separable state

Consider first composite states with two degrees of freedom, referred to as bipartite states. By a postulate of quantum mechanics these can be described as vectors in the tensor product space $H\_1\backslash otimes\; H\_2$. In this discussion we will focus on the case of the Hilbert spaces $H\_1$ and $H\_2$ being finite-dimensional.

Let $\backslash \{|\{a\_i\}\backslash rangle\backslash \}\_\{i=1\}^n\backslash subset\; H\_1$ and $\backslash \{|\{b\_j\}\backslash rangle\backslash \}\_\{j=1\}^m\; \backslash subset\; H\_2$ be orthonormal bases for $H\_1$ and $H\_2$, respectively. A basis for $H\_1\; \backslash otimes\; H\_2$ is then $\backslash \{|\{a\_i\}\backslash rangle\backslash otimes\; |\{b\_j\}\backslash rangle\backslash \}$, or in more compact notation $\backslash \{|a\_i\; b\_j\; \backslash rangle\backslash \}$. From the very definition of the tensor product, any vector of norm 1, i.e. a pure state of the composite system, can be written as

: $$

|\psi\rangle = \sum_{i,j} c_{i,j} (| a_i \rangle \otimes | b_j \rangle) =\sum_{i,j} c_{i,j} | a_i b_j \rangle,

where $c\_\{i,j\}$ is a constant.

If $|\backslash psi\backslash rangle$ can be written as a simple tensor, that is, in the form $|\backslash psi\backslash rangle\; =\; |\backslash psi\_1\backslash rangle\; \backslash otimes\; |\backslash psi\_2\backslash rangle$ with $|\backslash psi\; \_i\; \backslash rangle$ a pure state in the i-th space, it is said to be a product state, and, in particular, separable. Otherwise it is called entangled. Note that, even though the notions of product and separable states coincide for pure states, they do not in the more general case of mixed states.

Pure states are entangled if and only if their partial states are not pure. To see this, write the Schmidt decomposition of $|\backslash psi\backslash rangle$ as

:$|\backslash psi\backslash rangle=\backslash sum\_\{k=1\}^\{r\_\backslash psi\}\; \backslash sqrt\{p\_k\}\; (|u\_k\backslash rangle\backslash otimes|v\_k\backslash rangle),$

where $\backslash sqrt\{p\_k\}>0$ are positive real numbers, $r\_\backslash psi$ is the Schmidt rank of $|\backslash psi\backslash rangle$, and $\backslash \{|u\_k\backslash rangle\backslash \}\_\{k=1\}^\{r\_\backslash psi\}\backslash subset\; H\_1$ and $\backslash \{|v\_k\backslash rangle\backslash \}\_\{k=1\}^\{r\_\backslash psi\}\backslash subset\; H\_2$ are sets of orthonormal states in $H\_1$ and $H\_2$, respectively.

The state $|\backslash psi\backslash rangle$ is entangled if and only if $r\_\backslash psi>1$. At the same time, the partial state has the form

:$\backslash rho\_A\backslash equiv\; \backslash operatorname\{Tr\}\_B(|\backslash psi\backslash rangle\backslash !\backslash langle\backslash psi|)\; =\; \backslash sum\_\{k=1\}^\{r\_\backslash psi\}\; p\_k\; \backslash ,\; |u\_k\backslash rangle\backslash !\backslash langle\; u\_k|.$

It follows that $\backslash rho\_A$ is pure --- that is, is projection with unit-rank --- if and only if $r\_\backslash psi=1$, which is equivalent to $|\backslash psi\backslash rangle$ being separable.

Physically, this means that it is not possible to assign a definite (pure) state to the subsystems, which instead ought to be described as statistical ensembles of pure states, that is, as density matrices. A pure state $\backslash rho=|\backslash psi\backslash rangle\backslash !\backslash langle\backslash psi|$ is thus entangled if and only if the von Neumann entropy of the partial state $\backslash rho\_A\backslash equiv\backslash operatorname\{Tr\}\_B(\backslash rho)$ is nonzero.

