We studied in this paper the bipartite quantum systems in finite Hilbert space. We applied the measurement on the first subsystem only and studied how this measurement is affecting the entropy of the second subsystem. The analytic work carried out in this paper shows that the post measurement state is factorizable when measurement is applied on the first subsystem. This result shows that the disorder of the system is reduced and hence the entropy of the whole system is reduced as expected. However, it is also shown in this work that the entropy of the second subsystem of the post-measurement state could be greater than that of the original state. Increasing entropy after measurement means that the disorder in the second subsystem is increased. The information we gain from measuring the first subsystem reduces our information about the second subsystem rather than increasing it or at least keeping it the same as expected. This means that more factors may be required in order to calculate the probability of finding the second subsystem in a certain state.
We studied in this paper the bipartite quantum systems in finite Hilbert space. We applied the measurement on the first subsystem only and studied how this measurement is affecting the entropy of the second subsystem. The analytic work carried out in this paper shows that the post measurement state is factorizable when measurement is applied on the first subsystem. This result shows that the disorder of the system is reduced and hence the entropy of the whole system is reduced as expected. However, it is also shown in this work that the entropy of the second subsystem of the post-measurement state could be greater than that of the original state. Increasing entropy after measurement means that the disorder in the second subsystem is increased. The information we gain from measuring the first subsystem reduces our information about the second subsystem rather than increasing it or at least keeping it the same as expected. This means that more factors may be required in order to calculate the probability of finding the second subsystem in a certain state.
In this paper, we will discuss the possibility of increasing the Holevo value when measuring with post-selection. By measurement with post-selection we mean that some of the outcomes of the measurement are discarded, and then we work with the resulting states. Using the example of two mixed states, it will be shown that mutual information after performing post-selection is greater than their initial Holevo value. An increase in the Holevo value follows from the monotonicity property of the quantum relative entropy.
Abstract We propose an extension of the quantum entropy power inequality for finite dimensional quantum systems, and prove a conditional quantum entropy power inequality by using the majorization relation as well as the concavity of entropic functions also given by Audenaert et al (2016 J. Math. Phys . 57 052202). Here, we make particular use of the fact that a specific local measurement after a partial swap operation (or partial swap quantum channel) acting only on finite dimensional bipartite subsystems does not affect the majorization relation for the conditional output states when a separable ancillary subsystem is involved. We expect our conditional quantum entropy power inequality to be useful, and applicable in bounding and analyzing several capacity problems for quantum channels.
Given an arbitrary measurement over a system of interest, the outcome of a posterior measurement can be used for improving the statistical estimation of the system state after the former measurement. Here, we realize an informational-entropic study of this kind of (Bayesian) retrodicted quantum measurement formulated in the context of quantum state smoothing. We show that the (average) entropy of the system state after the retrodicted measurement (smoothed state) is bounded from below and above by the entropies of the first measurement when performed in a selective and nonselective standard predictive way, respectively. For bipartite systems the same property is also valid for each subsystem. Their mutual information, in the case of a former single projective measurement, is also bounded in a similar way. The corresponding inequalities provide a kind of retrodicted extension of Holevo bound for quantum communication channels. These results quantify how much information gain is obtained through retrodicted quantum measurements in quantum state smoothing. While an entropic reduction is always granted, in bipartite systems mutual information may be degraded. Relevant physical examples confirm these features.
Information obtained by a quantum measurement process performed on a physical system and the entropy change of the measured physical system are considered in detail. It is shown that the condition for the amount of information obtained by the quantum measurement process to be represented by the Shannon mutual entropy is that the intrinsic observable of the measured physical system commutes with the operational observable defined by the quantum measurement process. When some measurement outcome is obtained, the decrease of the Shannon entropy of the measured system is compared with that of the von Neumann entropy. Furthermore, a condition is established under which the amount of information that can be established by the quantum measurement process becomes equal to the decrease of the Shannon entropy of the measured physical system.
This paper is our attempt, on the basis of physical theory, to bring more clarification on the question “What is life?” formulated in the well-known book of Schrödinger in 1944. According to Schrödinger, the main distinguishing feature of a biosystem’s functioning is the ability to preserve its order structure or, in mathematical terms, to prevent increasing of entropy. However, Schrödinger’s analysis shows that the classical theory is not able to adequately describe the order-stability in a biosystem. Schrödinger also appealed to the ambiguous notion of negative entropy. We apply quantum theory. As is well-known, behaviour of the quantum von Neumann entropy crucially differs from behaviour of classical entropy. We consider a complex biosystem S composed of many subsystems, say proteins, cells, or neural networks in the brain, that is, S=(Si). We study the following problem: whether the compound system S can maintain “global order” in the situation of an increase of local disorder and if S can preserve the low entropy while other Si increase their entropies (may be essentially). We show that the entropy of a system as a whole can be constant, while the entropies of its parts rising. For classical systems, this is impossible, because the entropy of S cannot be less than the entropy of its subsystem Si. And if a subsystems’s entropy increases, then a system’s entropy should also increase, by at least the same amount. However, within the quantum information theory, the answer is positive. The significant role is played by the entanglement of a subsystems’ states. In the absence of entanglement, the increasing of local disorder implies an increasing disorder in the compound system S (as in the classical regime). In this note, we proceed within a quantum-like approach to mathematical modeling of information processing by biosystems—respecting the quantum laws need not be based on genuine quantum physical processes in biosystems. Recently, such modeling found numerous applications in molecular biology, genetics, evolution theory, cognition, psychology and decision making. The quantum-like model of order stability can be applied not only in biology, but also in social science and artificial intelligence.
Using the single-mode approximation, we study entanglement measures including two independent quantities; i.e., negativity and von Neumann entropy for a tripartite generalized Greenberger–Horne–Zeilinger (GHZ) state in noninertial frames. Based on the calculated negativity, we study the whole entanglement measures named as the algebraic average π 3 -tangle and geometric average Π 3 -tangle. We find that the difference between them is very small or disappears with the increase of the number of accelerated qubits. The entanglement properties are discussed from one accelerated observer and others remaining stationary to all three accelerated observers. The results show that there will always exist entanglement, even if acceleration r arrives to infinity. The degree of entanglement for all 1–1 tangles are always equal to zero, but 1–2 tangles always decrease with the acceleration parameter r. We notice that the von Neumann entropy increases with the number of the accelerated observers and S κ I ζ I (κ, ζ ∈ (A, B, C)) first increases and then decreases with the acceleration parameter r. This implies that the subsystem ρ κ I ζ I is first more disorder and then the disorder will be reduced as the acceleration parameter r increases. Moreover, it is found that the von Neumann entropies S ABCI, S ABICI and S AIBICI always decrease with the controllable angle θ, while the entropies of the bipartite subsystems S 2−2non (two accelerated qubits), S 2-1non (one accelerated qubit) and S 2-0non (without accelerated qubit) first increase with the angle θ and then decrease with it.