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Can the choices of measurement basis for entangled quantum systems be random

Daegene Song

Published Jun 25, 2004• 0 citations
OpenAlex

Abstract

It is shown that, with respect to the outcomes of two entangled quantum systems, the choices of measurement basis cannot be independent from each other. Two parties share maximally entangled states and bases chosen between $Z$ and $X$ with $P=1/2$ are used to perform measurements at each end. With the assumption that an element of physical reality for the choice made at one end is not a function of the outcome of an entangled state at the other end, we derive a contradiction to this assumption and show that elements of physical reality for the choice of measurement basis at each end cannot be independent from each other.

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It is shown that, with respect to the outcomes of two entangled quantum systems, the choices of measurement basis cannot be independent from each other. Two parties share maximally entangled states and bases chosen between ZZZ and XXX with P=1/2P=1/2P=1/2 are used to perform measurements at each end. With the assumption that an element of physical reality for the choice made at one end is not a function of the outcome of an entangled state at the other end, we derive a contradiction to this assumption and show that elements of physical reality for the choice of measurement basis at each end cannot be independent from each other.
An ensemble consisting on systems of two entangled spin 1/2 particles, all of them in the same global quantum state, are considered. The two spins are measured, each of them, on a fixed direction, at two randomly selected measurement times. Realism, plus locality and freedom of choice referred to these chosen times, are assumed. Then, from the sole correlation function related to the two measurements, without considering any Bell inequalities, a contradiction between these assumptions and quantum mechanics is found.
In classical mechanics, performing a measurement without reading the measurement outcome is equivalent to not exploiting the measurement at all. A nonselective measurement in the classical realm carries no information. Here we show that the situation is remarkably different when quantum mechanical systems are concerned. A nonselective measurement on one part of a maximally entangled pair can allow communication between two parties. In the proposed protocol, the signal is encoded in the choice of the measurement basis of one of the communicating parties, while the outcomes of the measurement are irrelevant for the communication and therefore may be discarded. Different choices for the (nonselective) measurement correspond to different signals. The implication of the study of measurements in quantum mechanics is considered. The scheme is studied in a Hilbert space of prime dimension.
We show that correlations inconsistent with any locally causal description can be a generic feature of measurements on entangled quantum states. Specifically, spatially separated parties who perform local measurements on a maximally entangled state using randomly chosen measurement bases can, with significant probability, generate nonclassical correlations that violate a Bell inequality. For n parties using a Greenberger-Horne-Zeilinger state, this probability of violation rapidly tends to unity as the number of parties increases. We also show that, even with both a randomly chosen two-qubit pure state and randomly chosen measurement bases, a violation can be found about 10% of the time. Among other applications, our work provides a feasible alternative for the demonstration of Bell inequality violation without a shared reference frame.
We examine the problem of exhibiting Bell nonlocality for a two-qudit entangled pure state using a randomly chosen set of mutually unbiased bases (MUBs). Interestingly, even if we employ only two-setting Bell inequalities, we find a significant chance of obtaining a Bell violation if the two parties are individually allowed to measure a sufficient number of MUBs. In particular, for the case of maximally entangled qutrits and ququarts, our numerical estimates indicate that we can obtain near-guaranteed Bell violation by considering only such Bell inequalities. The case of maximally entangled ququints is similar, albeit the chance of ending up with a successful trial decreases somewhat to approximately 99.8499.84%99.84. Upon a closer inspection, we find that even all these no-violation instances violate some more-setting Bell inequalities. These results suggest that the experimental tests of Bell nonlocality for these higher-dimensional entangled states remain viable even if the two parties do not share a common reference frame.
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.
A state-dependent proof of Bell's theorem without inequalities using the product state of any two maximally entangled states (Bell states) of two qubits for two observers in an ideal condition, each of which possesses two qubits, is proposed. It is different from the other proofs in which there exists a fundamental requirement that certain specific suitable Bell states have been chosen. Moreover, in any non-ideal situation, a common Bell inequality independent of the choices of the 16-product states is derived, which is used to test the contradiction between quantum mechanics and local reality theory in the reach of current experimental technology.