LOCC

LOCC, or local operations and classical communication, is a method in quantum information theory where a local (product) operation is performed on part of the system, and where the result of that operation is "communicated" classically to another part where usually another local operation is performed. An example of this is distinguishing two Bell pairs, such as the following:

LOCC paradigm: the parties are not allowed to exchange particles coherently. Only Local operations and classical communication is allowed

|\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. Assume that Alice measures the first qubit, and obtains the result 0. We still don't know which Bell pair we were given. Alice sends the result to Bob over a classical channel, where Bob measures the second qubit, also obtaining 0. Bob now knows that since the joint measurement outcome is $|0\rangle _{A}\otimes |0\rangle _{B}$ , then the pair given was $|\psi _{1}\rangle$ .

These measurements contrasts with nonlocal or entangled measurements, where a single measurement is performed in ${\mathbb {C}}^{{2n}}$ instead of the product space ${\mathbb {C}}^{2}\otimes {\mathbb {C}}^{n}$ .

Entanglement convertibility

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

\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 that local operations cannot increase the degree of entanglement. It is quite possible that converting between $|\psi \rangle$ and $|\phi\rangle$ 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 converted into another becomes negligible.

Catalytic conversion

Just after Nielsen published his theorem of majorization, the catalytic transformation by LOCC was proposed.^[2] Consider the states,

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

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

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

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

$k$	$\|\psi \rangle$	$\|\phi\rangle$
0	0.4	0.5
1	0.8	0.65
2	0.9	1.0
3	1.0	1.0

In the table, red color is put if $\sum _{i=0}^{k}\omega _{i}>\sum _{i=0}^{k}\omega '_{i}$ , green color is put if $\sum _{i=0}^{k}\omega _{i}<\sum _{i=0}^{k}\omega '_{i}$ , and white color is remained if $\sum _{i=0}^{k}\omega _{i}=\sum _{i=0}^{k}\omega '_{i}$ . After building up the table, one can easily to find out whether $|\psi \rangle$ and $|\phi\rangle$ are convertible by looking at the color in the $k$ direction. $|\psi \rangle$ can be converted into $|\phi\rangle$ by LOCC if the color are all green or white, and $|\phi\rangle$ can be converted into $|\psi \rangle$ 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 $|\psi \rangle |c\rangle$ and $|\phi \rangle |c\rangle$ ：

{\begin{aligned}|\psi \rangle |c\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{aligned}}

{\begin{aligned}|\phi \rangle |c\rangle &={\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{aligned}}

Similarly, we make up the table:

$k$	$\|\psi \rangle \|c\rangle$	$\|\phi \rangle \|c\rangle$
0	0.24	0.30
1	0.48	0.50
2	0.64	0.65
3	0.80	0.80
4	0.86	0.90
5	0.92	1.00
6	0.96	1.00
7	1.00	1.00

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

References

↑ M. A. Nielsen, Phys. Rev. Lett. 83, 436 - 439 (1999)
↑ D. Jonathan and M. B. Plenio, Phys. Rev. Lett. 83, 3566 (1999)

LOCC

Entanglement convertibility

Catalytic conversion

References

Further reading