The information provided by a classical measurement is unambiguously determined by the mutual information between the output results and the measured quantity. However, quantum mechanically there are at least two notions of information gathering which can be considered, one characterizing the information provided about the initial preparation, useful in communication, and the other characterizing the information about the final state, useful in state-preparation and control. Here we are interested in understanding the properties of these measures, and the information gathering capacities of quantum and classical measurements. We provide a partial answer to the question `in what sense does information gain increase with initial uncertainty?' by showing that, for classical and quantum measurements which are symmetric with respect to reversible transformations of the state space, the information gain regarding the initial state always increases with the observer's initial uncertainty. In addition, we calculate the capacity of all unitarily covariant and commutative permutation-symmetric measurements for obtaining classical information. While it is the von Neumann entropy of the effects which appears in the latter capacity, it is the subentropy which appears in the expression for the former.
The information provided by a classical measurement is unambiguously determined by the mutual information between the output results and the measured quantity. However, quantum mechanically there are at least two notions of information gathering which can be considered, one characterizing the information provided about the initial preparation, useful in communication, and the other characterizing the information about the final state, useful in state-preparation and control. Here we are interested in understanding the properties of these measures, and the information gathering capacities of quantum and classical measurements. We provide a partial answer to the question `in what sense does information gain increase with initial uncertainty?' by showing that, for classical and quantum measurements which are symmetric with respect to reversible transformations of the state space, the information gain regarding the initial state always increases with the observer's initial uncertainty. In addition, we calculate the capacity of all unitarily covariant and commutative permutation-symmetric measurements for obtaining classical information. While it is the von Neumann entropy of the effects which appears in the latter capacity, it is the subentropy which appears in the expression for the former.
It is shown that the entropy reduction (the information gain in the initial terminology) of an efficient (ideal or pure) quantum measurement coincides with the generalized quantum mutual information of a quantum-classical channel mapping an a priori state to the corresponding posteriori probability distribution of the outcomes of the measurement. As a result the entropy reduction is defined for arbitrary a priori states (not only for states with finite von Neumann entropy) and its analytical properties is studied in detail by using general properties of the quantum mutual information. By using this approach it is shown that the entropy reduction of an efficient quantum measurement is a nonnegative lower semicontinuous concave function on the set of all a priori states having continuous restrictions to subsets on which the von Neumann entropy is continuous. Monotonicity and subadditivity of the entropy reduction are also easily proved by this method. A simple continuity condition for the entropy reduction and for the mean posteriori entropy considered as functions of a pair (a priori state, measurement) is obtained. A characterization of an irreducible measurement (in the Ozawa sense) which is not efficient is considered in the Appendix.
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-
I compare the role of the information in classical and quantum dynamics by examining the relation between information flows in measurements and the ability of observers to reverse evolutions. I show that in the Newtonian dynamics reversibility is unaffected by the observer's retention of the information about the measurement outcome. By contrast-even though quantum dynamics is unitary, hence, reversible-reversing quantum evolution that led to a measurement becomes, in principle, impossible for an observer who keeps the record of its outcome. Thus, quantum irreversibility can result from the information gain rather than just its loss-rather than just an increase of the (von Neumann) entropy. Recording of the outcome of the measurement resets, in effect, initial conditions within the observer's (branch of) the Universe. Nevertheless, I also show that the observer's friend-an agent who knows what measurement was successfully carried out and can confirm that the observer knows the outcome but resists his curiosity and does not find out the result-can, in principle, undo the measurement. This relativity of quantum reversibility sheds new light on the origin of the arrow of time and elucidates the role of information in classical and quantum physics. Quantum discord appears as a natural measure of the extent to which dissemination of information about the outcome affects the ability to reverse the measurement.This article is part of a discussion meeting issue 'Foundations of quantum mechanics and their impact on contemporary society'.
