Better Way Of Separating Two CQ-States - Quantum Computing Stack Exchange most recent 30 from quantumcomputing.stackexchange.com 2019-11-11T20:42:51Z https://quantumcomputing.stackexchange.com/feeds/question/5396 https://creativecommons.org/licenses/by-sa/4.0/rdf https://quantumcomputing.stackexchange.com/q/5396 1 Better Way Of Separating Two CQ-States Hasan Iqbal https://quantumcomputing.stackexchange.com/users/2403 2019-02-07T01:02:15Z 2019-02-07T12:45:16Z <p>I have this cq-state:</p> <p><span class="math-container">$$\frac{1}{2} \times (|0\rangle \langle0|_A \otimes \rho^0_E + |1\rangle \langle1|_A \otimes \rho^1_E)$$</span></p> <p>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 <span class="math-container">$\rho^0_E$</span> and similarly, if Alice's measurement is 1, then Eve's knowledge about it is <span class="math-container">$\rho^1_E$</span>. Now, how do I measure how different these <span class="math-container">$\rho$</span>s are? How do I separate it from the system and measure their difference?</p> <p>I was using projectors on the standard basis to filter them out. But is there a standard way?</p> https://quantumcomputing.stackexchange.com/questions/5396/-/5405#5405 2 Answer by Josu Etxezarreta Martinez for Better Way Of Separating Two CQ-States Josu Etxezarreta Martinez https://quantumcomputing.stackexchange.com/users/2371 2019-02-07T10:00:51Z 2019-02-07T12:45:16Z <p>In quantum information theory, the standard way to obtain what it is called <em>reduced density operator</em> from a quantum system composed by several quantum states is to use the so-called <em>partial trace</em> operation. For the case where there are two quantum states, <span class="math-container">$\rho^{AB}$</span> can be reduced to <span class="math-container">$\rho^A$</span> and <span class="math-container">$\rho ^B$</span> in the following way:</p> <ul> <li><span class="math-container">$\rho^A=tr_B(\rho^{AB})$</span></li> <li><span class="math-container">$\rho^B=tr_A(\rho^{AB})$</span></li> </ul> <p>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 <em>fidelity</em> measurement is used usually in quantum information theory. Such the measure is defined as</p> <ul> <li><span class="math-container">$F(\rho,\sigma)=\sqrt{\rho^{1/2}\sigma\rho^{1/2}}$</span></li> </ul> <p>where <span class="math-container">$\rho$</span> and <span class="math-container">$\sigma$</span> are quantum density matrices. These things are explained in the <a href="https://en.wikipedia.org/wiki/Quantum_Computation_and_Quantum_Information" rel="nofollow noreferrer">Nielsen and Chuang</a> book, refer there for more information.</p>