# Better Way Of Separating Two CQ-States

I have this cq-state:

$$\frac{1}{2} \times (|0\rangle \langle0|_A \otimes \rho^0_E + |1\rangle \langle1|_A \otimes \rho^1_E)$$

Where Alice (A) is classical and an adversary Eve (E) has some knowledge about Alice's system. If Alice's measurement is 0, then Eve's knowledge is $$\rho^0_E$$ and similarly, if Alice's measurement is 1, then Eve's knowledge about it is $$\rho^1_E$$. Now, how do I measure how different these $$\rho$$s are? How do I separate it from the system and measure their difference?

I was using projectors on the standard basis to filter them out. But is there a standard way?

• you mean how do you do it with pen and paper on in the real world? There are many ways to quantify the distance between states, a common one being the fidelity
– glS
Feb 7 '19 at 10:47

In quantum information theory, the standard way to obtain what it is called reduced density operator from a quantum system composed by several quantum states is to use the so-called partial trace operation. For the case where there are two quantum states, $$\rho^{AB}$$ can be reduced to $$\rho^A$$ and $$\rho ^B$$ in the following way:

• $$\rho^A=tr_B(\rho^{AB})$$
• $$\rho^B=tr_A(\rho^{AB})$$

This operator acts like it is tracing out one of the systems from the whole density matrix that describes the whole system. Apart from that, in order to measure how different two quantum states are, the fidelity measurement is used usually in quantum information theory. Such the measure is defined as

• $$F(\rho,\sigma)=\sqrt{\rho^{1/2}\sigma\rho^{1/2}}$$

where $$\rho$$ and $$\sigma$$ are quantum density matrices. These things are explained in the Nielsen and Chuang book, refer there for more information.

• Hi, I edited out the link to the PDF version of the textbook and replaced it with the corresponding Wikipedia link. Such PDFs are often pirated versions of still-in-print textbooks and linking to them as such is discouraged on most SE sites. C.f. Chemistry: On the issue of Copyright & Stack Exchange Network Acceptable Use Policy. Feb 7 '19 at 12:49
• Hi @Martinez, thanks for your answer. But tracing out A subsystem would give me a mixture of rhos from two different cases. I wouldn't be able to distinguish between them. What I need is a way to take these two rhos out and measure how different they are. Feb 7 '19 at 19:16