Separability (quantum mechanics)

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

Separability for pure states

For the sake of simplicity, in the following all spaces are assumed to be finite-dimensional. First we consider pure states .

Separability is a property of composite quantum systems, that is, in the simplest (“bipartite”) case, an overall system 12 consisting of subsystems 1 and 2 . The quantum mechanical state spaces of the subsystems are the Hilbert spaces and with the respective orthonormal basis vectors and . The Hilbert space of the composite system is then the tensor product ${\displaystyle H_{1}}$ ${\displaystyle H_{2}}$ ${\displaystyle \{|{a_{i}}\rangle \}_{i=1}^{n}}$ ${\displaystyle \{|{b_{j}}\rangle \}_{j=1}^{m}}$

${\displaystyle H_{12}=H_{1}\otimes H_{2},}$

with the base , or in more compact notation . Any vector in (i.e., any pure state of system 12 ) can be written as . ${\displaystyle \{|{a_{i}}\rangle \otimes |{b_{j}}\rangle \}}$ ${\displaystyle \{|a_{i}b_{j}\rangle \}}$ ${\displaystyle H_{12}}$ ${\displaystyle |\psi \rangle =\Sigma _{i,j}c_{i,j}|a_{i}\rangle \otimes |b_{j}\rangle =\Sigma _{i,j}c_{i,j}|a_{i}b_{j}\rangle }$

If a pure state can be written in the form (where is a pure state of the subsystem ), it is called separable or product state . Otherwise the state is called entangled . ${\displaystyle |\psi \rangle \in H_{1}\otimes H_{2}}$ ${\displaystyle |\psi \rangle =|\psi _{1}\rangle \otimes |\psi _{2}\rangle }$ ${\displaystyle |\psi _{i}\rangle }$ ${\displaystyle i}$

Standard examples for a separable and an entangled state vector in are ${\displaystyle H_{12}=\mathbb {C} ^{2}\otimes \mathbb {C} ^{2}}$

${\displaystyle |00\rangle \doteq {\begin{pmatrix}1\\0\\0\\0\end{pmatrix}}}$       or.       ${\displaystyle (|00\rangle +|11\rangle )/{\sqrt {2}}\doteq {\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\0\\0\\1\end{pmatrix}},}$

whereby, as usual, is to be read as: "is represented by". ${\displaystyle \doteq }$

One sees,

• that in a purely separable state, each subsystem can be assigned its “own” state.
• that every pure separable state can be generated from every other state (e.g. off ) by local quantum-mechanical operations . ${\displaystyle |00\rangle }$

Both are not possible in a locked state. Appropriately generalized, this distinction can also be transferred to the case of mixed states.

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

Separability for 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 . ${\displaystyle \rho }$ ${\displaystyle H_{12}=H_{1}\otimes H_{2}}$

${\displaystyle \rho }$is separable when it with and states on and on are (each mixed states of the subsystems described), so that ${\displaystyle p_{k}\geq 0}$ ${\displaystyle p_{1}+p_{2}+\dots =1}$ ${\displaystyle \{\rho _{1}^{k}\}}$ ${\displaystyle H_{1}}$ ${\displaystyle \{\rho _{2}^{k}\}}$ ${\displaystyle H_{2}}$

${\displaystyle \rho =\sum _{k}p_{k}\rho _{1}^{k}\otimes \rho _{2}^{k}.}$

Otherwise it means entangled . ${\displaystyle \rho }$

The physical meaning of this mathematical definition is that a separable state can be understood as a mixture of product states . ${\displaystyle \rho _{1}^{k}\otimes \rho _{2}^{k}}$

• On the one hand, this implies that a separable state only describes classic correlations between the subsystems. (Because a product state describes independent (uncorrelated) systems and the correlations are given by the classic probability distribution.) ${\displaystyle p_{k}}$
• On the other hand, it follows that a separable state can be generated from any other state (e.g. off ) by means of local quantum mechanical operations and classical communication . (Using classic communication, both parties select an index according to the probability distribution and then generate (which is locally possible in each case) the product status .) ${\displaystyle |00\rangle }$ ${\displaystyle k}$ ${\displaystyle p_{k}}$ ${\displaystyle \rho _{k}^{1}\otimes \rho _{k}^{2}}$

From the above definition it is clear 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 called separable if it can be approximated as precisely as desired (in the track standard ) by states of the above form.

Separability for multi-party systems

The preceding discussion can easily be generalized for quantum systems consisting of many subsystems. If the system consists of sub-systems with a system Hilbert space , then a pure state is separable (more precisely: completely separable) if and only if it is from the form ${\displaystyle n}$ ${\displaystyle H_{i},i=1,\dots ,n}$ ${\displaystyle H_{1,\dots ,n}=H_{1}\otimes H_{2}\otimes \dots \otimes H_{n}}$

${\displaystyle |\psi \rangle =|\psi _{1}\rangle \otimes \cdots \otimes |\psi _{n}\rangle }$

is. Analogously, a mixed state is separable if it can be written as a convex sum of product states : ${\displaystyle \rho }$ ${\displaystyle H_{1..n}}$

${\displaystyle \rho =\sum _{k}p_{k}\rho _{1}^{k}\otimes \cdots \rho _{n}^{k}}$.

Separability Criteria

Conditions that can be easily checked and that meet all separable states are also referred to as separability criteria (necessary conditions for separability). Your injury for a given state can then be understood as evidence that the state is inseparable, i.e. entangled. The distinction between separable and entangled states is of great interest in quantum information theory, since only entangled states have quantum correlations and represent an important resource that enables methods such as quantum teleportation or quantum error correction .

