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Unformatted text preview: Negative Information in Quantum Mechanics Qingqing Yuan June 11, 2006 1 Introduction Classical information theory has a relatively long history which can be dated back to 1940’s when the notion of classical information was first introduced by Shannon. The important elements of classical information theory has been in place since the 1970’s. In contrast, quantum information theory aroused interests only from mid 1990’s and is still in its infancy, with many of key building blocks not well under- stood. We know that Shannon entropy measures the uncertainty associated with a classical probability distribution. Quantum states can also be described in a similar fashion, with density operators replacing probability distributions. Based on this observation, researchers are trying to define quantum information theory. The basic definition of the entropy of a quantum state ρ , corresponding to Shannon entropy in classical theory, is called Von Neumann entropy by the formula S ( ρ ) ≡ - tr ( ρ log ρ ) (1) Other basic elements in quantum information theory, such as the relative en- tropy, etc, then can be defined following the ones in classical information theory, by replacing Shannon entropy with von Neumann entropy. However, such naive replacement arouses problems. For instance, classically the conditional entropy is defined as the difference between the total entropy and the entropy of subsys- tem, i.e. H ( A | B ) = H ( AB )- H ( B ) . By replacing Shannon entropies with von Neumann one in this formula, we get S ( A | B ) = S ( A,B )- S ( B ) Such an approach has been strongly advocated. But we know that if we measure one subsystem, it will destroy the whole state. Then what’s the meaning of such a formula? Another more serious problem is that the conditional entropy can be negative when there is entanglement. 1 There are some work discussing negative information [3, 4, 2, 1, 5, 6]. In this project, we try to understand what the negative information means and the applications which take use of the negative information. In section 2 we will first give some definitions. Most notations we use here are adopted from the book of Nielsen and Chuang. The meaning of negative information will be discussed in Section 3. And in Section 4, we will review some interesting applications....
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- Fall '09
- Information Theory, negative information, Quantum entanglement, von Neumann entropy, Conditional Entropy