Probing the geometry of two-qubit state space by evolution
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Abstract
We provide explicit geometric description of state manifolds obtained from evolution governed by a four-parameter family of time-independent Hamiltonians. We cover most cases related to the real interacting two-qubit systems and discuss possible types of evolutions in terms of the defining parameters. The relevant description of the pure state spaces and their Riemannian geometry with the Fubini–Study metric is given. In particular, we analyze the modification of known geometry of quantum state manifold by the linear noncommuting perturbation of the Hamiltonian. Finally, we investigate the behavior of the entanglement for obtained families of states resulting from the unitary evolution.
Keywords
Geometry of quantum state space Fubini–Study metric Quantum evolution Compound systems Entanglement1 Introduction
The precise geometric description of the full state space of quantum system is crucial in studying its physical properties [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], especially for compound systems, where characterization of this space beyond the general property of convexity for multilevel-systems gets rather involved, even for bipartite systems. In general, such a quantum state space cannot be expected to form a smooth manifold. However, in the case of two-qubit system, on which we shall focus in the present work, the quantum state space is 15-dimensional convex set, and its pure state subset forms seven-dimensional sphere \(S^7\), where the separable pure states are located within \(S^7\) in the product of two Bloch spheres \(S^2 \times S^2\) [19].
On the other hand, there is an option to focus on distinguished subsets of states of quantum system, namely on orbits generated by unitary evolutions defined by physically relevant Hamiltonians which can be realized experimentally. Such a focus has been proven fruitful from various perspectives, such as the control theory [20, 21], the quantum brachistochrone problem [22, 23], the time-optimal evolution [5, 6, 7, 8, 21, 24, 25], or the question of Zermelo navigation [26, 27, 28].
The organization of the paper is as follows. In the next section, we obtain the explicit parametrizations of sets of quantum states generated by evolution of selected initial states and obtain relevant manifolds of the dimension depending on the initial states. We describe the Riemannian geometry of the obtained manifolds by introducing the relevant Fubini–Study metrics for each case. What is important, we focus on the question of control of such systems. Therefore, assuming the final time of evolution as a fixed \(t_{\mathrm{fin}}\) (for convenience we put \(t_{\mathrm{fin}}=1\)), we analyze the dependence of above manifolds and their geometry on the physical parameters entering the definition of the Hamiltonian. In Sect. 4, we study the changes in geometry resulting from a small perturbation in the original Hamiltonian by switching on an additional weak magnetic field along the first axis. Furthermore, in Sect. 5 we discuss the behavior of entanglement of states originating from the obtained manifolds using concurrence as the entanglement measure [35]. The concurrence changing from 0 to 1 in all cases indicates that it is not the intrinsic geometry of the manifold of quantum states and its dimension that determine the behavior of entanglement, but rather it is the specific location of such manifold relative to the torus \(S^2\times S^2\) of the separable states inside the \(S^7\).
2 Unitary transformation of two-qubit state
The normalization of an initial state means that \(\vert \eta _1\vert ^2+\vert \eta _2\vert ^2+\vert \eta _3\vert ^2+\vert \eta _4\vert ^2=1\). Moreover, the state (12) depends on four parameters (\(\omega \), \(\phi \), \(c_3\), \(c_+\)) satisfying some periodic conditions. These conditions, in turn, depend on the initial state coordinates \(\eta _j\).
- C1.For \(\eta _1=\eta _2=0\) and \(\eta _3\ne 0\), \(\eta _4\ne 0\), the state (12) takes the formIt is easy to see that this state depends only on \(c_+\) parameter and satisfies the following periodic condition$$\begin{aligned} \vert \psi ^{(0)}\rangle =\hbox {e}^{ic_3}\left( \eta _3 \hbox {e}^{-ic_+}\vert \psi _3^{(0)}\rangle +\eta _4\hbox {e}^{ic_+}\vert \psi _4^{(0)}\rangle \right) . \end{aligned}$$(13)$$\begin{aligned} \vert \psi ^{(0)}\left( c_++\pi \right) \rangle =-\vert \psi ^{(0)}\left( c_+\right) \rangle . \end{aligned}$$(14)
- C2.
