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AIP conference proceedings

Instantaneous and Local Nondemolition Measurement of Nonlocal Observables using Entangled States

Luca Marinatto

Published Jan 1, 2006• 0 citations
Publisher

Abstract

With reference to a system of n spin‐1/2 particles which are distributed between two space‐like separated parties, we present two procedures to perform an instantaneous and local measurement of any nondegenerate observable having as its eigenstates the orthonormal basis of n‐partite entangled states introduced in [S. Bose, V. Vedral, and P. L. Knight, Phys. Rev. A 57, 822 (1998)], whose vectors reduce to the Bell states for n = 2 and to the GHZ states for n = 3. The first procedure implements a genuine nondemolition measurement by resorting to two n‐particle entangled states as an additional resource. The second one distinguishes between those eigenstates, though without preserving them (i.e., it is a demolition measurement), but it requires only a single couple of additional particles in a (maximally entangled) Bell state. Both methods are instantaneous, in the sense that the outcome of the measurement process is completely determined as soon as the local operations, whose duration can be made as short as one wishes, have been independently performed in the two distant regions where the particles are located.

DOI
10.1063/1.2219366

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With reference to a system of n spin‐1/2 particles which are distributed between two space‐like separated parties, we present two procedures to perform an instantaneous and local measurement of any nondegenerate observable having as its eigenstates the orthonormal basis of n‐partite entangled states introduced in [S. Bose, V. Vedral, and P. L. Knight, Phys. Rev. A 57, 822 (1998)], whose vectors reduce to the Bell states for n = 2 and to the GHZ states for n = 3. The first procedure implements a genuine nondemolition measurement by resorting to two n‐particle entangled states as an additional resource. The second one distinguishes between those eigenstates, though without preserving them (i.e., it is a demolition measurement), but it requires only a single couple of additional particles in a (maximally entangled) Bell state. Both methods are instantaneous, in the sense that the outcome of the measurement process is completely determined as soon as the local operations, whose duration can be made as short as one wishes, have been independently performed in the two distant regions where the particles are located.
Although nondemolition, reliable, and instantaneous quantum measurements of some nonlocal variables are impossible, demolition reliable instantaneous measurements are possible for all variables. It is shown that this is correct also in the framework of the time-symmetric quantum formalism, i.e. nonlocal variables of composite quantum systems with quantum states evolving both forward and backward in time are measurable in a demolition way. The result follows from the possibility to reverse with certainty the time direction of a backward evolving quantum state. Demolition measurements of nonlocal backward evolving quantum states require remarkably small resources. This is so because the combined operation of time reversal and teleportation of a local backward evolving quantum state requires only a single quantum channel and no transmission of classical information.
A local measurement on one member of an entangled composite system is said to change the distant member's state instantaneously. Taken without qualification, this is misleading. A selective measurement changes the conditional state assigned to the remote subsystem once a local outcome is specified; a non-selective measurement, where that outcome is ignored, leaves the remote reduced density operator unchanged. This distinction reconciles quantum state update with operational no-signalling and provides the setting for Einstein–Podolsky–Rosen steering. This review develops measurement using density operators, completely positive instruments, Kraus representations, partial traces, conditional states, and steering assemblages. It derives the invariance of remote marginal statistics under local trace-preserving operations and contrasts this with Bell-local factorization. It examines the hierarchy among entanglement, steering, and Bell nonlocality, joint measurability, remote state preparation, and one-sided device-independent protocols. The article considers spacelike-separated measurements, microcausality in relativistic quantum theory, and ontological readings of selective state update offered by Everettian, Bohmian, collapse, and relational approaches. Decoherence is treated as the process through which local outcomes become records, while quantum Darwinism accounts for their redundant local accessibility. It highlights gaps in literature, and concludes that entanglement permits nonclassical conditional correlations, but local measurement supplies no controllable superluminal signal.
Bell-state measurements are essential ingredients in many protocols for quantum information processing, ranging from quantum teleportation and dense coding to entanglement distribution in quantum networks. Their power relies on the fact that they are measurements in an entangled basis of a two-particle system and that the used Bell-state basis can be generated from a single Bell state by local unitary transformations. How can these measurements be generalized to more particles? We develop a general framework for this state-to-measurement problem: We introduce a hierarchy of classes of measurement bases, distinguished by the local transformations the parties may use for their generation from a single state. This leads to a generalization of the concept of maximally entangleable (or weighted hypergraph) states and the identification of a novel maximally entangled basis of four qubits, being a candidate for data-hiding tasks or distillation protocols. Finally, we prove that not all forms of entanglement can be encoded in an entire measurement basis.
For certain joint measurements on a pair of spatially separated particles, we ask how much entanglement is needed to carry out the measurement exactly. For a class of orthogonal measurements on two qubits with partially entangled eigenstates, we present upper and lower bounds on the entanglement cost. The upper bound is based on a recent result by Berry [Phys. Rev. A 75, 032349 (2007)]. The lower bound, based on the entanglement production capacity of the measurement, implies that for almost all measurements in the class we consider, the entanglement required to perform the measurement is strictly greater than the average entanglement of its eigenstates. On the other hand, we show that for any complete measurement in $d\ifmmode\times\else\texttimes\fi{}d$ dimensions that is invariant under all local generalized Pauli operations, the cost of the measurement is exactly equal to the average entanglement of the states associated with the outcomes.
A complete, non-demolition procedure is established for measuring multi-qubit entangled states, such as the Bell-states and the GHZ-states, which is essential in certain processes of quantum communication, computation, and teleportation. No interaction between the individual parts of the entangled system, nor with any environment is required. A small probe (e.g. a single qubit) takes care of all interaction with the system, and is used repeatedly. The probe-qubit interaction is of the simplest form, and only this one type of interaction is required to perform a complete measurement. The process may be divided into elementary local operations and interactions, taking place sequentially as the probe visits each of the qubits. A shuttle mode is described, which may be repeated indefinitely. By the quantum Zeno effect, the entangled states can be maintained until released in a predictable state. This shuttle process is stable, and self-correcting, by virtue of the standard measurements performed repeatedly on the probe.
A simple protocol for complete and deterministic Bell state measurement is proposed. It consists of measurements of nonlocal spin product operators with the help of shared entanglement as an ancillary resource. The protocol realizes not only nonlocal Bell state measurement between a pair of distant qubits but also a complete Bell filter that transmits either one of the Bell states indicated by the measurement outcome. These schemes will be useful in quantum technologies where nonlocal Bell state measurement is indispensable.