the physical implementation, the limits and features of qubits implied by quantum information theory hold as all these systems are mathematically described by the same apparatus of density matrices over the complex numbers. Another important difference with quantum mechanics is that, while quantum mechanics often studies infinite-dimensional systems such as a harmonic oscillator, quantum information theory concerns both with continuous-variable systems and finite-dimensional systems. Entropy and information Entropy measures the uncertainty in the state of a physical system. Entropy can be studied from the point of view of both the classical and quantum information theories. Classical information theory Classical information is based on the concepts of information laid out by Claude Shannon. Classical information, in principle, can be stored in a bit of binary strings. Any system having two states is a capable bit. Shannon entropy Shannon entropy is the quantification of the information gained by measuring the value of a random variable. Another way of thinking about it is by looking at the uncertainty of a system prior to measurement. As a result, entropy, as pictured by Shannon, can be seen either as a measure of the uncertainty prior to making a measurement or as a measure of information gained after making said measurement. Shannon entropy, written as a functional of a discrete probability distribution, associated with events , can be seen as the average information associated with this set of events, in units of bits: This definition of entropy can be used to quantify the physical resources required to store the output of an information source. The ways of interpreting Shannon entropy discussed above are usually only meaningful when the number of samples of an experiment is large. Rényi entropy The Rényi entropy is a generalization of Shannon entropy defined above. The Rényi entropy of order r, written as a function of a discrete probability distribution, , associated with events , is defined as: for and . We arrive at the definition of Shannon entropy from Rényi when , of Hartley entropy (or max-entropy) when , and min-entropy when . Quantum information theory Quantum information theory is largely an extension of classical information theory to quantum systems. Classical information is produced when measurements of quantum systems are made. Von Neumann entropy One interpretation of Shannon entropy was the uncertainty associated with a probability distribution. When we want to describe the information or the uncertainty of a quantum state,