LOCC

The formal definition of the set of LOCC operations is complicated due to the fact that later local operations depend in general on all the previous classical communication and due to the unbounded number of communication rounds. For any finite number $r\backslash geq1$ one can define $\backslash operatorname\{LOCC\}\_r$, the set of LOCC operations that can be achieved with $r$ rounds of classical communication. The set becomes strictly larger whenever $r$ is increased and care has to be taken to define the limit of infinitely many rounds. In particular, the set LOCC is not topologically closed, that is there are quantum operations that can be approximated arbitrarily closely by LOCC but that are not themselves LOCC.{{cite journal |author1=Chitambar, E. |author2=Leung, D. |author3=Mancinska, L. |author4=Ozols, M. |author5=Winter, A. |title=Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask) |journal=Commun. Math. Phys. |volume=328 |page=303 |year=2012 |issue=1 |doi=10.1007/s00220-014-1953-9 |arxiv=1210.4583|bibcode=2014CMaPh.328..303C |s2cid=118478457 }}

A one-round LOCC $\backslash operatorname\{LOCC\}\_1$ is a quantum instrument $\backslash left\backslash \{\backslash mathcal\{E\}\_x\backslash right\backslash \}$, for which the trace-non-increasing completely positive maps (CPMs) $\backslash mathcal\{E\}\_x$ are local for all measurement results $x$, i.e., $\backslash mathcal\{E\}\_x\; =\; \backslash bigotimes\_\{j\}(\{\backslash cal\; E\}\_x^j)$ and there is one site $j=K$ such that only at $K$ the map $\backslash mathcal\{E\}\_x^K$ $\backslash mathcal\{E\}\_x\; =\; \backslash bigotimes\_\{j\backslash not=K\}(\{\backslash cal\; T\_j^x\})\backslash otimes\{\backslash cal\; E\}\_K$ is not trace-preserving. This means that the instrument can be realized by the party at site $K$ applying the (local) instrument $\backslash left\backslash \{\backslash mathcal\{E\}\_x^K\backslash right\backslash \}$ and communicating the classical result $x$ to all other parties, which then each perform (conditioned on $x$) trace-preserving (deterministic) local quantum operations $\{\backslash cal\; T\}\_x^j$.

Then $\backslash operatorname\{LOCC\}\_r$ are defined recursively as those operations that can be realized by following up an operation $\backslash operatorname\{LOCC\}\_\{r-1\}$ with a $\backslash operatorname\{LOCC\}\_1$-operation. Here it is allowed that the party which performs the follow-up operations depends on the result of the previous rounds. Moreover, we also allow "coarse-graining", i.e., discarding some of the classical information encoded in the measurement results (of all rounds).

The union of all $\backslash operatorname\{LOCC\}\_r$ operations is denoted by $\backslash operatorname\{LOCC\}\_\{\backslash mathbb\{N\}\}$ and contains instruments that can be approximated better and better with more LOCC rounds. Its topological closure $\backslash overline\{\backslash operatorname\{LOCC\}\}\_\{\backslash mathbb\{N\}\}$ contains all such operations.

It can be shown that all these sets are different:

:$\backslash operatorname\{LOCC\}\_r\backslash subset\; \backslash operatorname\{LOCC\}\_\{r+1\}\backslash subset\; \backslash operatorname\{LOCC\}\_\{\backslash mathbb\{N\}\}\backslash subset\backslash overline\{\backslash operatorname\{LOCC\}\}\_\{\backslash mathbb\{N\}\}$

The set of all LOCC operations is contained in the set $\backslash operatorname\{SEP\}$ of all separable operations. $\backslash operatorname\{SEP\}$ contains all operations that can be written using Kraus operators that have all product form, i.e.,

:$$

{\cal E} (\rho) = \sum_l K^l_1\otimes K^l_2\dots\otimes K_N \rho (K^l_1\otimes K^l_2\dots\otimes K_N)^\dagger,

with $\backslash sum\_l\; K^l\_1\backslash otimes\; K^l\_2\backslash dots\backslash otimes\; K\_N(K^l\_1\backslash otimes\; K^l\_2\backslash dots\backslash otimes\; K\_N)^\backslash dagger=1$. Not all operations in $\backslash operatorname\{SEP\}$ are LOCC,

:$\backslash overline\{\backslash operatorname\{LOCC\}\}\_\{\backslash mathbb\{N\}\}\backslash subset\; \backslash operatorname\{SEP\},$

i.e., there are examples that cannot be implemented locally even with infinite rounds of communication.