Formally, the embedding of a product of states into the product space is given by the Segre embedding.{{Cite journal |last1=Gharahi |first1=Masoud |last2=Mancini |first2=Stefano |last3=Ottaviani |first3=Giorgio |date=2020-10-01 |title=Fine-structure classification of multiqubit entanglement by algebraic geometry |journal=Physical Review Research |volume=2 |issue=4 |pages=043003 |doi=10.1103/PhysRevResearch.2.043003|arxiv=1910.09665 |bibcode=2020PhRvR...2d3003G |s2cid=204824024 |doi-access=free }} That is, a quantum-mechanical pure state is separable if and only if it is in the image of the Segre embedding.

For example, in a two-qubit space, where $H\_1=H\_2=\backslash mathbb\{C\}^2$, the states $|0\backslash rangle\backslash otimes|0\backslash rangle$, $|0\backslash rangle\backslash otimes|1\backslash rangle$, $|1\backslash rangle\backslash otimes|1\backslash rangle$, are all product (and thus separable) pure states, as is $|0\backslash rangle\backslash otimes|\backslash psi\backslash rangle$ with $|\backslash psi\backslash rangle\backslash equiv\backslash sqrt\{1/3\}|0\backslash rangle+\backslash sqrt\{2/3\}|1\backslash rangle$. On the other hand, states like $\backslash sqrt\{1/2\}|00\backslash rangle+\backslash sqrt\{1/2\}|11\backslash rangle$ or $\backslash sqrt\{1/3\}|01\backslash rangle+\backslash sqrt\{2/3\}|10\backslash rangle$ are not separable.

Consider the mixed state case. A mixed state of the composite system is described by a density matrix $\backslash rho$ acting on $H\_1\; \backslash otimes\; H\_2$. Such a state $\backslash rho$ is separable if there exist $p\_k\backslash geq\; 0$, $\backslash \{\; \backslash rho\_1^k\; \backslash \}$ and $\backslash \{\; \backslash rho\_2^k\; \backslash \}$ which are mixed states of the respective subsystems such that

:$$

\rho=\sum_k p_k \rho_1^k \otimes \rho_2^k

where

:$\backslash ;$

\sum_k p_k = 1.

Otherwise $\backslash rho$ is called an entangled state. We can assume without loss of generality in the above expression that $\backslash \{\; \backslash rho\_1^k\; \backslash \}$ and $\backslash \{\; \backslash rho\_2^k\; \backslash \}$ are all rank-1 projections, that is, they represent pure ensembles of the appropriate subsystems. It is clear from the definition that the family of separable states is a convex set.

Notice that, again from the definition of the tensor product, any density matrix, indeed any matrix acting on the composite state space, can be trivially written in the desired form, if we drop the requirement that $\backslash \{\; \backslash rho\_1^k\; \backslash \}$ and $\backslash \{\; \backslash rho\_2^k\; \backslash \}$ are themselves states and $\backslash ;\; \backslash sum\_k\; p\_k\; =\; 1.$ If these requirements are satisfied, then we can interpret the total state as a probability distribution over uncorrelated product states.

In terms of quantum channels, a separable state can be created from any other state using local actions and classical communication while an entangled state cannot.

When the state spaces are infinite-dimensional, density matrices are replaced by positive trace class operators with trace 1, and a state is separable if it can be approximated, in trace norm, by states of the above form.

If there is only a single non-zero $p\_k$, then the state can be expressed just as $\backslash rho\; =\; \backslash rho\_1\; \backslash otimes\; \backslash rho\_2\; ,$ and is called simply separable or product state. One property of the product state is that in terms of entropy,

: $S(\backslash rho)\; =\; S(\backslash rho\_1)\; +\; S(\backslash rho\_2).$

The above discussion generalizes easily to the case of a quantum system consisting of more than two subsystems. Let a system have n subsystems and have state space $H\; =\; H\_1\; \backslash otimes\; \backslash cdots\; \backslash otimes\; H\_n$. A pure state $|\; \backslash psi\; \backslash rangle\; \backslash in\; H$ is separable if it takes the form

:$|\; \backslash psi\; \backslash rangle\; =\; |\; \backslash psi\_1\; \backslash rangle\; \backslash otimes\; \backslash cdots\; \backslash otimes\; |\; \backslash psi\_n\; \backslash rangle\; .$

Similarly, a mixed state ρ acting on H is separable if it is a convex sum

:$\backslash rho\; =\; \backslash sum\_k\; p\_k\; \backslash rho\_1\; ^k\; \backslash otimes\; \backslash cdots\; \backslash otimes\; \backslash rho\_n\; ^k.$

Or, in the infinite-dimensional case, ρ is separable if it can be approximated in the trace norm by states of the above form.

