Entanglement characterization of a two-level system

== The Questions == The most fundamental object in [[quantum information]] theory is [[qubit]], which is essentially a two-level system, whose basis states are denoted by $\backslash left\backslash \{\; \backslash left|\; 0\; \backslash right\backslash rangle\; ,\backslash left|\; 1\; \backslash right\backslash rangle\; \backslash right\backslash \}$. The most general (pure) state is represented by $\backslash alpha\; \backslash left|\; 0\; \backslash right\backslash rangle\; +\backslash beta\; \backslash left|\; 1\; \backslash right\backslash rangle$, the complex numbers satisfy the normalization condition $\backslash left|\; \backslash alpha\; \backslash right|^\{2\}+\backslash left|\; \backslash beta\; \backslash right|^\{2\}=1$ . Suppose we now have two such qubits. The basis of the whole system becomes $\backslash left\backslash \{\; \backslash left|\; 00\; \backslash right\backslash rangle\; ,\backslash left|\; 01\; \backslash right\backslash rangle\; ,\backslash left|\; 10\; \backslash right\backslash rangle\; ,\backslash left|\; 11\; \backslash right\backslash rangle\; \backslash right\backslash \}$ and the most general (pure) state is of the form: :$\backslash left|\; \backslash rm\{General\}\; \backslash right\backslash rangle\; =\backslash alpha\; \backslash left|\; 00\; \backslash right\backslash rangle\; +\backslash beta\; \backslash left|\; 01\; \backslash right\backslash rangle\; +\backslash gamma\; \backslash left|\; 10\; \backslash right\backslash rangle\; +\backslash lambda\; \backslash left|\; 11\; \backslash right\backslash rangle$ (most general two-qubit pure state), where $\backslash left|\; \backslash alpha\; \backslash right|^\{2\}+\backslash left|\; \backslash beta\; \backslash right|^\{2\}+\backslash left|\; \backslash gamma\; \backslash right|^\{2\}+\backslash left|\; \backslash lambda\; \backslash right|^\{2\}=1$. Here we use a short-hand notation $\backslash left|\; 00\; \backslash right\backslash rangle\; \backslash equiv\; \backslash left|\; 0\; \backslash right\backslash rangle\; \backslash otimes\; \backslash left|\; 0\; \backslash right\backslash rangle$ to represent the [[tensor product]]. The main questions we ask in this article are : :'''Question (1) When is the state entangled?''' :'''Question (2) How is the entanglement quantified?''' Part of the reasonings in the following is also applicable to more general states for a qubit coupled with any other quantum systems in pure state. However, for pedagogical purposes, we shall focus on two qubits only. === Answer to Question (1) === To answer question (1), it may be better to ask when the state is not entangled? We shall define an non-entangled state as '''product state'''. By definition, a product state is of the form: :$\backslash left|\; \backslash rm\; \{Product\}\; \backslash right\backslash rangle\; =\backslash left(\; \backslash alpha\; \_\{1\}\backslash left|\; 0\; \backslash right\backslash rangle\; +\backslash beta\; \_\{2\}\backslash left|\; 1\; \backslash right\backslash rangle\; \backslash right)\backslash otimes\; \backslash left(\; \backslash alpha\; \_\{2\}\backslash left|\; 0\; \backslash right\backslash rangle\; +\backslash beta\; \_\{2\}\backslash left|\; 1\; \backslash right\backslash rangle\; \backslash right)$. We then ask under what circumstances (what values of $\backslash left\backslash \{\; \backslash alpha\; ,\backslash beta\; ,\backslash gamma\; ,\backslash lambda\; \backslash right\backslash \}$) the state $\backslash left|\; \backslash rm\; General\; \backslash right\backslash rangle$ becomes $\backslash left|\; \backslash rm\; Product\; \backslash right\backslash rangle$? This can be answered by a mathematical theorem, known as [[Schmidt decomposition]]. In our case, it is always possible to write our states as (the Schmidt form) :$\backslash left|\; \backslash rm\; Schmidt\; \backslash right\backslash rangle\; =\backslash sqrt\{\backslash mu\_\{1\}\}\backslash left|\; p\_\{1\}\; \backslash right\backslash rangle\; \backslash otimes\; \backslash left|\; p\_\{2\}\; \backslash right\backslash rangle\; +\backslash sqrt\{\backslash mu\_\{2\}\}\backslash left|\; q\_\{1\}\; \backslash right\backslash rangle\; \backslash otimes\; \backslash left|\; q\_\{2\}\; \backslash right\backslash rangle$, where it is important to note that $\backslash left\backslash langle\; p\_\{1,2\}\; \backslash right.\backslash left|\; p\_\{1,2\}\; \backslash right\backslash rangle\; =\backslash left\backslash langle\; q\_\{1,2\}\; \backslash right.\backslash left|\; q\_\{1,2\}\; \backslash right\backslash rangle\; =1$ and $\backslash left\backslash langle\; p\_\{1,2\}\; \backslash right.