- C3.For \(\eta _3=\eta _4=0\) and nonzero \(\eta _1\), \(\eta _2\), the family of states is defined by the parameters \(\omega \) and \(\phi \) as followswith the following periodic conditions$$\begin{aligned} \vert \psi ^{(0)}\rangle =\hbox {e}^{-ic_3}\left( \eta _1 \hbox {e}^{-i\omega }\vert \psi _1^{(0)}\rangle +\eta _2\hbox {e}^{i\omega }\vert \psi _2^{(0)}\rangle \right) , \end{aligned}$$(16)$$\begin{aligned} \vert \psi ^{(0)}\left( \omega +\pi ,\phi \right) \rangle= & {} -\vert \psi ^{(0)}\left( \omega ,\phi \right) \rangle ,\nonumber \\ \vert \psi ^{(0)}\left( \omega ,\phi +2\pi \right) \rangle= & {} \vert \psi ^{(0)}\left( \omega ,\phi \right) \rangle . \end{aligned}$$(17)
- C4.For \(\eta _1=0\) or \(\eta _2=0\) and \(\eta _3=0\) or \(\eta _4=0\), the family of states is defined by two parameters \(\phi \) and c, i.e.,where \(c=2c_3+(-1)^j c_++(-1)^{l+1}\omega \). Here, \(l=1,2\), \(j=3,4\). The states satisfy the following periodic conditions$$\begin{aligned} \vert \psi ^{(0)}\rangle =\hbox {e}^{-i\left( c_3+(-1)^{l+1}\omega \right) }\left( \eta _{l} \vert \psi _{l}^{(0)}\rangle +\eta _{j}\hbox {e}^{ic}\vert \psi _{j}^{(0)}\rangle \right) , \end{aligned}$$(18)$$\begin{aligned}&\vert \psi ^{(0)}\left( \phi +2\pi ,c\right) \rangle =\vert \psi ^{(0)}\left( \phi ,c\right) \rangle ,\nonumber \\&\vert \psi ^{(0)}\left( \phi ,c+2\pi \right) \rangle =\vert \psi ^{(0)}\left( \phi ,c\right) \rangle . \end{aligned}$$(19)
- C5.If \(\eta _1\), \(\eta _2\) are nonzero, and \(\eta _3=0\) or \(\eta _4=0\), then the family of states is defined by three parameters: \(\omega \), \(\phi \) and c. Namely,Here, \(c=2c_3+(-1)^j c_+\). In this case, the states satisfy the following periodic conditions$$\begin{aligned} \vert \psi ^{(0)}\rangle =\hbox {e}^{-ic_3}\left( \eta _1 \hbox {e}^{-i\omega }\vert \psi _{1}^{(0)}\rangle +\eta _2 \hbox {e}^{i\omega }\vert \psi _{2}^{(0)}\rangle +\eta _{j}\hbox {e}^{ic}\vert \psi _{j}^{(0)}\rangle \right) . \end{aligned}$$(20)$$\begin{aligned}&\vert \psi ^{(0)}\left( \omega +\pi ,\phi ,c+\pi \right) \rangle =-\vert \psi ^{(0)}\left( \omega ,\phi ,c\right) \rangle ,\nonumber \\&\vert \psi ^{(0)}\left( \omega ,\phi +2\pi ,c\right) \rangle =\vert \psi ^{(0)}\left( \omega ,\phi ,c\right) \rangle ,\nonumber \\&\vert \psi ^{(0)}\left( \omega ,\phi ,c+2\pi \right) \rangle =\vert \psi ^{(0)}\left( \omega ,\phi ,c\right) \rangle . \end{aligned}$$(21)
