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This paper provides a systematic study of the operational idea that a quantum “state” is only defined up to what can be distinguished by a chosen family of observables. Concretely, any von Neumann algebra of observables M induces an equivalence relation on pure and mixed states by declaring two preparations indiscernible when they give identical statistics for every observable in M . The corresponding quo tient, the Holevo space associated with M , is the effective (relational) state space of the experiment, explicitly dependent on the observer’s available measurements. We analyse the resulting geometry and topology of these quotients, and prove a context-complete classical repre sentation theorem: for every von Neumann algebra M there is a canonical lift a ↦ ̂ a to bounded continuous functions on the Holevo space, reproducing expectation values pointwise. In the commutative case this reduces to ordinary probability theory on the joint spectrum. The framework is illustrated in explicit examples, including position measurements of a free particle and polarisation measurements in the qubit, Einstein–Podolsky–Rosen (EPR), and Bell settings. In particular, in the EPR scenario Charlie’s joint observable defines a simplex of joint out come distributions, whereas the Alice/Bob marginal viewpoint collapses the effective description to a lower-dimensional space by “forgetting” the correlation parameter. We show that by varying the polariser settings, the indiscernibility classes become conjugated (and generically reshuffled), and different settings are typically incompatible at the level of observable algebras.
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This paper provides a systematic study of the operational idea that a quantum “state” is only defined up to what can be distinguished by a chosen family of observables. Concretely, any von Neumann algebra of observables M induces an equivalence relation on pure and mixed states by declaring two preparations indiscernible when they give identical statistics for every observable in M . The corresponding quo tient, the Holevo space associated with M , is the effective (relational) state space of the experiment, explicitly dependent on the observer’s available measurements. We analyse the resulting geometry and topology of these quotients, and prove a context-complete classical repre sentation theorem: for every von Neumann algebra M there is a canonical lift a ↦ ̂ a to bounded continuous functions on the Holevo space, reproducing expectation values pointwise. In the commutative case this reduces to ordinary probability theory on the joint spectrum. The framework is illustrated in explicit examples, including position measurements of a free particle and polarisation measurements in the qubit, Einstein–Podolsky–Rosen (EPR), and Bell settings. In particular, in the EPR scenario Charlie’s joint observable defines a simplex of joint out come distributions, whereas the Alice/Bob marginal viewpoint collapses the effective description to a lower-dimensional space by “forgetting” the correlation parameter. We show that by varying the polariser settings, the indiscernibility classes become conjugated (and generically reshuffled), and different settings are typically incompatible at the level of observable algebras.
In this article, we present a detailed analysis of two famous delayed choice experiments: Wheeler’s classic gedanken-experiment and the delayed quantum eraser. Our analysis shows that the outcomes of both experiments can be fully explained on the basis of the information collected during the experiments using textbook quantum mechanics only. At no point in the argument, information from the future is needed to explain what happens next. In fact, more is true: for both experiments, we show, in a strictly mathematical way, that a modified version in which the time-ordering of the steps is changed to avoid the delayed choice leads to exactly the same final state. In this operational sense, the scenarios are completely equivalent in terms of conclusions that can be drawn from their outcomes.
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In this article, we present a detailed analysis of two famous delayed choice experiments: Wheeler’s classic gedanken-experiment and the delayed quantum eraser. Our analysis shows that the outcomes of both experiments can be fully explained on the basis of the information collected during the experiments using textbook quantum mechanics only. At no point in the argument, information from the future is needed to explain what happens next. In fact, more is true: for both experiments, we show, in a strictly mathematical way, that a modified version in which the time-ordering of the steps is changed to avoid the delayed choice leads to exactly the same final state. In this operational sense, the scenarios are completely equivalent in terms of conclusions that can be drawn from their outcomes.
We present an analysis of the Frauchiger–Renner Gedankenexperiment from the point of view of the relational interpretation of quantum mechanics. Our analysis shows that the paradox obtained by Frauchiger and Renner disappears if one rejects promoting one agent’s certainty to another agent’s certainty when it cannot be validated by records from the past. A by-product of our analysis is an interaction-free detection scheme for the existence of such records.
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We present an analysis of the Frauchiger–Renner Gedankenexperiment from the point of view of the relational interpretation of quantum mechanics. Our analysis shows that the paradox obtained by Frauchiger and Renner disappears if one rejects promoting one agent’s certainty to another agent’s certainty when it cannot be validated by records from the past. A by-product of our analysis is an interaction-free detection scheme for the existence of such records.
