Quantitative relations between different measurement contexts

Ming Ji and Holger F. Hofmann

Graduate School of Advanced Science and Engineering, Hiroshima University, Kagamiyama 1-3-1, Higashi Hiroshima 739-8530, Japan

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In quantum theory, a measurement context is defined by an orthogonal basis in a Hilbert space, where each basis vector represents a specific measurement outcome. The precise quantitative relation between two different measurement contexts can thus be characterized by the inner products of nonorthogonal states in that Hilbert space. Here, we use measurement outcomes that are shared by different contexts to derive specific quantitative relations between the inner products of the Hilbert space vectors that represent the different contexts. It is shown that the probabilities that describe the paradoxes of quantum contextuality can be derived from a very small number of inner products, revealing details of the fundamental relations between measurement contexts that go beyond a basic violation of noncontextual limits. The application of our analysis to a product space of two systems reveals that the nonlocality of quantum entanglement can be traced back to a local inner product representing the relation between measurement contexts in only one system. Our results thus indicate that the essential nonclassical features of quantum mechanics can be traced back to the fundamental difference between quantum superpositions and classical alternatives.

Quantum contextuality proves that quantum systems cannot be described by a measurement independent reality. However, it is still quite a mystery how the quantum formalism can replace the conventional notion of reality with fundamental relations that do not require any pre-determined reality of observable physical properties. Here, we investigate how quantum superpositions define the relations between different measurement contexts and derive precise quantitative relations that directly contradict the identification of quantum state components with unobserved realities.

The quantitative relations between different measurement contexts are given by the inner products of the Hilbert space vectors that describe the measurement outcomes of each context. Usually, these inner products define measurement probabilities relating state preparation to measurement outcomes. By applying these relations to multiple contexts, we show that the inner products introduce precise quantitative relations between the measurement outcomes of different contexts, necessarily resulting in the paradoxical relations that are widely seen as proofs of quantum contextuality. This result also applies to quantum non-locality, where we can derive the probability of observing Hardy's paradox based on the inner product of two state vectors representing the outcomes of incompatible local measurements.

Our analysis demonstrates that both contextuality and quantum non-locality can be explained in terms of the fundamental quantitative relations between different measurement contexts described by the inner products between state vectors representing the outcomes of these measurement contexts. Moreover, it provides a unified approach providing precise quantitative relations between measurement outcomes of incompatible measurements. Our new approach may thus hold the key to a deeper understanding of the nature of reality at the quantum level.

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Cited by

[1] Ming Ji, Jonte R. Hance, and Holger F. Hofmann, "Tracing quantum correlations back to collective interferences", arXiv:2401.16769, (2024).

[2] Kengo Matsuyama, Ming Ji, Holger F. Hofmann, and Masataka Iinuma, "Quantum contextuality of complementary photon polarizations explored by adaptive input state control", Physical Review A 108 6, 062213 (2023).

[3] Holger F. Hofmann, "Sequential propagation of a single photon through five measurement contexts in a three-path interferometer", arXiv:2308.02086, (2023).

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