The problem of deciding whether a state is separable in general is sometimes called the separability problem{{anchor|quantum separability problem}} in quantum information theory. It is considered to be a difficult problem. It has been shown to be NP-hard in many cases Gurvits, L., Classical deterministic complexity of Edmonds’ problem and quantum entanglement, in Proceedings of the 35th ACM Symposium on Theory of Computing, ACM Press, New York, 2003.Sevag Gharibian, Strong NP-Hardness of the Quantum Separability Problem, Quantum Information and Computation, Vol. 10, No. 3&4, pp. 343-360, 2010. arXiv:0810.4507. and is believed to be so in general. Some appreciation for this difficulty can be obtained if one attempts to solve the problem by employing the direct brute force approach, for a fixed dimension. The problem quickly becomes intractable, even for low dimensions. Thus more sophisticated formulations are required. The separability problem is a subject of current research.

A separability criterion is a necessary condition a state must satisfy to be separable. In the low-dimensional (2 X 2 and 2 X 3) cases, the Peres-Horodecki criterion is actually a necessary and sufficient condition for separability. Other separability criteria include (but not limited to) the range criterion, reduction criterion, and those based on uncertainty relations.{{cite journal |last1=Hofmann |first1=Holger F. |last2=Takeuchi |first2=Shigeki |title=Violation of local uncertainty relations as a signature of entanglement |journal=Physical Review A |date=22 September 2003 |volume=68 |issue=3 |page=032103 |doi=10.1103/PhysRevA.68.032103|arxiv=quant-ph/0212090 |bibcode=2003PhRvA..68c2103H |s2cid=54893300 }}{{cite journal |last1=Gühne |first1=Otfried |title=Characterizing Entanglement via Uncertainty Relations |journal=Physical Review Letters |date=18 March 2004 |volume=92 |issue=11 |page=117903 |doi=10.1103/PhysRevLett.92.117903|arxiv=quant-ph/0306194 |pmid=15089173 |bibcode=2004PhRvL..92k7903G |s2cid=5696147 }}{{cite journal |last1=Gühne |first1=Otfried |last2=Lewenstein |first2=Maciej |title=Entropic uncertainty relations and entanglement |journal=Physical Review A |date=24 August 2004 |volume=70 |issue=2 |page=022316 |doi=10.1103/PhysRevA.70.022316|arxiv=quant-ph/0403219 |bibcode=2004PhRvA..70b2316G |s2cid=118952931 }}{{cite journal |last1=Huang |first1=Yichen |title=Entanglement criteria via concave-function uncertainty relations |journal=Physical Review A |date=29 July 2010 |volume=82 |issue=1 |page=012335 |doi=10.1103/PhysRevA.82.012335 |bibcode=2010PhRvA..82a2335H}} See Ref.{{cite journal|last1=Gühne|first1=Otfried|last2=Tóth|first2=Géza|title=Entanglement detection|journal=Physics Reports|volume=474|issue=1–6|pages=1–75|doi=10.1016/j.physrep.2009.02.004|arxiv = 0811.2803 |bibcode = 2009PhR...474....1G|year=2009 |s2cid=119288569 }} for a review of separability criteria in discrete variable systems.