- C6.For \(\eta _1=0\) or \(\eta _2=0\), and nonvanishing \(\eta _3\), \(\eta _4\), the family of states is defined by three parametersHere, \(c=2c_3+(-1)^{l+1} \omega \). In this case, we have the following periodic conditions$$\begin{aligned} \vert \psi ^{(0)}\rangle =\hbox {e}^{-i\left( c_3+(-1)^{l+1}\omega \right) }\left( \eta _{l}\vert \psi _{l}^{(0)}\rangle +\eta _3 \hbox {e}^{i\left( c-c_+\right) }\vert \psi _{3}^{(0)}\rangle +\eta _4 \hbox {e}^{i\left( c+c_+\right) }\vert \psi _{4}^{(0)}\rangle \right) . \nonumber \\ \end{aligned}$$(22)$$\begin{aligned}&\vert \psi ^{(0)}\left( \phi +2\pi ,c,c_+\right) \rangle =\vert \psi ^{(0)}\left( \phi ,c,c_+\right) \rangle ,\nonumber \\&\vert \psi ^{(0)}\left( \phi ,c+2\pi ,c_+\right) \rangle =\vert \psi ^{(0)}\left( \phi ,c,c_+\right) \rangle ,\nonumber \\&\vert \psi ^{(0)}\left( \phi ,c+\pi ,c_++\pi \right) \rangle =\vert \psi ^{(0)}\left( \phi ,c,c_+\right) \rangle . \end{aligned}$$(23)
- C7.In the general case, when all parameters \(\eta _1\), \(\eta _2\), \(\eta _3\) and \(\eta _4\) are nonzero, we have the state defined by expression (12) with the following periodic conditions$$\begin{aligned}&\vert \psi ^{(0)}\left( \omega +\pi ,\phi ,c_3+\pi /2,c_+\right) \rangle =i\vert \psi ^{(0)}\left( \omega ,\phi ,c_3,c_+\right) \rangle ,\nonumber \\&\vert \psi ^{(0)}\left( \omega +\pi ,\phi ,c_3,c_++\pi \right) \rangle =-\vert \psi ^{(0)}\left( \omega ,\phi ,c_3,c_+\right) \rangle ,\nonumber \\&\vert \psi ^{(0)}\left( \omega ,\phi +2\pi ,c_3,c_+\right) \rangle =\vert \psi ^{(0)}\left( \omega ,\phi ,c_3,c_+\right) \rangle ,\nonumber \\&\vert \psi ^{(0)}\left( \omega ,\phi ,c_3+\pi ,c_+\right) \rangle =-\vert \psi ^{(0)}\left( \omega ,\phi ,c_3,c_+\right) \rangle ,\nonumber \\&\vert \psi ^{(0)}\left( \omega ,\phi ,c_3+\pi /2,c_++\pi \right) \rangle =-i\vert \psi ^{(0)}\left( \omega ,\phi ,c_3,c_+\right) \rangle . \end{aligned}$$(24)
Let us study the Fubini–Study metric of these manifolds, \(\mathcal {M}_{\psi ^{(0)}}\)
3 The Fubini–Study metric of quantum state manifolds
- 1.
In the first case, the manifold is defined by the parameter \(c_+\in \left[ 0,\pi \right] \) and metric tensor is reduced to \(g_{c_+c_+}\) component with \(\eta _{34}^+=1\). This is the metric of the circle of the radius \(\gamma \sqrt{1-\left( \eta _{34}^{-}\right) ^2}/2\).
- 2.
In the second case, the manifold is defined by parameter \(\phi \in \left[ 0,2\pi \right] \) and metric tensor is reduced to \(g_{\phi \phi }^{(0)}\) with \(\eta _{12}^+=1\). This metric also describes the circle of the radius \(\gamma /2\).
- 3.
In this case, the manifold is two-parametric \(\theta \in \left[ 0,\pi \right] \), \(\phi '\in \left[ 0,2\pi \right] \) and is described by the metric tensor with components \(g_{\theta \theta }^{(0)}\), \(g_{\phi '\phi '}^{(0)}\), where \(\eta _{12}^+=1\). This means that it is the sphere of radius \(\gamma /2\).
