In quantum mechanics, outcomes of measurements on a state have a probabilistic interpretation while the evolution of the state is treated deterministically. Here we show that one can also treat the evolution as being probabilistic in nature and one can measure which unitary acted. In further analogy to states, one can also choose which basis of unitaries to measure. Likewise, one can give an information-theoretic interpretation to evolutions by defining the entropy of a completely positive map. This entropy gives the rate at which the informational content of the evolution can be compressed. One cannot compress this information and still have the evolution act on an unknown state, but we demonstrate a general scheme to do so probabilistically. This allows one to generalize super-dense coding to the sending of quantum information. One can also define the ``interaction-entanglement'' of a unitary, and concentrate this entanglement.
In quantum mechanics, outcomes of measurements on a state have a probabilistic interpretation while the evolution of the state is treated deterministically. Here we show that one can also treat the evolution as being probabilistic in nature and one can measure which unitary acted. In further analogy to states, one can also choose which basis of unitaries to measure. Likewise, one can give an information-theoretic interpretation to evolutions by defining the entropy of a completely positive map. This entropy gives the rate at which the informational content of the evolution can be compressed. One cannot compress this information and still have the evolution act on an unknown state, but we demonstrate a general scheme to do so probabilistically. This allows one to generalize super-dense coding to the sending of quantum information. One can also define the ``interaction-entanglement'' of a unitary, and concentrate this entanglement.
Information arises out of a communication game played between a sender and receiver of signals. For physical systems, the players may be designated as Nature and the scientist. The average information obtained from a quantum system is given by the von Neumann measurement scheme, which represents a generalization of thermodynamic entropy that is in perfect accord with common sense when considering a mixed state. While testing multiple copies of such a state can reveal information about the choice made by the sender, entropy for an unknown pure state is zero. A related puzzle is how the total information in the universe appears to be increasing over time.1 Zero entropy for an unknown pure state is reasonable inasmuch as once that state has been identified, no further information may be gained from examining its copies. But such an idea will not be reasonable if the game between sender and receiver consists of the former choosing one of a certain number of polarization states (say, for a photon) and supplying several copies to the receiver. In this case, measurements made on the copies reveal information regarding the sender’s choices. If the set of choices is infinite, the ‘information’ generated by the source is unbounded. From the point of view of the preparer of the states, information in the pure state is limited by the ‘relationship’ between source and receiver, and by the precision of the receiver’s measurement apparatus. If the sender chose a polarization state with which the apparatus was synchronized, the receiver could readily recognize the state. In recent papers,2, 3 I have been investigating information obtainable from an unknown pure state within the framework of communication between source and receiver. I propose a measure of entropy that covers both pure and mixed states. In general, entropy has two components. One is informational, related to the pure components of the quantum state, which can vary by receiver. The other component is thermodynamic and independent of the receiver. The increase of information over time is a consequence of the interplay between unitary and non-unitary evolution, which makes it possible to transform one type of information into the other. Such complementarity indicates that a fundamental duality is essential for information. For a two-component elementary mixed state, the most information to be obtained from each measurement is one bit. Each further measurement of identically prepared states will also yield one bit. For an unknown pure state, information represents the choice made by the source from an infinity of choices related to the values of probability amplitudes with respect to the basis components of the receiver’s measurement apparatus. For a two-component pure state, each measurement also provides a maximum of one bit of information. If the source has made available an unlimited number of identically prepared states, the receiver can obtain additional information from each measurement until the probability amplitudes have been correctly estimated. Once that has occurred, unlike the case with a mixed state, no further information will be obtained from testing additional copies of this pure state. These ideas have potential application in the field of quantum computing. Estimates by the receiver can be made by adjusting the basis vectors to move closer to the unknown pure state. In the most general case, the information obtainable from such a state in repeated experiments is potentially infinite. But if the observer learns the value of the pure state, the information associated with the states vanishes. This suggests a fundamental divide between objective and subjective information. This approach is consistent with the positivist view that one cannot speak of information associated with a system except in relation to an experimental arrangement together with the protocol for measurement. The experimental arrangement is thus integral to the amount of information that can be obtained. The informational measure I have proposed resolves the puzzle of entropy increase. We can suppose that, in the begin-
