Negative Entropy and Information in Quantum Mechanics

Abstract

A framework for a quantum mechanical information theory is introduced that is based entirely on density operators, and gives rise to a unified description of classical correlation and quantum entanglement. Unlike in classical (Shannon) information theory, quantum (von Neumann) conditional entropies can be negative when considering quantum entangled systems, a fact related to quantum nonseparability. The possibility that negative (virtual) information can be carried by entangled particles suggests a consistent interpretation of quantum informational processes. [S0031-9007(97) [4] have been obtained recently, quantum information is still puzzling in many respects. This is especially true for quantum teleportation and superdense coding, two purely quantum communication schemes devised recently [5]. Indeed, these dual processes which rely on the quantum correlation between the two members of a spatially separated Einstein-PodolskyRosen (EPR) pair are difficult to interpret in terms of information theory. We show in this Letter that these processes can be understood in a consistent way by exploiting a fundamental difference between Shannon theory [6] and an extended information theory that accounts for quantum entanglement. As we shall see, the latter allows for negative conditional entropy even though this is forbidden classically. This leads us to propose that such quantum informational processes can be described by diagramsmuch like particle physics reactions-involving particles carrying negative (virtual) information. By analogy with antiparticles, we refer to them as antiqubits.Previous attempts to describe quantum informational processes have generally relied on the formulas of classical information theory supplemented with quantum probabilities, not amplitudes. However, it has been realized since Schumacher [3] that the von Neumann entropy has an information-theoretical meaning, characterizing (asymptotically) the minimum amount of quantum resources required to code an ensemble of quantum states. This suggests that an extended information theory can be defined that explicitly takes quantum phases into account, as attempted in this Letter. The theory described here characterizes multipartite quantum systems using only density operators and von Neumann entropies. It includes Shannon theory as a special case but describes quantum entanglement as well, thereby providing a unified treatment of classical and quantum information. To be specific, let us consider a composite system consisting of two (classical or quantum) variables, A and B, and outline in parallel the classical and our quantum informationtheoretic treatment of it. In classical information theory, we define the Shannon entropy of A [6],where the variable A takes on value a with probability p͑a͒. It is interpreted as the uncertainty about A [an analogous definition holds for H͑B͒]. The quantum analog is the von Neumann entropy S͑r A ͒ of a quantum source A described by the density operator r A ,where Tr A denotes the trace over the degrees of freedom ...

Total correlations and mutual information

“…Thus, we see that random variables A and B are not independent and therefore there exist classical correlations between qubits A and B in the state (5). Assume now, according to the statement (i), that total correlations of a bipartite quantum state are quantified by quantum mutual information which is defined in formal analogy to classical mutual information as…”

Quantum downconversion and multipartite entanglement via a mesoscopic superconducting quantum interference device ring

“…Nevertheless, it is still useful for gaining an appreciation of the correlations that are present between the components of the system in which we are interested. The second entropic quantity we shall use is the entanglement measure introduced by Adami and Cerf, 22 which for a subsystem A of the total system T is given by…”

“…To the limits of the computational power available to us, we extend our theoretical investigation of such downconversion processes to that of generating four output photons, each at a quarter of the frequency of the input photon ͑i.e., a four photon downconversion͒. Within the Adami-Cerf criterion [22][23][24] we show that entanglement exists between any one of these output oscillator modes and the rest of the system ͑i.e., SQUID ring and input oscillator mode͒. This implies that all the output modes will be entangled with each other, as well as to the rest of the coupled system.…”

Nonlocality and the Correlation of Measurement Bases

“…This is rather puzzling because there seems to exist faster-than-light influencing yet superluminal signalling is not allowed. Another puzzling feature related to entanglement is the negativity shown in the conditional entropy of entangled quantum systems [6]. While a number of interpretations have been made with regard to this negativity [7], this issue is still considered to be unsettled [8].…”

“…While a number of interpretations have been made with regard to this negativity [7], this issue is still considered to be unsettled [8]. Due to this negativity, Cerf and Adami [6] have proposed to interpret entanglement as a qubit and anti-qubit correlation where anti-qubit is a qubit traveling backwards in time.…”