values: Applications in Quantum State Reconstruction and Entanglement Detection

Abstract

Weak measurements introduced by Aharonov, Albert and Vaidman (AAV) can provide informations about the system with minimal back action. Weak values of product observables (commuting) or higher moments of an observable are informationally important in the sense that they are useful to resolve some paradoxes, realize strange quantum effects, reconstruct density matrices, etc. In this work, we show that it is possible to access the higher moment weak values of an observable using weak values of that observable with pairwise orthogonal post-selections. Although the higher moment weak values of an observable are inaccessible with Gaussian pointer states, our method allows any pointer state. We have calculated product weak values in a bipartite system for any given pure and mixed pre selected states. Such product weak values can be obtained using only the measurements of local weak values (which are defined as single system weak values in a multi-partite system). As an application, we use higher moment weak values and product weak values to reconstruct unknown quantum states of single and bipartite systems, respectively. Further, we give a necessary separability criteria for finite dimensional systems using product weak values and certain class of entangled states violate this inequality by cleverly choosing the product observables and the post selections. By such choices, positive partial transpose (PPT) criteria can be achieved for these classes of entangled states. Robustness of our method which occurs due to inappropriate choices of quantum observables and noisy post-selections is also discussed here. Our method can easily be generalized to the multi-partite systems.

References in corpus (33)

• Quantum entanglement
• Entanglement detection
• Direct Measurement of the Quantum Wavefunction
• Experimental joint weak measurement on a photon pair as a probe of Hardy's Paradox
• Complex weak values in quantum measurement
• Nonperturbative theory of weak pre- and post-selected measurements
• Direct measurement of general quantum states using weak measurement
• Revisiting Hardy's Paradox: Counterfactual Statements, Real Measurements, Entanglement and Weak Values
• Direct observation of Hardy's paradox by joint weak measurement with an entangled photon pair
• Amplification of Angular Rotations Using Weak Measurements
• Experimental Realization of the Quantum Box Problem
• Entropic uncertainty relations and entanglement
• Fast light, slow light, and phase singularities: a connection to generalized weak values
• Direct measurement of the density matrix of a quantum system
• A double-slit `which-way' experiment on the complementarity--uncertainty debate
• Extracting joint weak values with local, single-particle measurements
• State tomography via weak measurements
• Measuring measurement--disturbance relationships with weak values
• Practical measurement of joint weak values and their connection to the annihilation operator
• Modular values and weak values of quantum observables
• Measuring Non-Hermitian Operators via Weak Values
• Weak measurements with orbital angular momentum pointer states
• Can a Future Choice Affect a Past Measurement's Outcome?
• Directly Observing Momentum Transfer in Twin-Slit "Which-Way" Experiments
• Using weak values to experimentally determine "negative probabilities" in a two-photon state with Bell correlations
• Products of weak values: uncertainty relations, complementarity and incompatibility
• Quantum precision thermometry with weak measurement
• Measuring average of non-Hermitian operator with weak value in a Mach-Zehnder interferometer
• Strengthened Bell inequalities for orthogonal spin directions
• Full characterization of modular values for two-dimensional systems
• Universal detection of entanglement in two-qubit states using only two copies
• Complex counterpart of variance in quantum measurements for pre- and post-selected systems
• Direct Measurement Methods of Density Matrix of an Entangled Quantum State

Cited by in corpus (1)

• Uncertainty Relations in Pre- and Post-Selected Systems