LOCC are the "free operations" in the resource theories of entanglement: Entanglement cannot be produced from separable states with LOCC and if the local parties in addition to being able to perform all LOCC operations are also furnished with some entangled states, they can realize more operations than with LOCC alone.

LOCC operations are useful for state preparation, state discrimination, and entanglement transformations.

Alice and Bob are given a quantum system in the product state $|00\backslash rangle\; =\; |0\backslash rangle\_A\backslash otimes\; |0\backslash rangle\_B$. Their task is to produce the separable state $\backslash rho=\backslash frac\{1\}\{2\}|00\backslash rangle\backslash langle00|+\backslash frac\{1\}\{2\}|11\backslash rangle\backslash langle11|$. With local operations alone this cannot be achieved, since they cannot produce the (classical) correlations present in $\backslash rho$. However, with LOCC (with one round of communication) $\backslash rho$ can be prepared: Alice throws an unbiased coin (that shows heads or tails each with 50% probability) and flips her qubit (to $|1\backslash rangle\_A$) if the coin shows "tails", otherwise it is left unchanged. She then sends the result of the coin-flip (classical information) to Bob who also flips his qubit if he receives the message "tails". The resulting state is $\backslash rho$. In general, all separable states (and only these) can be prepared from a product states with LOCC operations alone.

Given two quantum states $\backslash psi$ on a bi- or multipartite Hilbert space $\{\backslash cal\; H\}=\{\backslash cal\; H\}\_A\backslash otimes\{\backslash cal\; H\}\_B\backslash otimes\backslash dots\{\backslash cal\; H\}\_Z$, the task is to determine which one of two (or more) possible states $\backslash psi\_1,\; \backslash psi\_2$ it is. As a simple example, consider the two Bell states

:$$

|\psi_1\rangle = \frac{1}{\sqrt{2}}\left(|0\rangle_A\otimes|0\rangle_B + |1\rangle_A\otimes|1\rangle_B\right)

:$$

|\psi_2\rangle = \frac{1}{\sqrt{2}}\left(|0\rangle_A\otimes|1\rangle_B + |1\rangle_A\otimes|0\rangle_B\right)

Let's say the two-qubit system is separated, where the first qubit is given to Alice and the second is given to Bob. Without communication, Alice and Bob cannot distinguish the two states, since for all local measurements all measurement statistics are exactly the same (both states have the same reduced density matrix). E.g., assume that Alice measures the first qubit, and obtains the result 0. Since this result is equally likely to occur (with probability 50%) in each of the two cases, she does not gain any information on which Bell pair she was given and the same holds for Bob if he performs any measurement. But now let Alice send her result to Bob over a classical channel. Now Bob can compare his result to hers and if they are the same he can conclude that the pair given was $|\backslash psi\_1\backslash rangle$, since only this allows for a joint measurement outcome $|0\backslash rangle\_A\backslash otimes|0\backslash rangle\_B$. Thus with LOCC and two measurements these two states can be distinguished perfectly. Note that with global (nonlocal or entangled) measurements, a single measurement (on the joint Hilbert space) is sufficient to distinguish these two (mutually orthogonal) states.