A pure state on exactly then separable if it is a product state. This can be checked on the basis of the reduced state in one of the two subsystems: for pure separable states the reduced state is also pure, that is, its Von Neumann entropy disappears. That is, a pure state is separable if and only if or is (both equations are equivalent via the Schmidt decomposition ). ${\displaystyle \rho _{12}}$ ${\displaystyle H_{1}\otimes H_{2}}$ ${\displaystyle S}$ ${\displaystyle \rho _{12}}$ ${\displaystyle S(\rho _{1})=0}$ ${\displaystyle S(\rho _{2})=0}$

The question of whether a given mixed state is separable ( separability problem ) is generally difficult to answer ( NP severity ). The common separability criteria are easy to check, but only partially solve the problem, that is, they cannot decide for all states whether they are entangled. ${\displaystyle \rho _{12}}$

Examples of such criteria are the fulfillment of a Bell's inequality or the Peres-Horodecki criterion , which states that the density matrix of a separable state remains positive under partial transposition. More generally, it can be formulated that the density matrix of a separable state must remain positive in one of the subsystems using every positive mapping : ${\displaystyle T}$

${\displaystyle (1\otimes T)\rho \geq 0}$.

In general (i.e. for not necessarily separable states) this only applies to completely positive maps . The validity of the above inequality for all positive maps is necessary and sufficient for separability. ${\displaystyle T}$ ${\displaystyle T}$

Other Separabilitätskriterien arising from the so-called entanglement witnesses ( entanglement witnesses ) or entanglement dimensions .

A general algorithm for solving the separability problem was presented in 2011. He uses semidefinite programming to decide whether the given state has a symmetrical extension to N systems, that is, whether there is a state for all N such that the reduced state on systems 1 "and" j "is the same for all j the state is. All separable states have for all N such a symmetrical extension. for each entangled state, there is a N such that no such extension exists. ${\displaystyle \rho _{12}}$ ${\displaystyle \rho _{123\dots N}}$ ${\displaystyle \rho _{1j}}$ ${\displaystyle \rho _{12}}$

literature

• Gernot Alber and M. Freyberger: Quantum Correlations and Bell's Inequalities . In: Physical sheets . tape 55 , no. 10 , 1999, p. 24 , doi : 10.1002 / phbl.19990551006 .
• Asher Peres : Quantum Theory: Concepts and Methods . Kluwer Academic, 1995, ISBN 0-7923-3632-1 ( springer.com ).
• Jürgen Audretsch: Entangled world. Fascination with the quanta . Wiley-VCH, 2002.
• Eckert et al .: Entanglement Properties of Composite Quantum Systems . In: Quantum Information Processing . Th.Beth and G. Leuchs (eds.), Wiley-VCH, 2003.
• R. Horodecki, P. Horodecki, M. Horodecki & K. Horodecki: Quantum entanglement . In: Rev. Mod. Phys. tape 81 , 2009, p. 865-942 , doi : 10.1103 / RevModPhys.81.865 , arxiv : quant-ph / 0702225 .

Web links

• M. Lewenstein et al .: Separability and distillability in composite quantum systems - a primer . J. Mod. Opt. 47, 2481 (2000). arxiv : quant-ph / 0006064
• M., P. and R. Horodecki: Mixed-State Entanglement and Quantum Communication . In: G. Alber et al. (Ed.): Quantum Information, Springer, 2001, arxiv : quant-ph / 0109124
• Measurement of separable and entangled photonic states

Individual evidence

1. ↑ L. Gurvits: Classical complexity and quantum entanglement . In: J. Comput. Syst. Sci. tape 69 , 2004, pp. 448-484 , doi : 10.1016 / j.jcss.2004.06.003 , arxiv : quant-ph / 0201022 .
2. ↑ A matrix in which the transposition is only formed with respect to one of the two subsystems is called the partial transposition of a matrix on . Let and be orthonormal bases of or and be the matrix elements in the basis , then for the matrix partially transposed with respect to , that . The linear mapping is often referred to as partial transposition . is an example of a "positive but not entirely positive" figure. (cf. e.g. Horodecki et al. Phys. Lett. A 223 , 1 (1996)) ${\displaystyle M}$ ${\displaystyle H_{1}\otimes H_{2}}$ ${\displaystyle H_{1},H_{2}}$ ${\displaystyle \{e_{i}\}}$ ${\displaystyle \{f_{i}\}}$ ${\displaystyle H_{1}}$ ${\displaystyle H_{2}}$ ${\displaystyle M_{ij,kl}}$ ${\displaystyle \{e_{i}\otimes f_{j}\}}$ ${\displaystyle H_{1}}$ ${\displaystyle M^{T_{1}}}$ ${\displaystyle (M^{T_{1}})_{ij,kl}=M_{kj,il}}$ ${\displaystyle T_{1}\colon M\to M^{T_{1}}}$ ${\displaystyle T_{1}}$
3. ↑ Michał Horodecki, Paweł Horodecki, Ryszard Horodecki: Separability of mixed states: necessary and sufficient conditions . In: Physics Letters A . tape 223 , 1996, pp. 1-8 , doi : 10.1016 / S0375-9601 (96) 00706-2 , arxiv : quant-ph / 9605038 .
4. ^ FGL Brandao, M. Christandl, J. Yard: A quasipolynomial-time algorithm for the quantum separability problem . In: ACM (Ed.): Proceedings of the 43rd annual ACM symposium on Theory of Computing . 2011, p. 343-352 , doi : 10.1145 / 1993636.1993683 , arxiv : 1011.2751 .
5. ↑ AC Doherty, PA Parrilo, FM Spedalieri: A complete family of separability criteria . In: Phys. Rev. A . tape 69 , 2004, pp. 022308 , doi : 10.1103 / PhysRevA.69.022308 , arxiv : quant-ph / 0308032 .