In physics, the von Neumann entropy, named after John von Neumann, is a measure of the statistical uncertainty within a description of a quantum system. It extends the concept of Gibbs entropy from classical statistical mechanics to quantum statistical mechanics, and it is the quantum counterpart of the Shannon entropy from classical information theory. For a quantum-mechanical system described by a density matrix , the von Neumann entropy is
where denotes the trace and denotes the matrix version of the natural logarithm. If the density matrix is written in a basis of its eigenvectors as
then the von Neumann entropy is merely
In this form, S can be seen as the Shannon entropy of the eigenvalues, reinterpreted as probabilities.
The von Neumann entropy and quantities based upon it are widely used in the study of quantum entanglement.
Fundamentals
In quantum mechanics, probabilities for the outcomes of experiments made upon a system are calculated from the quantum state describing that system. Each physical system is associated with a vector space, or more specifically a Hilbert space. The dimension of the Hilbert space may be infinite, as it is for the space of square-integrable functions on a line, which is used to define the quantum physics of a continuous degree of freedom. Alternatively, the Hilbert space may be finite-dimensional, as occurs for spin degrees of freedom. A density operator, the mathematical representation of a quantum state, is a positive semi-definite, self-adjoint operator of trace one acting on the Hilbert space of the system. A density operator that is a rank-1 projection is known as a pure quantum state, and all quantum states that are not pure are designated mixed. Pure states are also known as wavefunctions. Assigning a pure state to a quantum system implies certainty about the outcome of some measurement on that system (i.e., for some outcome ). The state space of a quantum system is the set of all states, pure and mixed, that can be assigned to it. For any system, the state space is a convex set: Any mixed state can be written as a convex combination of pure states, though not in a unique way. The von Neumann entropy quantifies the extent to which a state is mixed.
The prototypical example of a finite-dimensional Hilbert space is a qubit, a quantum system whose Hilbert space is 2-dimensional. An arbitrary state for a qubit can be written as a linear combination of the Pauli matrices, which provide a basis for self-adjoint matrices:
where the real numbers are the coordinates of a point within the unit ball and
The von Neumann entropy vanishes when is a pure state, i.e., when the point lies on the surface of the unit ball, and it attains its maximum value when is the maximally mixed state, which is given by .
Properties Some properties of the von Neumann entropy:
is zero if and only if represents a pure state. is maximal and equal to for a maximally mixed state, being the dimension of the Hilbert space. is invariant under changes in the basis of , that is, , with a unitary transformation. is concave, that is, given a collection of positive numbers which sum to unity and density operators , we have
is additive for independent systems. Given two density matrices describing independent systems A and B, we have
is strongly subadditive. That is, for any three systems A, B, and C:
This automatically means that is subadditive:
Below, the concept of subadditivity is discussed, followed by its generalization to strong subadditivity.
Subadditivity
If are the reduced density matrices of the general state , then
The right hand inequality is known as subadditivity, and the left is sometimes known as the triangle inequality. While in Shannon's theory the entropy of a composite system can never be lower than the entropy of any of its parts, in quantum theory this is not the case; i.e., it is possible that , while . This is expressed by saying that the Shannon entropy is monotonic but the von Neumann entropy is not. For example, take the Bell state of two spin-1/2 particles:
This is a pure state with zero entropy, but each spin has maximum entropy when considered individually, because its reduced density matrix is the maximally mixed state. This indicates that it is an entangled state; the use of entropy as an entanglement measure is discussed further below.
Strong subadditivity
The von Neumann entropy is also strongly subadditive. Given three Hilbert spaces, A, B, C,
By using the proof technique that establishes the left side of the triangle inequality above, one can show that the strong subadditivity inequality is equivalent to the following inequality:
where , etc. are the reduced density matrices of a density matrix . If we apply ordinary subadditivity to the left side of this inequality, we then find
By symmetry, for any tripartite state , each of the three numbers is less than or equal to the sum of the other two.
Minimum Shannon entropy Given a quantum state and a specification of a quantum measurement, we can calculate the probabilities for the different possible results of that measurement, and thus we can find the Shannon entropy of that probability distribution. A quantum measurement can be specified mathematically as a positive operator valued measure, or POVM. In the simplest case, a system with a finite-dimensional Hilbert space and measurement with a finite number of outcomes, a POVM is a set of positive semi-definite matrices on the Hilbert space that sum to the identity matrix,
The POVM element is associated with the measurement outcome , such that the probability of obtaining it when making a measurement on the quantum state is given by
A POVM is rank-1 if all of the elements are proportional to rank-1 projection operators. The von Neumann entropy is the minimum achievable Shannon entropy, where the minimization is taken over all rank-1 POVMs.
