Fundamental studies of quantum measurements and their capacity to acquire information are typically based on scenarios in which the full Hilbert space of the measured quantum system is open to measurement interactions. In this work, we consider a class of incomplete quantum measurements — quantum subspace measurements (QSM's) — for which all measurement interactions are restricted to an arbitrary but specified subspace of the measured system Hilbert space. We define QSM's formally through a condition on the measurement Hamiltonian, obtain forms for the post-measurement states and positive operators (POVM elements) associated with QSM's acting in a specified subspace, and upper bound the accessible information for such measurements. Characteristic features of QSM's are identified and discussed.
Fundamental studies of quantum measurements and their capacity to acquire information are typically based on scenarios in which the full Hilbert space of the measured quantum system is open to measurement interactions. In this work, we consider a class of incomplete quantum measurements — quantum subspace measurements (QSM's) — for which all measurement interactions are restricted to an arbitrary but specified subspace of the measured system Hilbert space. We define QSM's formally through a condition on the measurement Hamiltonian, obtain forms for the post-measurement states and positive operators (POVM elements) associated with QSM's acting in a specified subspace, and upper bound the accessible information for such measurements. Characteristic features of QSM's are identified and discussed.
In the general theory of quantum measurement, one associates a positive-semidefinite operator on a d-dimensional Hilbert space to each of the n possible outcomes of an arbitrary measurement. In the special case of a projective measurement, these operators are pairwise Hilbert-Schmidt orthogonal, but when n>d, orthogonality is restricted by positivity. This restriction allows us to more precisely state the quantum adage: information gain of a system is always accompanied by unavoidable disturbance. Specifically, we investigate three properties of a measurement with L\"uders rule updating: its disturbance, a measure of how the expected postmeasurement state deviates from the input; its measurement strength, a measure of the intrinsic information producing capacity of the measurement; and its orthogonality, a measure of the degree to which the measurement operators differ from an orthonormal set. These quantities satisfy an information-disturbance tradeoff relation that highlights the additional role played by orthogonality. Finally, we assess several classes of measurements on these grounds and identify symmetric informationally complete quantum measurements as the unique quantum analogs of a perfectly informative and nondisturbing classical ideal measurement.
Quantum Measurement problem studied in Information Theory approach of systems selfdescription which exploits the information acquisition incompleteness for the arbitrary information system. The studied model of measuring system (MS) consist of measured state S environment E and observer O processing input S signal. O considered as the quantum object which interaction with S,E obeys to Schrodinger equation (SE). MS incomplete or restricted states for O derived by the algebraic QM formalism which exploits Segal and C∗-algebras. From Segal theorem for systems subalgebras it's shown that such restricted states VO=∣Oj><Oj∣ describes the classical random 'pointer' outcomes Oj observed by O in the individual events. The 'preferred' basis ∣Oj> defined by O state decoherence via O - E interactions.
In the general theory of quantum measurement, one associates a positive semidefinite operator on a d-dimensional Hilbert space to each of the n possible outcomes of an arbitrary measurement. In the special case of a projective measurement, these operators are pairwise Hilbert--Schmidt orthogonal, but when n>d, orthogonality is restricted by positivity. This restriction has consequences which are not present from a classical perspective; in particular, we find it allows us to more precisely state the old quantum adage that it is impossible to extract information from a system without disturbance. Restricting attention throughout to the Luders state updating rule, we consider three properties of a measurement: its disturbance, a measure of how the expected post-measurement state deviates from the input state, its purity gain, a measure of the expected reduction of uncertainty resulting from measurement, and its orthogonality, a measure of the degree to which the measurement operators differ from an orthonormal set. We show that these quantities satisfy a simple algebraic relation amounting to an expression of an information-disturbance trade-off. Finally, we assess several classes of measurements on these grounds and identify symmetric informationally complete quantum measurements as the unique quantum analogs of a perfectly informative and nondisturbing classical ideal measurement.
This study proposes a new approach to quantum state recovery following measurement. Specifically, we introduce a special operation that transfers the probability amplitude of the quantum state into its orthogonal complement. This operation is followed by a measurement performed on this orthogonal subspace, enabling the undisturbed original quantum state to be regained. Remarkably, this recovery is achieved without dependence of the post-measurement operation on the measurement outcome, thus allowing the recovery without historical dependence. This constitutes a highly nontrivial phenomenon. From the operational perspective, as the no-cloning theorem forbids perfect and probabilistic cloning of arbitrary quantum states, and traditional post-measurement reversal methods typically rely on operations contingent on the measurement outcomes, it questions fundamental assumptions regarding the necessity of historic dependence. From an informational perspective, since this recovery method erases the information about the measurement outcome, it's intriguing that the information can be erased without accessing the measurement outcome. These results imply the operational and informational non-triviality formulated in a direct-sum Hilbert space framework.
Article 104 established the finite causal-record capacity of a resource-bounded causal diamond as a foundational FCLET postulate. Article 105 gives that postulate its complete operational and mathematical closure. The capacity of a spacetime domain is not identified with the formal dimension of an unrestricted Hilbert space, the cardinality of a bare field configuration space or an observer-independent count of abstract states. FCLET capacity is the maximum number of physically admissible states that remain operationally distinguishable under a specified causal domain, energy budget, proper duration, bandwidth, measurement class, decoding architecture and error threshold. Let be a causal diamond and let denote its complete resource specification. The restricted measurement class defines an operational distinguishability pseudometric on the admissible state family . The central capacity definition is where is the maximal number of pairwise -distinguishable admissible states. This formulation resolves the mathematical difference between exact equivalence, finite-resolution distinguishability and decoder-induced record classes. Exact indistinguishability generates a genuine quotient space. Finite-resolution capacity is governed by packing and covering numbers. Decoder-defined equivalence generates the discrete record classes physically registered by the observer. Article 105 proves that operational capacity is finite whenever the resource-constrained state space is totally bounded in the operational metric. A positive accessible Hamiltonian with compact resolvent, finite energy, bounded spectral access and finite measurement resolution supplies a direct sufficient condition: The occupied information content is defined by the mutual information actually established between an encoded source variable and a registered record: Capacity utilization and FCLET load are then and The resulting chain provides the covariant operational source architecture required by FCLET latency dynamics.
We consider the question of characterizing the incompatibility of sets of high-dimensional quantum measurements. We introduce the concept of measurement incompatibility in subspaces. That is, starting from a set of measurements that is incompatible, one considers the set of measurements obtained by projection onto any strict subspace of fixed dimension. We identify three possible forms of incompatibility in subspaces: (i) incompressible incompatibility---measurements that become compatible in every subspace, (ii) fully compressible incompatibility---measurements that remain incompatible in every subspace, and (iii) partly compressible incompatibility---measurements that are compatible in some subspace and incompatible in another. For each class, we discuss explicit examples. Finally, we present some applications of these ideas. First, we show that joint measurability and coexistence are two inequivalent notions of incompatibility in the simplest case of qubit systems. Second, we highlight the implications of our results for tests of quantum steering.