Entanglement conversion

Nielsen ^[3] 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 ${\displaystyle d}$ with particle states ${\displaystyle |\psi \rangle }$ and ${\displaystyle |\phi \rangle }$ with Schmidt decompositions

${\displaystyle |\psi \rangle =\sum _{i}{\sqrt {\omega _{i}}}|i_{A}\rangle \otimes |i_{B}\rangle }$

${\displaystyle |\phi \rangle =\sum _{i}{\sqrt {\omega _{i}'}}|i_{A}'\rangle \otimes |i_{B}'\rangle }$

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

${\displaystyle \sum _{i=1}^{k}\omega _{i}\leq \sum _{i=1}^{k}\omega _{i}'}$

In more concise notation:

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

This is a more restrictive condition than that local operations cannot increase entanglement measures. It is quite possible that ${\displaystyle |\psi \rangle }$ and ${\displaystyle |\phi \rangle }$ 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 ${\displaystyle d}$ if all Schmidt coefficients are non-zero then the probability of one set of coefficients majorising the other becomes negligible. Therefore, for large ${\displaystyle d}$ 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.^[4] 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.^[5]

Going beyond LOCC: Catalytic conversion

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.^[6] 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" ${\displaystyle |c\rangle }$ 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,

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

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

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

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

${\displaystyle k}$	${\displaystyle |\psi \rangle }$	${\displaystyle |\phi \rangle }$
0	0.4	0.5
1	0.8	0.75
2	0.9	1.0
3	1.0	1.0

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

${\displaystyle {\begin{aligned}|\psi \rangle |c\rangle &={\sqrt {0.24}}|00\rangle \mid \uparrow \uparrow \rangle +{\sqrt {0.24}}|11\rangle \mid \uparrow \uparrow \rangle +{\sqrt {0.16}}|00\rangle \mid \downarrow \downarrow \rangle +{\sqrt {0.16}}|11\rangle \mid \downarrow \downarrow \rangle \\&+{\sqrt {0.06}}|22\rangle \mid \uparrow \uparrow \rangle +{\sqrt {0.06}}|33\rangle \mid \uparrow \uparrow \rangle +{\sqrt {0.04}}|22\rangle \mid \downarrow \downarrow \rangle +{\sqrt {0.04}}|33\rangle \mid \downarrow \downarrow \rangle \end{aligned}}}$

${\displaystyle {\begin{aligned}|\phi \rangle |c\rangle &={\sqrt {0.30}}|00\rangle \mid \uparrow \uparrow \rangle +{\sqrt {0.20}}|00\rangle \mid \downarrow \downarrow \rangle +{\sqrt {0.15}}|11\rangle \mid \uparrow \uparrow \rangle +{\sqrt {0.15}}|22\rangle \mid \uparrow \uparrow \rangle \\&+{\sqrt {0.10}}|11\rangle \mid \downarrow \downarrow \rangle +{\sqrt {0.10}}|22\rangle \mid \downarrow \downarrow \rangle \end{aligned}}}$

Similarly, we make up the table:

${\displaystyle k}$	${\displaystyle |\psi \rangle |c\rangle }$	${\displaystyle |\phi \rangle |c\rangle }$
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 ${\displaystyle k}$ direction are all green or white, therefore, according to the Nielsen's theorem, ${\displaystyle |\psi \rangle |c\rangle }$ is possible to be converted into ${\displaystyle |\phi \rangle |c\rangle }$ by the LOCC. The catalyst state ${\displaystyle |c\rangle }$ is taken away after the conversion. Finally we find ${\displaystyle |\psi \rangle {\overset {|c\rangle }{\rightarrow }}|\phi \rangle }$ 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.^[7] In more detail, a pure state ${\displaystyle |\psi \rangle }$ can be converted into another pure state ${\displaystyle |\phi \rangle }$ via catalytic LOCC if and only if

${\displaystyle S(\psi ^{A})\geq S(\phi ^{A})}$,

where ${\displaystyle S}$ is the von Neumann entropy, and ${\displaystyle \psi ^{A}}$ and ${\displaystyle \phi ^{A}}$ are the reduced states of ${\displaystyle |\psi \rangle }$ and ${\displaystyle |\phi \rangle }$, 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.