Separable state

In quantum mechanics is defined as the state of a composite system as separable if it is not crossed, that is, if it can be written as a mixture of product states.

Separability for pure states

For simplicity, all spaces are assumed to finite below. First, we consider pure states.

Separability is a property of composite quantum systems, that is, in the simplest ( " bipartite " ) case, one of the sub-systems 1 and 2 existing overall system 12, the quantum-mechanical state spaces of the subsystems are the Hilbert space, and with the respective orthonormal basis vectors and. The Hilbert space of the composite system is then the tensor product

To the base, or in a more compact notation. Each vector in (ie, each pure state of the system 12) can be written as.

If a pure state can be written in the form ( with a pure state of the subsystem ), it is called separable or product condition. Otherwise, it is called entangled state.

Standard examples of a separable and an entangled state vector are in

Where, as usual, is to be read as: " is represented by ".

It can be seen

• That you can assign to each subsystem "own " state in a pure separable state.
• That every separable pure state ( eg off) can be produced by local quantum mechanically permissible operations from any other state.

Both are not possible in an entangled state. Generalized Fits can be transferred to the case of mixed states this distinction.

The foregoing discussion can be generalized without any significant changes to the case of infinite-dimensional systems.

Separability of mixed states

Now we consider the case of mixed states. A mixed state of the composite quantum system 12 is described by a density matrix, which acts on the Hilbert space.

Is separable when it is on and with conditions and ( each describing the states of the mixed sub-systems ), so that

Otherwise called entangled.

The physical meaning of this mathematical definition is that a separable state can be regarded as a mixture of product states.

• This implies, firstly, that a separable state describes only classical correlations between the subsystems. ( For a product state describes independent ( uncorrelated ) systems and the correlations are given by the classical probability distribution. )
• Second, it follows that a separable state by means of local quantum mechanically allowed operations and classical communication from any other state can ( eg off) produce. (Using classical communication both parties choose an index according to the probability distribution, and then generate (which is always possible locally ) the product state. )

It is clear from the above definition that the separable states form a convex set.

If the state spaces are infinite dimensional, density matrices are replaced by positive trace class operators with trace 1. A state is then called separable if it ( in the trace norm) can be approximated by states of the above form with arbitrary precision.

Separability of multi-party systems

The foregoing discussion can be easily generalized for consisting of many subsystems quantum systems. When the system is sub-systems with system Hilbert space, then a pure state if and separable (more precisely completely separable ) when the form of the

Is. Analog is a mixed state to be separable if it can be written as a convex sum of product states:

Separabilitätskriterien

A pure state is precisely then separable, when the entropy of the reduced states disappears, i.e., when or (both equations are equivalent to the Schmidt decomposition ).

The question whether a given mixed state is separable ( Separabilitätsproblem ), is in general difficult to answer ( NP-hardness ). The distinction between separable and entangled states in quantum information theory is of great interest, because only entangled states quantum correlations exhibit and represent an important resource, the method allows as quantum teleportation.

A Separabilitätskriterium is a (slightly verifiable ) condition satisfying any separable state ( necessary condition for separability ). The breach of such a condition is sufficient for the detection of entanglement. Examples of such criteria are the fulfillment of Bell 's inequality or the Peres - Horodecki criterion, which states that the density matrix of a separable state remains positive under partial transposition. General can be formulated that the density matrix of a separable state of each positive image must remain positive in one of the subsystems using:

In general (ie not necessary for separable states), this is valid only for positive pictures. The validity of the above inequality for all positive images is necessary and sufficient for separability.

Other Separabilitätskriterien result from the so-called entanglement witnesses ( entanglement witnesses ) or from entanglement measure.