The following subsections use the formalism and theoretical framework developed in the articles
bra–ket notation and
mathematical formulation of quantum mechanics.
Pure states Consider two arbitrary quantum systems and , with respective
Hilbert spaces and . The Hilbert space of the composite system is the
tensor product : H_A \otimes H_B. If the first system is in state | \psi \rangle_A and the second in state | \phi \rangle_B, the state of the composite system is : |\psi\rangle_A \otimes |\phi\rangle_B. States of the composite system that can be represented in this form are called separable states, or
product states. However, not all states of the composite system are separable. Fix a
basis \{|i \rangle_A\} for and a basis \{|j \rangle_B\} for . The most general state in is of the form : |\psi\rangle_{AB} = \sum_{i,j} c_{ij} |i\rangle_A \otimes |j\rangle_B. This state is separable if there exist vectors [c^A_i], [c^B_j] so that c_{ij}= c^A_i c^B_j, yielding |\psi\rangle_A = \sum_{i} c^A_{i} |i\rangle_A and |\phi\rangle_B = \sum_{j} c^B_{j} |j\rangle_B. It is inseparable if for any vectors [c^A_i],[c^B_j] at least for one pair of coordinates c^A_i,c^B_j we have c_{ij} \neq c^A_ic^B_j. If a state is inseparable, it is called an 'entangled state'. : \rho = \sum_i w_i \rho_i^A \otimes \rho_i^B, where the are positively valued probabilities and the \rho_i^As and \rho_i^Bs are themselves mixed states (density operators) on the subsystems and respectively. In other words, a state is separable if it is a probability distribution over uncorrelated states, or product states. By writing the density matrices as sums of pure ensembles and expanding, we may assume without loss of generality that \rho_i^A and \rho_i^B are themselves pure ensembles. A state is then said to be entangled if it is not separable. In general, finding out whether or not a mixed state is entangled is considered difficult. The general bipartite case has been shown to be
NP-hard. For the and cases, a necessary and sufficient criterion for separability is given by the famous
Positive Partial Transpose (PPT) condition.
Reduced density matrices The idea of a reduced density matrix was introduced by
Paul Dirac in 1930. Consider as above systems and each with a Hilbert space . Let the state of the composite system be : |\Psi \rangle \in H_A \otimes H_B. As indicated above, in general there is no way to associate a pure state to the component system . However, it still is possible to associate a density matrix. Let : \rho_T = |\Psi\rangle \; \langle\Psi|. which is the
projection operator onto this state. The state of is the
partial trace of over the basis of system : : \rho_A \ \stackrel{\mathrm{def}}{=}\ \sum_j^{N_B} \left( I_A \otimes \langle j|_B \right) \left( |\Psi\rangle \langle\Psi| \right)\left( I_A \otimes |j\rangle_B \right) = \hbox{Tr}_B \; \rho_T. The sum occurs over N_B := \dim(H_B) and I_A the identity operator in H_A. is sometimes called the reduced density matrix of on subsystem . Colloquially, we "trace out" or "trace over" system to obtain the reduced density matrix on . The setting in which this perspective is most evident is that of "distant labs", i.e., two quantum systems labelled "A" and "B" on each of which arbitrary
quantum operations can be performed, but which do not interact with each other quantum mechanically. The only interaction allowed is the exchange of classical information, which combined with the most general local quantum operations gives rise to the class of operations called
LOCC (local operations and classical communication). These operations do not allow the production of entangled states between systems A and B. But if A and B are provided with a supply of entangled states, then these, together with LOCC operations can enable a larger class of transformations. If Alice and Bob share an entangled state, Alice can tell Bob over a telephone call how to reproduce a quantum state |\Psi\rangle she has in her lab. Alice performs a joint measurement on |\Psi\rangle together with her half of the entangled state and tells Bob the results. Using Alice's results Bob operates on his half of the entangled state to make it equal to |\Psi\rangle. Since Alice's measurement necessarily erases the quantum state of the system in her lab, the state |\Psi\rangle is not copied, but transferred: it is said to be "
teleported" to Bob's laboratory through this protocol.