Holevo χ quantity
If are density operators and is a collection of positive numbers which sum to unity , then
is a valid density operator, and the difference between its von Neumann entropy and the weighted average of the entropies of the is bounded by the Shannon entropy of the :
Equality is attained when the supports of the – the spaces spanned by their eigenvectors corresponding to nonzero eigenvalues – are orthogonal. The difference on the left-hand side of this inequality is known as the Holevo χ quantity and also appears in Holevo's theorem, an important result in quantum information theory.
Change under time evolution
Unitary The time evolution of an isolated system is described by a unitary operator:
Unitary evolution takes pure states into pure states, and it leaves the von Neumann entropy unchanged. This follows from the fact that the entropy of is a function of the eigenvalues of .
Measurement A measurement upon a quantum system will generally bring about a change of the quantum state of that system. Writing a POVM does not provide the complete information necessary to describe this state-change process. To remedy this, further information is specified by decomposing each POVM element into a product:
The Kraus operators , named for Karl Kraus, provide a specification of the state-change process. They are not necessarily self-adjoint, but the products are. If upon performing the measurement the outcome is obtained, then the initial state is updated to
An important special case is the Lüders rule, named for Gerhart Lüders. Translated by K. A. Kirkpatrick as If the POVM elements are projection operators, then the Kraus operators can be taken to be the projectors themselves:
If the initial state is pure, and the projectors have rank 1, they can be written as projectors onto the vectors and , respectively. The formula simplifies thus to
We can define a linear, trace-preserving, completely positive map, by summing over all the possible post-measurement states of a POVM without the normalisation:
It is an example of a quantum channel, and can be interpreted as expressing how a quantum state changes if a measurement is performed but the result of that measurement is lost. Channels defined by projective measurements can never decrease the von Neumann entropy; they leave the entropy unchanged only if they do not change the density matrix. A quantum channel will increase or leave constant the von Neumann entropy of every input state if and only if the channel is unital, i.e., if it leaves fixed the maximally mixed state. An example of a channel that decreases the von Neumann entropy is the amplitude damping channel for a qubit, which sends all mixed states towards a pure state.
Thermodynamic meaning The quantum version of the canonical distribution, the Gibbs states, are found by maximizing the von Neumann entropy under the constraint that the expected value of the Hamiltonian is fixed. A Gibbs state is a density operator with the same eigenvectors as the Hamiltonian, and its eigenvalues are
where T is the temperature, is the Boltzmann constant, and Z is the partition function. The von Neumann entropy of a Gibbs state is, up to a factor , the thermodynamic entropy.
Generalizations and derived quantities Conditional entropy Let be a joint state for the bipartite quantum system AB. Then the conditional von Neumann entropy is the difference between the entropy of and the entropy of the marginal state for subsystem B alone:
This is bounded above by . In other words, conditioning the description of subsystem A upon subsystem B cannot increase the entropy associated with A.
Quantum mutual information can be defined as the difference between the entropy of the joint state and the total entropy of the marginals:
which can also be expressed in terms of conditional entropy:
Relative entropy
Let and be two density operators in the same state space. The relative entropy is defined to be
The relative entropy is always greater than or equal to zero; it equals zero if and only if . Unlike the von Neumann entropy itself, the relative entropy is monotonic, in that it decreases (or remains constant) when part of a system is traced over:
Entanglement measures Just as energy is a resource that facilitates mechanical operations, entanglement is a resource that facilitates performing tasks that involve communication and computation. The mathematical definition of entanglement can be paraphrased as saying that maximal knowledge about the whole of a system does not imply maximal knowledge about the individual parts of that system. If the quantum state that describes a pair of particles is entangled, then the results of measurements upon one half of the pair can be strongly correlated with the results of measurements upon the other. However, entanglement is not the same as "correlation" as understood in classical probability theory and in daily life. Instead, entanglement can be thought of as potential correlation that can be used to generate actual correlation in an appropriate experiment. The state of a composite system is always expressible as a sum, or superposition, of products of states of local constituents; it is entangled if this sum cannot be written as a single product term. Entropy provides one tool that can be used to quantify entanglement. If the overall system is described by a pure state, the entropy of one subsystem can be used to measure its degree of entanglement with the other subsystems. For bipartite pure states, the von Neumann entropy of reduced states is the unique measure of entanglement in the sense that it is the only function on the family of states that satisfies certain axioms required of an entanglement measure. It is thus known as the entanglement entropy.
It is a classical result that the Shannon entropy achieves its maximum at, and only at, the uniform probability distribution . Therefore, a bipartite pure state is said to be a maximally entangled state if the reduced state of each subsystem of is the diagonal matrix