There are quantum states that cannot be distinguished with LOCC operations.{{cite journal |author=Charles H. Bennett |author2=David P. DiVincenzo |author3=Christopher A. Fuchs |author4=Tal Mor |author5=Eric Rains |author6=Peter W. Shor |author7=John A. Smolin |author8=William K. Wootters |title=Quantum nonlocality without entanglement |journal=Phys. Rev. A |volume=59 |pages=1070–1091 |year=1999 |issue=2 |doi=10.1103/PhysRevA.59.1070 |arxiv= quant-ph/9804053|bibcode=1999PhRvA..59.1070B |s2cid=15282650 }}

While LOCC cannot generate entangled states out of product states, they can be used to transform entangled states into other entangled states. The restriction to LOCC severely limits which transformations are possible.

Nielsen {{cite journal |author=M. A. Nielsen |title=Conditions for a Class of Entanglement Transformations |journal=Phys. Rev. Lett. |volume=83 |pages=436–439 |year=1999 |issue=2 |doi=10.1103/PhysRevLett.83.436 |arxiv=quant-ph/9811053|bibcode=1999PhRvL..83..436N |s2cid=17928003 }} has derived a general condition to determine whether one pure state of a bipartite quantum system may be transformed into another using only LOCC. Full details may be found in the paper referenced earlier, the results are sketched out here.

Consider two particles in a Hilbert space of dimension $d$ with particle states $|\backslash psi\backslash rangle$ and $|\backslash phi\backslash rangle$ with Schmidt decompositions

:$$

|\psi\rangle=\sum_i\sqrt{\omega_i}|i_A\rangle\otimes|i_B\rangle

:$$

|\phi\rangle=\sum_i\sqrt{\omega_i'}|i_A'\rangle\otimes|i_B'\rangle

The $\backslash sqrt\{\backslash omega\_i\}$'s are known as Schmidt coefficients. If they are ordered largest to smallest (i.e., with $\backslash omega\_1>\backslash omega\_d$) then $|\backslash psi\backslash rangle$ can only be transformed into $|\backslash phi\backslash rangle$ using only local operations if and only if for all $k$ in the range $1\backslash leq\; k\; \backslash leq\; d$

:$$

\sum_{i=1}^k\omega_i\leq\sum_{i=1}^k\omega_i'

In more concise notation:

:$$

|\psi\rangle\rightarrow|\phi\rangle\quad\text{iff}\quad\omega \prec \omega'

This is a more restrictive condition than that local operations cannot increase entanglement measures. It is quite possible that $|\backslash psi\backslash rangle$ and $|\backslash phi\backslash rangle$ have the same amount of entanglement but converting one into the other is not possible and even that conversion in either direction is impossible because neither set of Schmidt coefficients majorises the other. For large $d$ if all Schmidt coefficients are non-zero then the probability of one set of coefficients majorising the other becomes negligible. Therefore, for large $d$ the probability of any arbitrary state being convertible into another via LOCC becomes negligible.

The operations described so far are deterministic, i.e., they succeed with probability 100%. If one is satisfied by probabilistic transformations, many more transformations are possible using LOCC.{{cite journal |author=Guifré Vidal |title=Entanglement Monotones |journal=J. Mod. Opt. |volume=47 |page=355 |year=2000 |issue=2–3 |doi=10.1080/09500340008244048 |arxiv=quant-ph/9807077|bibcode=2000JMOp...47..355V |s2cid=119347961 }} These operations are called stochastic LOCC (SLOCC). In particular for multi-partite states the convertibility under SLOCC is studied to gain a qualitative insight into the entanglement properties of the involved states.{{cite journal |author1=G. Gour |author2=N. R. Wallach |title=Classification of Multipartite Entanglement of All Finite Dimensionality |journal=Phys. Rev. Lett. |volume=111 |page=060502 |year=2013 |issue=6 |doi=10.1103/PhysRevLett.111.060502 |arxiv=1304.7259|bibcode=2013PhRvL.111f0502G |pmid=23971544 |s2cid=1570745 }}

