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An Invitation to Quantum Incompatibility
Teiko Heinosaari, Takayuki Miyadera, Mario Ziman
TL;DR
The paper examines how incompatibility of physical devices unifies quantum impossibility statements and relates to nonclassical phenomena such as Bell violations and steering. It surveys incompatibility in operational theories and quantum devices, including its formulation, quantification, applications, and order-theoretic characterization. The review also identifies unresolved questions about the tightness of bounds and the physical origins of quantum incompatibility.
Problem
Quantum incompatibility underlies several quantum impossibility statements and nonclassical effects, but its central aspects span measurements, channels, operational theories, and information-processing settings.
Method
The paper provides a concise overview of incompatibility through general operational formulations, quantification, quantum observables and channels, process observables, and order-theoretic characterization.
Results
Incompatibility connects to Bell-inequality violations, quantum steering, and universal broadcasting, while compatible observables cannot violate any Bell inequality and incompatible observables are necessary and sufficient for steering.
Takeaways & Limitations
Incompatibility offers a common framework for understanding no-cloning, no information without disturbance, Bell violations, and steering across quantum devices.
Takeaways & Limitations
The paper does not provide a satisfactory explanation of why quantum theory is as incompatible as it is and calls for a general framework to investigate this question.
Abstract
from arXiv · showhide
In the context of a physical theory, two devices, A and B, described by the theory are called incompatible if the theory does not allow the existence of a third device C that would have both A and B as its components. Incompatibility is a fascinating aspect of physical theories, especially in the case of quantum theory. The concept of incompatibility gives a common ground for several famous impossibility statements within quantum theory, such as ``no-cloning'' and ``no information without disturbance''; these can be all seen as statements about incompatibility of certain devices. The purpose of this paper is to give a concise overview of some of the central aspects of incompatibility.
1. Introduction
Quantum incompatibility began with uncertainty and complementarity, and now links measurement constraints to Bell violations, impossibility results, and broader operational theories. The paper surveys these connections and central aspects across quantum devices.
- Quantum measurements can be incompatible when they cannot be implemented simultaneously, a notion rooted in Heisenberg’s uncertainty principle and Bohr’s complementarity.
- Only incompatible measurements enable Bell-inequality violations, which are used to assess experimental settings for quantum-information tasks such as quantum cryptography.
- Like entanglement, incompatibility is non-classical, can be destroyed by noise, and supports analogous notions such as breaking channels and robustness.
- Incompatibility generalizes from measurement collections to input-output devices, unifying impossibility statements including no-cloning and no information without disturbance.
- Operational theories provide a general setting for incompatibility: classical theories make all devices compatible, while quantum theory contains maximally incompatible device pairs.
- The paper gives a concise overview covering general operational formulations, quantification, quantum observables and channels, process observables, and order-theoretic characterizations.
2. Incompatibility in operational theories
Incompatibility concerns whether multiple input-output devices can be jointly implemented by one device. The section develops this idea operationally, quantifies it through noise robustness, and connects it to cloning, broadcasting, geometry, and descriptive compatibility.
- Preliminary definition of incompatibility: Devices A and B are compatible when a two-output joint device C has A and B as its marginals; otherwise, they are incompatible.This formalizes joint implementation using one input for devices that would otherwise require separate inputs.
- Quantification of incompatibility: A compatibility region contains the mixing parameters for which adding trivial devices makes the resulting devices compatible.Trivial devices ignore the input and produce fixed outputs, allowing incompatible devices to be approximated by compatible mixtures.
- Quantum examples: Finite-dimensional position and momentum observables become more incompatible as dimension d increases, while quantum theory also contains maximally incompatible infinite-dimensional pairs.The standard position and momentum observables on L2(R) are cited as maximally incompatible; finite-dimensional optimality questions remain open.
- Broadcasting: If unknown states can be copied, every finite collection of devices is compatible; therefore, any theory containing incompatible devices cannot contain a copying machine.Concatenating devices with a copying machine produces a joint multi-output device.
- Geometry and descriptive compatibility: Compatible device pairs form a convex set, and descriptive compatibility permits separate implementation with overlapping outputs even when simultaneous implementation is unavailable.For observables, operational and descriptive compatibility coincide because measurement outcomes can be duplicated.
3. Incompatibility of quantum observables
Quantum observables are compatible exactly when they can arise as marginals of a single joint observable. The section develops sufficient and informational criteria for incompatibility, and connects joint measurability with coexistence, steering, and Bell-inequality violations.
- Joint observability: A joint observable outputs tuples of outcomes, while each original observable is obtained by ignoring all but its corresponding component.Thus, a finite collection is compatible precisely when its observables are marginals of one joint observable.
- Compatibility criteria: For finite outcome observables, positivity of the operator J_n(M_1(x_1), ..., M_n(x_n)) for every outcome tuple is sufficient for compatibility.Commuting effects guarantee this positivity, but the condition also covers compatible observables beyond commuting sets.
