Difference between revisions of "Majorizes"
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===Bipartite LOCC=== | ===Bipartite LOCC=== | ||
| − | A well-known result of Nielsen<ref>M. A. Nielsen. Conditions for a class of entanglement transformations. Phys. Rev. Lett., 83:439, 1999.</ref> says that a bipartite pure state $|\psi\rangle \in \mathbb{C}^m \otimes \mathbb{C}^n$ can be converted into another state $|\phi\rangle$ via [[LOCC]] if and only if the squared Schmidt coefficients of $|\phi\rangle$ majorize the squared Schmidt coefficients of $|\psi\rangle$. Thus we can determine whether or not we can convert $|\psi\rangle$ to $|\phi\rangle$ via LOCC as follows: | + | A well-known result of Nielsen<ref>M. A. Nielsen. Conditions for a class of entanglement transformations. ''Phys. Rev. Lett.'', 83:439, 1999.</ref> says that a bipartite pure state $|\psi\rangle \in \mathbb{C}^m \otimes \mathbb{C}^n$ can be converted into another state $|\phi\rangle$ via [[LOCC]] if and only if the squared Schmidt coefficients of $|\phi\rangle$ majorize the squared Schmidt coefficients of $|\psi\rangle$. Thus we can determine whether or not we can convert $|\psi\rangle$ to $|\phi\rangle$ via LOCC as follows: |
<pre<noinclude></noinclude>> | <pre<noinclude></noinclude>> | ||
>> phi = [[RandomStateVector|RandomStateVector(9)]]; % generate two random states in C^3 \otimes C^3 | >> phi = [[RandomStateVector|RandomStateVector(9)]]; % generate two random states in C^3 \otimes C^3 | ||
| Line 52: | Line 52: | ||
1 | 1 | ||
</pre<noinclude></noinclude>> | </pre<noinclude></noinclude>> | ||
| + | |||
| + | ===Majorization criterion for separability=== | ||
| + | The [[majorization criterion]] says that every separable state $\rho_{AB}$ is such that each of $\rho_A$ and $\rho_B$ majorize $\rho_{AB}$<ref>M. A. Nielsen and J. Kempe. Separable states are more disordered globally than locally. ''Phys. Rev. Lett.'', 86:5184, 2001. E-print: [http://arxiv.org/abs/quant-ph/0011117 arXiv:quant-ph/0011117]</ref>. Thus the following code constructs the maximally-entangled quantum state and uses the majorization criterion to verify that it is entangled: | ||
| + | <pre<noinclude></noinclude>> | ||
| + | >> v = MaxEntangled(3); | ||
| + | >> rho = v*v'; | ||
| + | >> Majorizes(PartialTrace(rho),rho) | ||
| + | |||
| + | ans = | ||
| + | |||
| + | 0 | ||
| + | </pre<noinclude></noinclude>> | ||
| + | |||
| + | Note that the majorization criterion is strictly weaker than the partial transpose criterion (and even the [[reduction criterion]] <ref>T. Hiroshima. Majorization criterion for distillability of a bipartite quantum state. ''Phys. Rev. Lett.'', 91:057902, 2003. E-print: [http://arxiv.org/abs/quant-ph/0303057 arXiv:quant-ph/0303057]</ref>) and thus those alternate tests should be used in most situations instead. | ||
==References== | ==References== | ||
<references /> | <references /> | ||
Revision as of 15:12, 4 March 2014
| Majorizes | |
| Determines whether or not a vector or matrix majorizes another | |
| Other toolboxes required | none |
|---|---|
Majorizes is a function that determines whether or not one vector or matrix weakly majorizes another vector or matrix. That is, given d-dimensional vectors $A$ and $B$, it checks whether or not
\[ \sum_{i=1}^k a_i^{\downarrow} \geq \sum_{i=1}^k b_i^{\downarrow} \quad \text{for } k=1,\dots,d,\]
where $a^{\downarrow}_i$ and $b^{\downarrow}_i$ are the elements of $A$ and $B$, respectively, sorted in decreasing order.
In the case of matrices, it is said that $A$ majorizes $B$ if the vector of $A$'s singular values majorizes the vector of $B$'s singular values. If the two vectors or matrices are of different sizes, the smaller one is padded with zeros appropriately so that they are comparable.
Syntax
- M = Majorizes(A,B)
Argument descriptions
- A: Either a vector or a matrix.
- B: Either a vector or a matrix.
Examples
A simple example
It is straightforward to see that the vector $(3,0,0)$ majorizes the vector $(1,1,1)$, which we can verify as follows:
>> Majorizes([3,0,0],[1,1,1])
ans =
1
Bipartite LOCC
A well-known result of Nielsen[1] says that a bipartite pure state $|\psi\rangle \in \mathbb{C}^m \otimes \mathbb{C}^n$ can be converted into another state $|\phi\rangle$ via LOCC if and only if the squared Schmidt coefficients of $|\phi\rangle$ majorize the squared Schmidt coefficients of $|\psi\rangle$. Thus we can determine whether or not we can convert $|\psi\rangle$ to $|\phi\rangle$ via LOCC as follows:
>> phi = RandomStateVector(9); % generate two random states in C^3 \otimes C^3 >> psi = RandomStateVector(9); >> Majorizes(SchmidtDecomposition(phi).^2,SchmidtDecomposition(psi).^2) ans = 0
The above code shows that the conversion $|\psi\rangle \stackrel{LOCC}{\rightarrow} |\phi\rangle$ is impossible. On the other hand, the following code shows that the maximally-entangled pure state can be converted to a random pure state via LOCC:
>> phi = RandomStateVector(9); >> psi = MaxEntangled(3); % maximally-entangled state in C^3 \otimes C^3 >> Majorizes(SchmidtDecomposition(phi).^2,SchmidtDecomposition(psi).^2) ans = 1
Majorization criterion for separability
The majorization criterion says that every separable state $\rho_{AB}$ is such that each of $\rho_A$ and $\rho_B$ majorize $\rho_{AB}$[2]. Thus the following code constructs the maximally-entangled quantum state and uses the majorization criterion to verify that it is entangled:
>> v = MaxEntangled(3);
>> rho = v*v';
>> Majorizes(PartialTrace(rho),rho)
ans =
0
Note that the majorization criterion is strictly weaker than the partial transpose criterion (and even the reduction criterion [3]) and thus those alternate tests should be used in most situations instead.
References
- ↑ M. A. Nielsen. Conditions for a class of entanglement transformations. Phys. Rev. Lett., 83:439, 1999.
- ↑ M. A. Nielsen and J. Kempe. Separable states are more disordered globally than locally. Phys. Rev. Lett., 86:5184, 2001. E-print: arXiv:quant-ph/0011117
- ↑ T. Hiroshima. Majorization criterion for distillability of a bipartite quantum state. Phys. Rev. Lett., 91:057902, 2003. E-print: arXiv:quant-ph/0303057