- 4.Here, we have also two-parametric manifold defined by parameters \(\phi \in \left[ 0,2\pi \right] \), \(c\in \left[ 0,2\pi \right] \) and described by the following metric tensorAs we can see, the components of the metric tensor do not depend on the parameters \(\phi \) and c. This means that the manifold is flat. Taking into account periodic conditions (17), we conclude that it is a torus.$$\begin{aligned} g_{\phi \phi }^{(0)}=\frac{\gamma ^2}{4}\vert \eta _l\vert ^2,\quad g_{\phi c}^{(0)}=0,\quad g_{c c}^{(0)}=\gamma ^2\vert \eta _l\vert ^2\vert \eta _j\vert ^2. \end{aligned}$$(33)
- 5.In the fifth case, the manifold is three-parametric and defined by the parameters \(\theta \in \left[ 0,\pi \right] \), \(\phi '\in \left[ 0,2\pi \right] \), \(c'\in \left[ 0,2\pi \right] \). In the diagonal form, the metric tensor components \(g_{\theta \theta }^{(0)}\), and \(g_{\phi '\phi '}^{(0)}\) are defined by expression (32) and other component takes the formwhere \(c'\) is related to the parameter c from (20) by the following formula$$\begin{aligned} g_{c' c'}^{(0)}=\gamma ^2\eta _{12}^+\vert \eta _j\vert ^2, \end{aligned}$$(34)The manifold, which we obtain here, is a product of the sphere of radius \(\gamma \sqrt{\eta _{12}^+}/2\) in parameters \(\theta \), \(\phi '\) and of the circle of radius \(\gamma \sqrt{\eta _{12}^+}\vert \eta _j\vert \) in parameter \(c'\).$$\begin{aligned} c=-\frac{\eta _{12}^-}{\eta _{12}^+\sin ^2\theta }\omega '+\frac{1}{2}\cos \theta \phi '+c'. \end{aligned}$$(35)
- 6.In the case C6, we obtain a manifold with the metric tensor in the diagonal formand the component \(g_{c_+'c_+'}^{(0)}\) defined by the expression (32). Therefore, we obtain a three-parameter manifold defined by \(\phi \in \left[ 0,2\pi \right] \), \(c'\in \left[ 0,2\pi \right] \), \(c_+'\in \left[ 0,\pi \right] \). To diagonalize this metric, we use the following transformation$$\begin{aligned} g_{\phi \phi }^{(0)}=\frac{\gamma ^2}{4}\vert \eta _l\vert ^2,\quad g_{c' c'}^{(0)}=\gamma ^2\vert \eta _l\vert ^2\frac{\left( \eta _{34}^+\right) ^2-\left( \eta _{34}^-\right) ^2}{\eta _{34}^+-\left( \eta _{34}^-\right) ^2}, \end{aligned}$$(36)where c is defined as for the state (22). So, this manifold can be expressed by a circle of radius \(\gamma \vert \eta _l\vert /2\) in parameter \(\phi \) and torus in parameter \(c'\), \(c_+'\).$$\begin{aligned} c=c',\quad c_+=\frac{\vert \eta _l\vert ^2\eta _{34}^-}{\eta _{34}^+-\left( \eta _{34}^-\right) ^2}c'+c_+', \end{aligned}$$(37)
- 7.
In the general case, the metric is defined by expression (28) or (32). This manifold consists of two submanifolds, namely a sphere of radius \(\gamma \sqrt{\eta _{12}^+}/2\) in parameters \(\theta \in \left[ 0,\pi \right] \), \(\phi '\in \left[ 0,2\pi \right] \) and a torus in parameters \(c_3'\in \left[ 0,\pi \right] \), \(c_+'\in \left[ 0,\pi \right] \).
4 The Fubini–Study metric of quantum state manifold with perturbation
Dimensions and parametrization of the state manifolds
Case | dim\({\mathcal {M}}_{\vert \psi ^{(0)}\rangle }\) | Parameters |
---|---|---|
C1 | 1 | \(c_+\) |
C2 | 1 | \(\phi \) |
C3 | 2 | \(\omega \), \(\phi \) |
C4 | 2 | \(\phi \), c |
C5 | 3 | \(\omega \), \(\phi \), c |
C6 | 3 | \(\phi \), c, \(c_+\) |
C7 | 4 | \(\omega \), \(\phi \), \(c_3\) \(c_+\) |
5 Entanglement characterization of two-qubit quantum state manifolds \({\mathcal {M}}_{\vert \psi ^{(0)}\rangle }\)