If entangled states are available as a resource, these together with LOCC allow a much larger class of transformations. This is the case even if these resource states are not consumed in the process (as they are, for example, in quantum teleportation). Thus transformations are called entanglement catalysis.{{cite journal |author1=D. Jonathan |author2=M. B. Plenio |title=Entanglement-assisted local manipulation of pure quantum states |journal=Phys. Rev. Lett. |volume=83 |year=1999 |issue=17 |pages=3566–3569 |doi=10.1103/PhysRevLett.83.3566 |arxiv=quant-ph/9905071|bibcode=1999PhRvL..83.3566J |s2cid=392419 }} In this procedure, the conversion of an initial state to a final state that is impossible with LOCC is made possible by taking a tensor product of the initial state with a "catalyst state" $|c\backslash rangle$ and requiring that this state is still available at the end of the conversion process. I.e., the catalyst state is left unchanged by the conversion and can then be removed, leaving only the desired final state. Consider the states,

:$|\backslash psi\backslash rangle=\backslash sqrt\{0.4\}|00\backslash rangle+\backslash sqrt\{0.4\}|11\backslash rangle+\backslash sqrt\{0.1\}|22\backslash rangle+\backslash sqrt\{0.1\}|33\backslash rangle$

:$|\backslash phi\backslash rangle=\backslash sqrt\{0.5\}|00\backslash rangle+\backslash sqrt\{0.25\}|11\backslash rangle+\backslash sqrt\{0.25\}|22\backslash rangle$

:$|c\backslash rangle=\backslash sqrt\{0.6\}\backslash mid\backslash uparrow\backslash uparrow\backslash rangle+\backslash sqrt\{0.4\}\backslash mid\backslash downarrow\backslash downarrow\backslash rangle$

These states are written in the form of Schmidt decomposition and in a descending order. We compare the sum of the coefficients of $|\backslash psi\backslash rangle$ and $|\backslash phi\backslash rangle$

:::

In the table, red color is put if $\backslash sum\_\{i=0\}^k\backslash omega\_i>\backslash sum\_\{i=0\}^k\backslash omega\text{'}\_i$, green color is put if , and white color is remained if $\backslash sum\_\{i=0\}^k\backslash omega\_i=\backslash sum\_\{i=0\}^k\backslash omega\text{'}\_i$. After building up the table, one can easily to find out whether $|\backslash psi\backslash rangle$ and $|\backslash phi\backslash rangle$ are convertible by looking at the color in the $k$ direction. $|\backslash psi\backslash rangle$ can be converted into $|\backslash phi\backslash rangle$ by LOCC if the color are all green or white, and $|\backslash phi\backslash rangle$ can be converted into $|\backslash psi\backslash rangle$ by LOCC if the color are all red or white. When the table presents both red and green color, the states are not convertible.

Now we consider the product states $|\backslash psi\backslash rangle\; |c\backslash rangle$ and $|\backslash phi\backslash rangle\; |c\backslash rangle$：

:

\sqrt{0.24}|00\rangle\mid\uparrow\uparrow\rangle+\sqrt{0.24}|11\rangle\mid\uparrow\uparrow\rangle+

\sqrt{0.16}|00\rangle\mid\downarrow\downarrow\rangle+\sqrt{0.16}|11\rangle\mid\downarrow\downarrow\rangle\\

&+\sqrt{0.06}|22\rangle\mid\uparrow\uparrow\rangle+\sqrt{0.06}|33\rangle\mid\uparrow\uparrow\rangle+

\sqrt{0.04}|22\rangle\mid\downarrow\downarrow\rangle+\sqrt{0.04}|33\rangle\mid\downarrow\downarrow\rangle\end{align}

:

\sqrt{0.30}|00\rangle\mid\uparrow\uparrow\rangle+\sqrt{0.20}|00\rangle\mid\downarrow\downarrow\rangle+

\sqrt{0.15}|11\rangle\mid\uparrow\uparrow\rangle+\sqrt{0.15}|22\rangle\mid\uparrow\uparrow\rangle\\