- Compatibility criteria: If an observable contains a sufficiently sharp effect, compatibility with another observable is constrained because the effect’s unsharpness is measured by ||M_1(x) − M_1(x)^2||.This quantity vanishes exactly when the effect is a projection.
- Measurement uncertainty relations: A violation of the stated uncertainty inequality is sufficient to establish incompatibility between the observables under consideration.The same inequality is also necessary for incompatibility for two or three complementary qubit observables.
- Information and incompatibility: An observable that identifies selected known states from a single outcome is necessarily incompatible with an informationally complete observable that identifies an unknown state from full statistics.A joint device would otherwise perform both state-identification tasks, which the passage states is impossible.
- Coexistence: Joint measurability, coexistence, and weak coexistence form a strict hierarchy, although they coincide for some classes such as pairs with one discrete extreme observable.The hierarchy is jointly measurable ⇒ coexistent ⇒ weakly coexistent, while coexistence does not imply joint measurability in general.
- Bipartite settings: An assemblage generated by observables is steerable if and only if those observables are jointly measurable.Compatible observables cannot violate Bell inequalities, whereas every incompatible binary pair enables Bell-CHSH violation; whether all incompatible sets do so remains open.
4. Incompatibility of other quantum devices
The paper extends incompatibility beyond observables to quantum channels, instruments, and process observables, connecting these devices to cloning, disturbance, and process-measurement phenomena.
- 4.1. Incompatibility of quantum channels: Universal broadcasting is equivalent to jointly implementing two identity channels, so its impossibility means that the identity channels are incompatible.Broadcasting requires both reduced output channels to reproduce every input state exactly.
- 4.1. Incompatibility of quantum channels: A channel can be incompatible with itself: two identity channels cannot be jointly implemented because quantum information cannot be copied.This contrasts with observables and reflects the non-copyability of quantum information.
- 4.2. Conjugate channels: Conjugate channels quantify an information–disturbance trade-off: near-perfect decoding of one output makes the conjugate channel nearly completely depolarizing.If the input information can be almost entirely retrieved from one channel, little information flows to its environment through the conjugate channel.
- 4.3. Incompatibility of quantum observable and channel: A quantum instrument records measurement probabilities and transforms the measured state conditionally on the observed outcome.For outcome x, the probability is tr[Ix(ϱ)] and the conditioned state is obtained by normalizing Ix(ϱ).
- 4.3. Incompatibility of quantum observable and channel: General quantum instruments need not have the simple decomposable structure of the instruments introduced earlier.This marks a scope boundary for the preceding instrument examples.
- 4.3. Incompatibility of quantum observable and channel: Sequential measurement implements an observable together with a disturbed version of the second observable, which must be compatible with the first.When the original observables are compatible, the later measurement can be chosen to account for the first measurement’s disturbance.
- 4.4. No information without disturbance: No information without disturbance states that a unitary channel is incompatible with every nontrivial observable, while restricted orthogonal-state ensembles can permit information extraction without disturbance.In B92, eavesdropping information necessarily produces detectable disturbance, whereas orthogonal encodings provide an exception under restricted inputs.
5. Order theoretic characterization of quantum incompatibility
The paper recasts incompatibility through a preorder of devices, showing that compatibility is preserved under simulation and is naturally a property of equivalence classes. For observables and channels, this order yields concrete characterizations and least-disturbing representatives.
- 5.1. Preordering of devices: The simulation relation D1 ⪰ D2 is a preorder because devices can simulate themselves and simulation is transitive.It need not be a partial order because distinct devices can simulate each other.
- 5.1. Preordering of devices: Compatibility is preserved when each member of a compatible pair is replaced by a device it can simulate.This follows by composing the original joint device with the subsequent simulation devices.
- 5.1. Preordering of devices: Incompatibility is naturally studied on equivalence classes, because equivalent devices have exactly the same compatibility relations.The induced preorder becomes a partial order on these classes.
- 5.2. Preordering of quantum observables: Observables are compatible exactly when a common observable exists above every member in the smearing order.For rank-1 observables, compatibility occurs if and only if the observables are equivalent.
- 5.3. Preordering of quantum channels: For channels, two channels are compatible if and only if each is below the conjugate-channel class of the other.Conjugate channels are therefore characterized as compatible channel counterparts in the induced order.
- 5.3. Preordering of quantum channels: Each observable M has a least-disturbing M-channel ΛM, and every other compatible channel is obtained from it by subsequent processing.More noise in M permits less disturbance in ΛM, yielding a qualitative noise–disturbance relation.
6. Outlook
The paper concludes that quantum incompatibility is well studied but remains conceptually and technically incomplete. In particular, process-measurement incompatibility is largely unexplored, and its physical or informational origins lack a satisfactory general explanation.
- 6. Outlook: Incompatibility of measurements of quantum processes remains an unexplored research area despite broader progress in understanding quantum incompatibility.The paper notes that this area may affect future quantum applications because of qualitative and quantitative differences.
- 6. Outlook: The presentation derives incompatibility from its mathematical framework, while the physical or informational origins of quantum incompatibility remain unresolved.The authors call for a general framework to explain why quantum theory is as incompatible as it is.