Fedor Herbut

On mutual information in multipartite quantum states and equality in strong subadditivity of entropy

The challenge of equality in the strong subadditivity inequality of entropy is approached via a general additivity of correlation information in terms of nonoverlapping clusters of subsystems in multipartite states (density operators). A family of tripartite states satisfying equality is derived.

On EPR-Type Entanglement in the Experiments of Scully et al. I. The Micromaser Case and Delayed-Choice Quantum Erasure

Delayed-choice erasure is investigated in two-photon two-slit experiments that are generalizations of the micromaser experiment of Scully et al. (Nature 351:111–116, 1991). Applying quantum mechanics to the localization detector, it is shown that erasure with delayed choice in the sense of Scully, has an analogous structure as simple erasure. The description goes beyond probabilities. The EPR-type disentanglement, consisting in two mutually incompatible distant measurements, is used as a general framework in both parts of this study. Two simple coherence cases are shown to emerge naturally, and they are precisely the two experiments of Scully et al. The treatment seems to require the relative-reality-of-unitarily-evolving-state (RRUES) approach. Besides insight in the experiments, this study has also the goal of insight in quantum mechanics. The question is if the latter can be more than just a “book-keeping device” for calculating probabilities as Scully et al. modestly and cautiously claim.

Quantum probability law from 'environment-assisted invariance' in terms of pure-state twin unitaries

Zureks derivation of the Born rule from envariance (environment-assisted invariance) is retraced in an attempt to present a more detailed derivation, which is extended to encompass the trace-rule form of the quantum probability law in its full generality. The investigation begins by a review presentation of Schmidt decomposition, and a detailed and complete theory of twin unitaries, which are the other face of envariance, and which stand in close connection with twin observables. The trace rule tr(Eρ), with E being an event (projector) and ρ a quantum state (density operator), is derived in five stages. The first three achieve the same as Zureks derivation of Borns rule, but in a different way, using the full power of twin unitaries. Stage 4 extends the Zurek-like derivation to states (ray projectors) that are not eigenstates of the density operator. Finally, stage 5 utilizes the assumption of continuity of the probability law in the density operators to extend the trace rule to isolated (not correlated) systems.

On EPR-type Entanglement in the Experiments of Scully et al. II. Insight in the Real Random Delayed-choice Erasure Experiment

It was pointed out in the first part of this study (Herbut in Found. Phys. 38:1046–1064, 2008) that EPR-type entanglement is defined by the possibility of performing any of two mutually incompatible distant, i.e., direct-interaction-free, measurements. They go together under the term ‘EPR-type disentanglement’. In this second part, quantum-mechanical insight is gained in the real random delayed-choice erasure experiment of Kim et al. (Phys. Rev. Lett. 84:1–5, 2000) by a relative-reality-of-unitarily-evolving-state (RRUES) approach (explained in the first part). Finally, it is shown that this remarkable experiment, which performs, by random choice, two incompatible measurements at the same time, is actually an EPR-type disentanglement experiment, closely related to the micromaser experiment discussed in the first part.

Subsystem measurement in unitary quantum measurement theory with redundant entanglement

Measurement of a degenerate (or non-degenerate) discrete observable is investigated in the framework of quantum measurement theory short of collapse, i.e. premeasurement theory, based on a unitary evolution operator that includes the measurement interaction between object and measuring instrument. A pointer observable with eigen-projectors of, in general, many (or even infinitely) dimensional ranges is introduced as a new approach. It leads to redundant entanglement in the final state. As the first main result, the basic dynamical relation of the approach is derived. It is shown to be equivalent to the calibration condition, which is known to define general exact measurement. The latter is given a practical form. Complete measurement (premeasurement with objectification or collapse), which is in some sense implied by the premeasurement theory, performed on a subsystem of a bipartite object in a pure state is studied with particular attention to its effect on the opposite, interactionally unaffected subsys...

Derivation of the quantum probability law from minimal non-demolition measurement

One more derivation of the quantum probability rule is presented in order to shed more light on the versatile aspects of this fundamental law. It is shown that the change of state in minimal quantum non-demolition measurement, also known as ideal measurement, implies the probability law in a simple way. Namely, the very requirement of minimal change of state, put in proper mathematical form, gives the well-known Luders formula, which contains the probability rule.

On bipartite pure-state entanglement structure in terms of disentanglement

Schrodinger’s disentanglement [E. Schrodinger, Proc. Cambridge Philos. Soc. 31, 555 (1935)], i.e., remote state decomposition, as a physical way to study entanglement, is carried one step further with respect to previous work in investigating the qualitative side of entanglement in any bipartite state vector. Remote measurement (or, equivalently, remote orthogonal state decomposition) from previous work is generalized to remote linearly independent complete state decomposition both in the nonselective and the selective versions. The results are displayed in terms of commutative square diagrams, which show the power and beauty of the physical meaning of the (antiunitary) correlation operator inherent in the given bipartite state vector. This operator, together with the subsystem states (reduced density operators), constitutes the so-called correlated subsystem picture. It is the central part of the antilinear representation of a bipartite state vector, and it is a kind of core of its entanglement structure...

On compatibility and improvement of different quantum state assignments

When Alice and Bob have different quantum knowledges or state assignments (density operators) ρA and ρB, respectively, for one and the same specific individual system, then the problems of compatibility and pooling arise. The so-called first Brun–Finkelstein–Mermin (BFM) condition for compatibility is reobtained in terms of possessed or sharp (i.e., probability one) properties. The second BFM condition is shown to be generally invalid in an infinite-dimensional state space. An argument leading to a procedure of improvement of ρA on account of ρB and vice versa is presented.

The role of coherence entropy of physical twin observables in entanglement

The concept of physical twin observables (PTO) for bipartite quantum states, introduced and proved relevant for quantum information theory in recent work, is substantially simplified. The relation of observable and state is studied in detail from the point of view of coherence entropy. Properties of this quantity are further explored. It is shown that, besides for pure states, also for a class of mixed states, quantum discord (measure of entanglement) can be expressed through the coherence entropy of a PTO complete in relation to the state.

State-Dependent Implication and Equivalence in Quantum Logic

Ideal occurrence of an event (projector) leads to the known change of a state (density operator) into (the Luders state). It is shown that two events and give the same Luders state if and only if the equivalence relation is valid. This relation determines equivalence classes. The set of them and each class, are studied in detail. It is proved that the range projector of the Luders state can be evaluated as , where denotes the greatest lower bound, and is the null projector of . State-dependent implication extends absolute implication (which, in turn, determines the entire structure of quantum logic). and are investigated in a closely related way to mutual benefit. Inherent in the preorder is the state-dependent equivalence , defining equivalence classes in a given Boolean subalgebra. The quotient set, in which the classes are the elements, has itself a partially ordered structure, and so has each class. In a complete Boolean subalgebra, both structures are complete lattices. Physical meanings are discussed.