The concept of entropy rst arose in the study of thermodynamics and statistical me- chanics. In information theory the so-called Shannon entropy plays a signi cant role, which measures how much information we gain, on average, when we learn the value of a random variable. An alternative interpretation is that the entropy of the random variable quanti es the amount of uncertainty about the random variable before we learn its value. The concept was introduced by Claude E. Shannon in his 1948 paper "A Mathematical Theory of Communication". In the rst part of the present work we consider this fundamental measure of information and show some of its properties, such as non-negativity. The relative entropy or the Kullback-Leibler divergence is a very useful measure of the distance between two prob- ability distributions. After introducing its de nition and other notions (joint entropy, conditional entropy or mutual information) we can prove additional assertions about the Shannon entropy and present statements about the relations between the above men- tioned quantities. In quantum information theory, there is a quantity, namely von Neumann entropy, which corresponds to Shannon entropy and originates from it. In the section Von Neu- mann entropy we will introduce this quantity which is a continuous functional on the states, or on their mathematical representatives, the density operators, and it ful ls in- equalities such as subadditivity and strong subaddititvity. Just as with the Shannon entropy, it is extremely useful to de ne a quantum version of the relative entropy, which is one of the most important numerical quantities appearing in quantum information theory. It is used as a measure of distinguishability between quantum states. Among others its properties are jointly convexity, monotonicity which are speci cally convenient. Transformations on structures which preserve operations, quantities or relations among elements appear in many areas of mathematics. These are the so-called preservers which show up also in physics and even in chemistry where they are usually called symmetries. Therefore, the study of preserver transformations is an important area of research. The third section of this work is based on a paper, in which Lajos Molnár described the structure every bijective map on the space of all density operators on a given Hilbert space, which preserves the relative entropy. This result is closely related to Wigner's famous theorem on the form of quantum mechanical symmetry transformations. The above theorem of Molnár can be extended into two directions. On the one hand, the result holds also in the case when the transformation under consideration acts on the space of all positive semide nite operators not only on density operators (i.e., we omit the condition of unit trace). On the other hand the statement remains valid even if we omit the strongly restrictive condition that the transformation is bijective. The second result is a joint work with Prof. Lajos Molnár which appeared in the journal Linear Algebra and Its Applications.
In this chapter, we discuss several information measures that are important for quantifying the amount of information and correlations that are present in quantum systems. The first fundamental measure that we introduce is the von Neumann entropy (or simply quantum entropy ). It is the quantum generalization of the Shannon entropy, but it captures both classical and quantum uncertainty in a quantum state.1 The quantum entropy gives meaning to the notion of an information qubit . This notion is different from that of the physical qubit, which is the description of a quantum state of an electron or a photon. The information qubit is the fundamental quantum informational unit of measure, determining how much quantum information is present in a quantum system. The initial definitions here are analogous to the classical definitions of entropy, but we soon discover a radical departure from the intuitive classical notions from the previous chapter: the conditional quantum entropy can be negative for certain quantum states. In the classical world, this negativity simply does not occur, but it takes on a special meaning in quantum information theory. Pure quantum states that are entangled have stronger-than-classical correlations and are examples of states that have negative conditional entropy. The negative of the conditional quantum entropy is so important in quantum information theory that we even have a special name for it: the coherent information. We discover that the coherent information obeys a quantum data-processing inequality, placing it on a firm footing as a particular informational measure of quantum correlations. We then define several other quantum information measures, such as quantum mutual information, that bear similar definitions as in the classical world, but with Shannon entropies replaced with quantum entropies. This replacement may seem to make quantum entropy somewhat trivial on the surface, but a simple calculation reveals that a maximally entangled state on two qubits registers two bits of quantum mutual information (recall that the largest the mutual information can be in the classical world is one bit for the case of two maximally correlated bits). We then discuss several entropy inequalities that play an important role in quantum information processing: the monotonicity of quantum relative entropy, strong subadditivity, the quantum data-processing inequalities, and continuity of quantum entropy.
It is well known that a Shannon-based definition of information entropy leads in the classical case to the Boltzmann entropy. It is tempting to regard the von Neumann entropy as the corresponding quantum-mechanical definition. But the latter is problematic from the quantum information point of view. Consequently, we introduce a new definition of entropy that reflects the inherent uncertainty of quantum-mechanical states. We derive for it an explicit expression, and discuss some of its general properties. We distinguish between the minimum uncertainty entropy of pure states, and the excess statistical entropy of mixtures.