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arXiv:2507.10027 · Journal of Mathematical Physics

Indiscernibility of quantum states

Jan van Neerven, Marijn Waaijer

Published May 1, 2026• 0 citations
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Abstract

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 quotient, 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 representation theorem: for every von Neumann algebra M there is a canonical lift a↦â 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 outcome 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.

DOI
10.1063/5.0302377

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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 quotient, 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 representation theorem: for every von Neumann algebra M there is a canonical lift a↦â 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 outcome 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.
The origin and basis of the notion of quantum contextuality is identified in the Copenhagen approach to quantum mechanics, where context is automatically invoked by its requirement that the experimental arrangement involved in any measurements or set of measurements be taken into account while, in general, the outcome of a measurement may depend on other measurements immediately preceding or jointly performed on the same system. For Bohr, the specification of the experimental situation of any measurement is essential to its significance in light of complementarity and the omnipresence of the quantum of action in physics; for Heisenberg, the incompatibility of pairs of sharp measurements belonging to different situations coheres with both the completeness of the quantum state as an objective physical description and the principle of indeterminacy. Here, context in the Copenhagen approach is taken to be the equivalence class of experimental arrangements corresponding to a set of compatible measurements of quantum observables in standard quantum mechanics; the associated form of contextuality in quantum mechanics arises via the non-commutativity in general of sharp observables, proven by von Neumann, that can appear, providing different contexts. This notion is related to theoretical situations explored later by Bell, by Kochen and Specker, and by others in relation to the classification of hidden-variables theories and elsewhere in physics. This article is part of the theme issue ‘Contextuality and probability in quantum mechanics and beyond’.
Recent developments in the formalisation of quantum causal structures have made it possible to test and compare hypotheses about causal structure empirically, rather than being a-priori assumptions. Such differences in causal structure may be leveraged to distinguish between the processes they belong to, akin to distinguishing between quantum states known to belong to different eigenspaces of an observable. So how far this analogy can be pushed? Can causal order be interpreted as a kind of observable? Can it be measured? To this end we construct a completely operational definition of observables in terms of `discrimination tasks'. This begins with some base set of classes (analogous to a quantum observable's eigenspaces) before we impose three conditions on how these classes relate under discrimination tasks. These conditions recover the properties of a standard observable, letting us apply this definition to non-state entities such as quantum processes, which naturally encode causal structure. These conditions can be described in plain language as (1) `members of any class are perfectly distinguishable from members of any other' (i.e. our classes are `sharp'), (2) `entities perfectly distinguishable pairwise are all perfectly distinguishable by a joint intervention', and (3) `if all members of a class are perfectly distinguishable from an entity, then the whole class is perfectly distinguishable from it jointly'. By analysing classes of quantum processes having strict causal orders, we find causal classes which are sharp but violate condition 2, and classes which violate condition 3. This implies that causal order cannot be an observable. We note cases in the literature which claim to prove the opposite, and discuss them. We find their analyses presume special circumstances equivalent to conditions (2) and (3), and so do not contradict our own findings.
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We introduce a general witness-based framework for quantifying and operationally activating continuous-variable (CV) resources into discrete-variable (DV) bipartite entanglement or Einstein- Podolsky-Rosen (EPR) steering. For the three standard CV resource theories associated with Wigner negativity (WN), genuine non-Gaussianity (GNG), and standard non-Gaussianity (SNG), we define infinite families of bounded-witness monotones indexed by box constraints on the witness operators. For closed convex free sets, these monotones are faithful, strongly monotonic under free instruments, Lipschitz continuous, and convex. For closed nonconvex free sets, we show that faithfulness requires a two-copy lift and formulate the corresponding strong-monotonicity statement in the lifted theory. We further construct witness-dependent completely positive trace-preserving (CPTP) measure-and-prepare channels whose outputs are two-qubit Werner states. For the representative case n = m = 1, the optimal entanglement and EPR steering attainable within this witness-dependent activation family are exactly proportional to the underlying monotones. We illustrate the framework with odd-parity states, pure-loss single-photon states, and Gottesman- Kitaev-Preskill (GKP) states, and derive explicit lower bounds for pure-state GNG and SNG. More broadly, our results show that closed CV free sets admit witness-based quantifiers with a direct operational interpretation in terms of experimentally accessible DV correlations.
What are the minimal requirements for a physically reasonable mathematical model for quantum theory or a potential extension? In addressing this question, rather than relying on the generalized probabilistic theories or other usual approaches, we propose a reset based on the following fundamental features: transition probabilities, which are typical of quantum theory, and continuous reversible dynamical processes (e.g., represented by a Lie group), which play the same crucial role in classical as well as in quantum physics. In doing so, we get back to the very elementary transition probability space framework originally introduced by Bogdan Mielnik in 1968 and combine it with the continuous reversibility condition. The primary class of valid spaces arises from the irreducible atomic JBW algebras, which encompass the irreducible atomic von Neumann algebras and the Jordan matrix algebras. The von Neumann algebras represent the model of common quantum theory. Beyond these, there is a non-standard valid space, where the exceptional Lie group E6E_6E6​ acts transitively. While E6E_6E6​ has been proposed as a candidate for internal symmetries in particle physics, we highlight that this model deviates from standard quantum theory in critical ways. Specifically, it fails to guarantee the existence of post-measurement states in all situations. Though the above postulates are powerful, they still allow for some physically meaningless models. A further feature of quantum theory (the correspondence between the pure states and the minimal projections) leads us to an additional requirement that rules out these models. However, a complete classification of the valid models remains one of the open issues, pointed out in the paper, which is intended for the mathematical and physical experts, inspiring them to tackle these problems.
Two basic ingredients in the quantum formalism are the concepts of states and observables, and apart from the average value (which includes the Born probability as a paramountly important case), the variance is the most fundamental and prominent quantity arising from the coupling of a state and an observable. Given the simplicity and ubiquity of the variance, it seems surprising that we can still exploit the variance to gain conceptually different insight into the quantum substratum. The purpose of the present work is to advocate the following idea: By inputting a general quantum state as the observable in the variance and incorporating a resolution of identity induced by coherent states, we obtain a mathematically simple and physically intuitive method for quantifying nonclassicality. We reveal its fundamental properties and highlight its connection with phase-space quantification of nonclassicality. The idea further suggests a whole family of appealing quantifiers of nonclassicality involving deep mathematical subtlety and an intriguing conjecture with physical significance. Some prototypical examples are worked out to illustrate the concept. This approach of regarding states as observables presents an opportunity for investigating quantum features from an alternative perspective of uncertainty.