<p>It is often mathematically very convenient to work with sub-normalized states independently of the question whether they have a direct physical meaning.</p>

<p>However, you can always see them as states that are logically 'AND' with a certain event. For example, consider the c-q state</p>

<p>$\rho_{XB} = \sum_x |x\rangle\!\langle x|_X \otimes \rho_B^x$</p>

<p>where $\rho_B^x$ is sub-normalized so that the whole state is normalized. Then the trace of $\rho_B^x$ just gives the probability of finding the subsystem $B$ in the renormalized, conditional state $\rho_B^x / \text{tr}(\rho_B^x)$. </p>

<p>For example, the probability of measuring a certain event (POVM element) $E_B$ <strong>AND</strong> x is simply given by $\text{tr}(\rho_B^x E_B)$ and we never needed to renormalize the state to write this down.</p>

CC BY-SA 3.0
2014-10-26T21:17:52.533Z
2014-10-26T21:25:05.813Z
2014-10-26T21:25:05.813Z
Marco Tomamichel
<p>By basic postulates of QM, any state of a system is described by a normalised density operator. Now i fail to see why people study sub-normalized states ( e.g.: In  generalised fidelity etc). I'd be glad to know any physical interpretations of the same. </p>

CC BY-SA 3.0
2014-10-26T16:09:26.343Z
I am sorry! It seems i don't have permission to add this topic to quantum computing. I request those who have the accesses please add it.
Vamsi Krishna
CC BY-SA 3.0
2014-10-26T16:33:46.090Z
Subnormalized density operators are not only very useful in studying quantum computation, but in studying quantum physics as well.
Peter Shor 
<quantum-computing><quantum-information>
Why are sub-normalized states studied in quantum computation?
Vamsi Krishna
# Why are sub-normalized states studied in quantum computation?

(Asked by: username_0 on 2014-10-26)

By basic postulates of QM, any state of a system is described by a normalised density operator. Now i fail to see why people study sub-normalized states ( e.g.: In generalised fidelity etc). I'd be glad to know any physical interpretations of the same.

**Tags:** quantum-computing, quantum-information

> **Comments:**
> **username_0**: I am sorry! It seems i don't have permission to add this topic to quantum computing. I request those who have the accesses please add it.  
> **username_1**: Subnormalized density operators are not only very useful in studying quantum computation, but in studying quantum physics as well.  

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## 1 answers

### Answer by username_2 on 2014-10-26. Score: 6 | Accepted answer

It is often mathematically very convenient to work with sub-normalized states independently of the question whether they have a direct physical meaning.

However, you can always see them as states that are logically 'AND' with a certain event. For example, consider the c-q state

$\rho\_{XB} = \sum\_x |x\rangle\!\langle x|\_X \otimes \rho\_B^x$

where $\rho\_B^x$ is sub-normalized so that the whole state is normalized. Then the trace of $\rho\_B^x$ just gives the probability of finding the subsystem $B$ in the renormalized, conditional state $\rho\_B^x / \text{tr}(\rho\_B^x)$.

For example, the probability of measuring a certain event (POVM element) $E\_B$ **AND** x is simply given by $\text{tr}(\rho\_B^x E\_B)$ and we never needed to renormalize the state to write this down.


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