&+\sqrt{0.10}|11\rangle\mid\downarrow\downarrow\rangle+\sqrt{0.10}|22\rangle\mid\downarrow\downarrow\rangle\end{align}

Similarly, we make up the table:

:::

The color in the $k$ direction are all green or white, therefore, according to the Nielsen's theorem, $|\backslash psi\backslash rangle\; |c\backslash rangle$ is possible to be converted into $|\backslash phi\backslash rangle\; |c\backslash rangle$ by the LOCC. The catalyst state $|c\backslash rangle$ is taken away after the conversion. Finally we find $|\backslash psi\backslash rangle\backslash overset\{|c\backslash rangle\}\{\backslash rightarrow\}|\backslash phi\backslash rangle$ by the LOCC.

If correlations between the system and the catalyst are allowed, catalytic transformations between bipartite pure states are characterized via the entanglement entropy.{{Cite journal|last1=Kondra|first1=Tulja Varun|last2=Datta|first2=Chandan|last3=Streltsov|first3=Alexander|date=2021-10-05|title=Catalytic Transformations of Pure Entangled States|url=https://link.aps.org/doi/10.1103/PhysRevLett.127.150503|journal=Physical Review Letters|volume=127|issue=15|page=150503|doi=10.1103/PhysRevLett.127.150503 |pmid=34678004 |arxiv=2102.11136|bibcode=2021PhRvL.127o0503K |s2cid=237532098 }} In more detail, a pure state $$

|\psi\rangle

can be converted into another pure state $$

|\phi\rangle

via catalytic LOCC if and only if

S(\psi^A) \geq S(\phi^A)

,

where $$

S

is the von Neumann entropy, and $$

\psi^A

and $$

\phi^A

are the reduced states of $$

|\psi\rangle

and $$

|\phi\rangle

, respectively. In general, the conversion is not exact, but can be performed with an arbitrary accuracy. The amount of correlations between the system and the catalyst can also be made arbitrarily small.

A broader class of transformations can be achieved through the use of quantum batteries.{{Cite journal |last=Alhambra |first=Álvaro M. |date=2019 |title=Entanglement fluctuation theorems |url=https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.012317 |journal=Physical Review A |volume=100 |issue=1 |page=012317 |doi=10.1103/PhysRevA.100.012317|arxiv=1709.06139 }}{{Citation |last1=Ganardi |first1=Ray |title=Second Law of Entanglement Manipulation with Entanglement Battery |date=2024-05-17 |url=https://arxiv.org/abs/2405.10599 |access-date=2024-12-07 |arxiv=2405.10599 |last2=Kondra |first2=Tulja Varun |last3=Ng |first3=Nelly H. Y. |last4=Streltsov |first4=Alexander}} These are ancillary systems introduced during the transformation process, akin to quantum catalysts, with the condition that they retain their entanglement upon completion of the procedure. The incorporation of quantum batteries significantly extends the scope of LOCC, enhancing the ability to manipulate quantum states. In the asymptotic regime, where a large number of identical copies of the initial state are available, this approach enables reversible interconversion between any two entangled states. This leads to the formulation of a second law of entanglement manipulation, analogous to the second law of thermodynamics. Previously, the existence of a framework leading to the second law of entanglement manipulation has been identified as a major open problem in quantum information science.{{Cite web |title=Reversible entanglement manipulation – Open Quantum Problems |url=https://oqp.iqoqi.oeaw.ac.at/reversible-entanglement-manipulation |access-date=2024-12-07 |language=en-US}}

• Quantum teleportation
• Unextendible product basis

• https://quantiki.org/wiki/locc-operations
• {{cite journal | author = Nielsen M. A. | year = 1999| title = Conditions for a class of entanglement transformations | arxiv = quant-ph/9811053 | journal = Phys. Rev. Lett. | volume = 83 | issue = 2| pages = 436–439 | doi = 10.1103/physrevlett.83.436 | bibcode = 1999PhRvL..83..436N | s2cid = 17928003}}

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