Entanglement swapping is variant of teleportation that allows two parties that have never interacted to share an entangled state. The swapping protocol begins with two EPR sources. One source emits an entangled pair of particles A and B, while the other emits a second entangled pair of particles C and D. Particles B and C are subjected to a measurement in the basis of Bell states. The state of the remaining particles, A and D, collapses to a Bell state, leaving them entangled despite never having interacted with each other. An interaction between a qubit of A and a qubit of B can be realized by first teleporting A's qubit to B, then letting it interact with B's qubit (which is now a LOCC operation, since both qubits are in B's lab) and then teleporting the qubit back to A. Two maximally entangled states of two qubits are used up in this process. Thus entangled states are a resource that enables the realization of quantum interactions (or of quantum channels) in a setting where only LOCC are available, but they are consumed in the process. There are other applications where entanglement can be seen as a resource, e.g., private communication or distinguishing quantum states.
Classification of entanglement Not all quantum states are equally valuable as a resource. One method to quantify this value is to use an
entanglement measure that assigns a numerical value to each quantum state. However, it is often interesting to settle for a coarser way to compare quantum states. This gives rise to different classification schemes. Most entanglement classes are defined based on whether states can be converted to other states using LOCC or a subclass of these operations. The smaller the set of allowed operations, the finer the classification. Important examples are: • If two states can be transformed into each other by a local unitary operation, they are said to be in the same
LU class. This is the finest of the usually considered classes. Two states in the same LU class have the same value for entanglement measures and the same value as a resource in the distant-labs setting. There is an infinite number of different LU classes (even in the simplest case of two qubits in a pure state). • If two states can be transformed into each other by local operations including measurements with probability larger than 0, they are said to be in the same 'SLOCC class' ("stochastic LOCC"). Qualitatively, two states \rho_1 and \rho_2 in the same SLOCC class are equally powerful, since one can transform each into the other, but since the transformations \rho_1\to\rho_2 and \rho_2\to\rho_1 may succeed with different probability, they are no longer equally valuable. E.g., for two pure qubits there are only two SLOCC classes: the entangled states (which contains both the (maximally entangled) Bell states and weakly entangled states like |00\rangle+0.01|11\rangle) and the separable ones (i.e., product states like |00\rangle). • Instead of considering transformations of single copies of a state (like \rho_1\to\rho_2) one can define classes based on the possibility of multi-copy transformations. E.g., there are examples when \rho_1\to\rho_2 is impossible by LOCC, but \rho_1\otimes\rho_1\to\rho_2 is possible. A very important (and very coarse) classification is based on the property whether it is possible to transform an arbitrarily large number of copies of a state \rho into at least one pure entangled state. States that have this property are called
distillable. These states are the most useful quantum states since, given enough of them, they can be transformed (with local operations) into any entangled state and hence allow for all possible uses. It came initially as a surprise that not all entangled states are distillable; those that are not are called "
bound entangled".
Entropy In this section, the entropy of a mixed state is discussed as well as how it can be viewed as a measure of quantum entanglement.
Definition In classical
information theory , the
Shannon entropy, is associated to a probability distribution, p_1, \cdots, p_n, in the following way: : H(p_1, \cdots, p_n ) = - \sum_i p_i \log_2 p_i. Since a mixed state is a probability distribution over an ensemble, this leads naturally to the definition of the
von Neumann entropy: If the overall system is pure, 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 a classical result that the Shannon entropy achieves its maximum at, and only at, the uniform probability distribution . : \begin{bmatrix} \frac{1}{n}& & \\ & \ddots & \\ & & \frac{1}{n}\end{bmatrix}. For mixed states, the reduced von Neumann entropy is not the only reasonable entanglement measure.
Entanglement measures Entanglement measures quantify the amount of entanglement in a (often viewed as a bipartite) quantum state. As aforementioned,
entanglement entropy is the standard measure of entanglement for pure states (but no longer a measure of entanglement for mixed states). For mixed states, there are some entanglement measures in the literature
Quantum field theory The
Reeh–Schlieder theorem of
quantum field theory is sometimes interpreted as saying that entanglement is omnipresent in the
quantum vacuum. == Applications ==