In continuous variable systems, the Peres-Horodecki criterion also applies. Specifically, Simon {{cite journal|last1=Simon|first1=R.|title=Peres-Horodecki Separability Criterion for Continuous Variable Systems|journal=Physical Review Letters|volume=84|issue=12|pages=2726–2729|doi=10.1103/PhysRevLett.84.2726|arxiv = quant-ph/9909044 |bibcode = 2000PhRvL..84.2726S|pmid=11017310|year=2000|s2cid=11664720 }} formulated a particular version of the Peres-Horodecki criterion in terms of the second-order moments of canonical operators and showed that it is necessary and sufficient for $1\backslash oplus1$-mode Gaussian states (see Ref.{{cite journal|last1=Duan|first1=Lu-Ming|last2=Giedke|first2=G.|last3=Cirac|first3=J. I.|last4=Zoller|first4=P.|title=Inseparability Criterion for Continuous Variable Systems|journal=Physical Review Letters|volume=84|issue=12|pages=2722–2725|doi=10.1103/PhysRevLett.84.2722|arxiv = quant-ph/9908056 |bibcode = 2000PhRvL..84.2722D|pmid=11017309|year=2000 |s2cid=9948874 }} for a seemingly different but essentially equivalent approach). It was later found {{cite journal|last1=Werner|first1=R. F.|last2=Wolf|first2=M. M.|title=Bound Entangled Gaussian States|journal=Physical Review Letters|volume=86|issue=16|pages=3658–3661|doi=10.1103/PhysRevLett.86.3658|arxiv = quant-ph/0009118 |bibcode = 2001PhRvL..86.3658W|pmid=11328047|year=2001 |s2cid=20897950 }} that Simon's condition is also necessary and sufficient for $1\backslash oplus\; n$-mode Gaussian states, but no longer sufficient for $2\backslash oplus2$-mode Gaussian states. Simon's condition can be generalized by taking into account the higher order moments of canonical operators {{cite journal|last1=Shchukin|first1=E.|last2=Vogel|first2=W.|title=Inseparability Criteria for Continuous Bipartite Quantum States|journal=Physical Review Letters|volume=95|issue=23|doi=10.1103/PhysRevLett.95.230502|arxiv = quant-ph/0508132 |bibcode = 2005PhRvL..95w0502S|pmid=16384285|year=2005|page=230502 |s2cid=28595936 }}{{cite journal|last1=Hillery|first1=Mark|last2=Zubairy|first2=M.Suhail|title=Entanglement Conditions for Two-Mode States|journal=Physical Review Letters|volume=96|issue=5|doi=10.1103/PhysRevLett.96.050503|arxiv = quant-ph/0507168 |bibcode = 2006PhRvL..96e0503H|pmid=16486912|page=050503|year=2006|s2cid=43756465 }} or by using entropic measures.{{cite journal|last1=Walborn|first1=S.|last2=Taketani|first2=B.|last3=Salles|first3=A.|last4=Toscano|first4=F.|last5=de Matos Filho|first5=R.|title=Entropic Entanglement Criteria for Continuous Variables|journal=Physical Review Letters|volume=103|issue=16|doi=10.1103/PhysRevLett.103.160505|arxiv = 0909.0147 |bibcode = 2009PhRvL.103p0505W|pmid=19905682|page=160505|year=2009|s2cid=10523704 }}{{cite journal |last1=Yichen Huang |title=Entanglement Detection: Complexity and Shannon Entropic Criteria |journal=IEEE Transactions on Information Theory |date=October 2013 |volume=59 |issue=10 |pages=6774–6778 |doi=10.1109/TIT.2013.2257936|s2cid=7149863 }}

Quantum mechanics may be modelled on a projective Hilbert space, and the categorical product of two such spaces is the Segre embedding. In the bipartite case, a quantum state is separable if and only if it lies in the image of the Segre embedding.

Jon Magne Leinaas, Jan Myrheim and Eirik Ovrum in their paper "Geometrical aspects of entanglement"{{Cite journal |last1=Leinaas |first1=Jon Magne |last2=Myrheim |first2=Jan |last3=Ovrum |first3=Eirik |date=2006-07-19 |title=Geometrical aspects of entanglement |url=https://link.aps.org/doi/10.1103/PhysRevA.74.012313 |journal=Physical Review A |language=en |volume=74 |issue=1 |page=012313 |doi=10.1103/PhysRevA.74.012313 |issn=1050-2947|arxiv=quant-ph/0605079 |bibcode=2006PhRvA..74a2313L }} describe the problem and study the geometry of the separable states as a subset of the general state matrices. This subset have some intersection with the subset of states holding Peres-Horodecki criterion. In this paper, Leinaas et al. also give a numerical approach to test for separability in the general case.

Testing for separability in the general case is an NP-hard problem. Leinaas et al. formulated an iterative, probabilistic algorithm for testing if a given state is separable. When the algorithm is successful, it gives an explicit, random, representation of the given state as a separable state. Otherwise it gives the distance of the given state from the nearest separable state it can find.

• Entanglement witness

{{reflist}}

• [https://phweb.technion.ac.il/~stateseparator/index.html "StateSeparator" web-app]

Category:Quantum information science

Category:Quantum states