Probabilistic unitary maps and probabilistic unitary quantum channels are introduced, many quantum information applications, including (unambiguous) teleportation, can be described as probabilistic unitary quantum channels. Some properties of probabilistic unitary maps and probabilistic unitary quantum channels are derived. The property of a probabilistic unitary quantum channel ensures certain simple form for the measurements involved in realizing the probabilistic unitary quantum channel. The relation between a probabilistic unitary quantum channel and a uniformly entangled state is established. The combined operation of a noisy quantum channel and the error correction operation is a probabilistic unitary channel, from this point of view the condition for the errors to be correctable is easily derived. The condition for the errors caused by a noisy channel to be correctable with any nonzero probability is also obtained. Dense coding with a partially entangled state can also be viewed as a probabilistic unitary quantum channel when all messages are required to be transmitted with equal probability of success, the maximal achievable probability of success is derived and the optimum protocol is also obtained. 1
We present a new interpretation of the terms superposition, entanglement, and measurement that appear in quantum mechanics. We hypothesize that the structure of the wave function for a quantum system at the sub-Planck scale has a deterministic cyclic structure. Each cycle comprises a sequential succession of the eigenstates that comprise a given wave function. Between unitary operations or measurements on the wave function, the sequential arrangement of the current eigenstates chosen by the system is immaterial, but once chosen it remains fixed until another unitary operation or measurement changes the wave function. The probabilistic aspect of quantum mechanics is interpreted by hypothesizing a measurement mechanism which acts instantaneously but the instant of measurement is chosen randomly by the classical measuring apparatus over a small but finite interval from the time the measurement apparatus is activated. At the instant the measurement is made, the wave function irrevocably collapses to a new state (erasing some of the past quantum information) and continues from thereon in that state till changed by a unitary operation or a new measurement.
This preprint presents a closed, non-unitary evolution–measurement framework describing a cyclic process of distinction, probabilistic localization, reflexive stabilization, integration, and compression. Unlike standard quantum mechanics, where unitary Schrödinger evolution is taken as fundamental and measurement is treated as external, the present model treats measurement as an intrinsic structural step within the dynamics themselves. The framework introduces attention-modulated open-system evolution, explicit normalization-aware measurement probabilities, and a reflexive fixed-point construction for stabilized states. Physical observables arise through projection and partial tracing, yielding a reduced density matrix interpreted as physical reality. An internal, accumulated notion of time—distinct from the external evolution parameter—is also defined, allowing attention to modulate experienced temporal flow. The model is formulated at a formal and exploratory level, without assuming a specific physical Hamiltonian or experimental realization. It is intended as a conceptual and mathematical contribution to the foundations of quantum mechanics, particularly in relation to measurement theory, non-unitary dynamics, and information-theoretic structure. No claims of empirical validation are made.
An analysis of quantum measurement is presented that relies on an information-theoretic description of quantum entanglement. In a consistent quantum information theory of entanglement, entropies (uncertainties) conditional on measurement outcomes can be negative, implying that measurement can be described via unitary, entropy-conserving, interactions, while still producing randomness in a measurement device. In such a framework, quantum measurement is not accompanied by a wave-function collapse, or a quantum jump. The theory is applied to the measurement of incompatible variables, giving rise to a stronger entropic uncertainty relation than heretofore known. It is also applied to standard quantum measurement situations such as the Stern-Gerlach and double-slit experiments to illustrate how randomness, inherent in the conventional quantum probabilities, arises in a unitary framework. Finally, the present view clarifies the relationship between classical and quantum concepts.
A general theory of quantum-mechanical measurement is presented. It is shown that some general conditions should be satisfied for any apparatus to carry out the ideal measurement, and the reduction of state is an inevitable consequence of the action of such an apparatus. The statistical operator (in weak sense) is defined uniquely for ensembles prepared with respect to the measured observables, without recourse to the expectation value hypothesis. The probabilistic interpretation (the Born ansatz) comes out as a natural explanation of physical contents of the statistical operator after the measurement, rather than an axiomatic presumption of quantum mechanics.