\backslash left|\; q\_\{1,2\}\; \backslash right\backslash rangle\; =0$. The coefficients are assumed to be real and satisfy $\{\backslash ;\; \backslash mu\_\{1\}+\backslash mu\_\{2\}\; =\; 1\}$. Thanks to the bi-orthogonal form, the [[reduced density matrix]] $\backslash rho\; \_\{1,2\}=Tr\_\{2,1\}\backslash left|\; \backslash rm\; Schmidt\; \backslash right\backslash rangle\; \backslash left\backslash langle\; \backslash rm\; Schmidt\; \backslash right|=\backslash mu\_\{1\}\backslash left|\; p\_\{1,2\}\; \backslash right\backslash rangle\; \backslash left\backslash langle\; p\_\{1,2\}\; \backslash right|+\backslash mu\_\{2\}\backslash left|\; q\_\{1,2\}\; \backslash right\backslash rangle\; \backslash left\backslash langle\; q\_\{1,2\}\; \backslash right|$ can be immediately determined. By comparing the corresponding reduced density matrices taken from $\backslash left|\; \backslash rm\{Product\}\; \backslash right\backslash rangle\; \backslash left\backslash langle\; \backslash rm\{Product\}\; \backslash right|$, we conclude that :$\backslash left|\; \backslash rm\{General\}\; \backslash right\backslash rangle$ is a product state ''if and only if'' $\backslash ;\; \backslash mu\_\{1\}=1$ and $\backslash ;\; \backslash mu\_\{2\}=0$. Otherwise, it is an entangled state. Once we find a relationship between $\backslash left\backslash \{\; \backslash mu\_\{1\},\backslash mu\; \_\{2\}\; \backslash right\backslash \}$ and $\backslash left\backslash \{\; \backslash alpha\; ,\backslash beta\; ,\backslash gamma\; ,\backslash lambda\; \backslash right\backslash \}$, we are able to answer question (1). Here we note that $\backslash mu$'s are the same for both $\backslash rho\_1$ and $\backslash rho\_2$. Let us consider $\backslash rho\_1$, which is a 2x2 matrix: :. The eigenvectors are $\backslash left\backslash \{\; \backslash left|\; p\_\{1\}\; \backslash right\backslash rangle\; ,\backslash left|\; q\_\{1\}\; \backslash right\backslash rangle\; \backslash right\backslash \}$ and the corresponding eigenvalues are $\backslash left\backslash \{\; \backslash mu\; \_\{1\},\backslash mu\; \_\{2\}\; \backslash right\backslash \}$. The condition $\backslash mu\; \_\{1\}=1$ and $\backslash mu\; \_\{2\}=0$ is equivalent to say the determinant $\backslash det\; \backslash left(\; \backslash rho\; \_\{1\}\; \backslash right)=0$ vanishes. This gives the above result explicitly (which is surprisingly simple) in terms of the parameters $\backslash left\backslash \{\; \backslash alpha\; ,\backslash beta\; ,\backslash gamma\; ,\backslash lambda\; \backslash right\backslash \}$: :$\backslash left|\; \backslash rm\{General\}\; \backslash right\backslash rangle$ is a product state ''if and only if'' $\backslash left|\; \backslash alpha\; \backslash lambda\; -\backslash beta\; \backslash gamma\; \backslash right|=0$. === Answer to Question (2) === The second question we ask is how entanglement can be quantified. Before jumping into the details, perhaps, we should first ask what is the '''purpose''' of quantifying entanglement? First, entanglement is the essential feature of the quantum theory. This property is also known as [[non-locality]], where the concept was originated from the seminal paper A. Einstein, B. Podolsky, and N. Rosen, Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47 777 (1935). of Einstein, Podolsky and Rosen (EPR) in 1935. In short, non-locality is a property where a local disturbance cannot affect instantaneously, or at a speed faster then light, an distant object. Quantum mechanics, or more precisely, the measurement part, is non-local. For example, if we have a pair electrons in the spin-singlet state $\backslash left(\; \backslash left|\; \backslash uparrow\; \backslash downarrow\; \backslash right\backslash rangle\; -\backslash left|\; \backslash downarrow\; \backslash uparrow\; \backslash right\backslash rangle\; \backslash right)/\backslash sqrt\{2\}$ and separated far apart, then the measurement on one electron spin will, according to the [[orthodox interpretation]], necessarily cause the wavefunction to "collapse" to a product state e.g. $\backslash left|\; \backslash uparrow\; \backslash downarrow\; \backslash right\backslash rangle$. However, the idea of wavefunction collapse is purely a theoretical construct. No noticeable signal is transmitted when one of the distant entangled pair is measured. The bottom line here is that quantum entanglement does suggest different results from any local hidden variable theories. This is made possible by John Bell's famous [[Bell's inequality]] found in 1964. Non-locality is an important property of entanglement, should we quantify entanglement based on the degree the Bell's inequality is violated? Although it is known N. Gisin, Phys. Lett. A 154, 201 (1991) that all bipartite entangled pure state violate Bell's inequality, We would immediately run into trouble if we consider [[mixed states]] as well. It is because some mixed entangled states S. Popescu, Phys. Rev. Lett. 72, 797 - 799 (1994) do not violate any Bell's inequality. {{stub}} [[Category:Entanglement]]