- 1.In the case C1, the concurrence takes the formFor \(\eta _3=\vert \eta _3\vert \) and \(\eta _4=\vert \eta _4\vert \hbox {e}^{i\chi }\), where \(\chi \in \left[ 0,2\pi \right] \). We obtain the maximally entangled state if \(c_+=1/4\left[ (2n+1)\pi -2\chi \right] \), where \(n\in \mathbb {Z}\).$$\begin{aligned} C=\sqrt{\vert \eta _3\vert ^4+\vert \eta _4\vert ^4-2\vert \eta _3\vert ^2\vert \eta _4\vert ^2\cos \left( 4c_++2\chi \right) }. \end{aligned}$$(67)
- 2.The squared concurrence in the case C2 takes simple form$$\begin{aligned} C=\vert \cos \phi \vert . \end{aligned}$$(68)
- 3.For the C3 family of states, the manifold is defined by two parameters. The entanglement of the states is described by the following expressionSimilarly, as in the previous case C1, we put \(\eta _1=\vert \eta _1\vert \) and \(\eta _2=\vert \eta _2\vert \hbox {e}^{i\chi }\). As we can see, regardless of the initial state, the maximally entangled state is obtained when \(\phi =0\) and \(\omega =1/4\left[ (2n+1)\pi -2\chi \right] \).$$\begin{aligned} C= & {} \left[ \left( \vert \eta _1\vert ^4+\vert \eta _2\vert ^4-2\vert \eta _1\vert ^2\vert \eta _2\vert ^2\cos \left( 4\omega +2\chi \right) \right) \cos ^2\phi +4\vert \eta _1\vert ^2\vert \eta _2\vert ^2\sin ^2\phi \right. \nonumber \\&\quad \left. -\,4\vert \eta _1\vert \vert \eta _2\vert \left( \vert \eta _1\vert ^2-\vert \eta _2\vert ^2\right) \cos \left( 2\omega +\chi \right) \sin \phi \cos \phi \right] ^{1/2}. \end{aligned}$$(69)
- 4.Here, we also put \(\eta _l=\vert \eta _l\vert \) and \(\eta _j=\vert \eta _j\vert \hbox {e}^{i\chi }\) and obtain the expression for concurrenceSo, the conditions for maximally entangled are the following: \(\phi =0\) and \(c=1/2\left[ (2n+1)\pi -2\chi \right] \) for even \(l+j\), \(c=1/2\left[ 2\pi n-2\chi \right] \) for odd \(l+j\), and, respectively, for \(\phi =\pi \).$$\begin{aligned} C=\sqrt{\vert \eta _l\vert ^4\cos ^2\phi +\vert \eta _j\vert ^4-2(-1)^{l+j}\vert \eta _l\vert ^2\vert \eta _j\vert ^2\cos \left( 2c+2\chi \right) \cos \phi }. \end{aligned}$$(70)
- 5.For the C5 family of states, to simplify calculations, we analyze the case when \(\eta _1=\eta _2\) and we put \(\eta _1=\vert \eta _1\vert \), \(\eta _j=\vert \eta _j\vert \hbox {e}^{i\chi }\). The squared concurrence takes finally the formWe collect the conditions for preparation of maximally entangled states in Table 2.$$\begin{aligned} C= & {} \left[ \left( -2\vert \eta _1\vert ^2\sin \phi +(-1)^j\vert \eta _j\vert ^2\cos \left( 2c+2\chi \right) \right) ^2\right. \nonumber \\&\left. +\left( -2\vert \eta _1\vert ^2\sin 2\omega \cos \phi +(-1)^j\vert \eta _j\vert ^2\sin \left( 2c+2\chi \right) \right) ^2\right] ^{1/2}. \end{aligned}$$(71)
- 6.We shall use similar simplifications in the case C6. Here, we also put \(\eta _3=\eta _4\) and \(\eta _l=\vert \eta _l\vert \), \(\eta _3=\vert \eta _3\vert \hbox {e}^{i\chi }\). Then, the squared concurrence takes the formThe conditions for preparation of maximally entangled states for the C6-family are presented in Table 3.$$\begin{aligned} C= & {} \left[ \left( (-1)^{l+1}\vert \eta _l\vert ^2\cos \phi -2\vert \eta _3\vert ^2\sin \left( 2c+2\chi \right) \sin 2c_+\right) ^2\right. \nonumber \\&\left. + \,4\vert \eta _3\vert ^4\cos ^2\left( 2c+2\chi \right) \sin ^22c_+ \right] ^{1/2}. \end{aligned}$$(72)
- 7.In the C7 case, we assume that \(\eta _1=\eta _2=\vert \eta _1\vert \), \(\eta _3=\eta _4=\vert \eta _3\vert \hbox {e}^{i\chi }\) what yields the squared concurrence in the formThe conditions defining maximally entangled states are collected in Table 4.$$\begin{aligned} C= & {} \left[ \left( 2\vert \eta _1\vert ^2\sin \phi +2\vert \eta _3\vert ^2\sin 2c_+\sin \left( 4c_3+2\chi \right) \right) ^2\right. \nonumber \\&\left. +\left( -2\vert \eta _1\vert ^2\sin 2\omega \cos \phi +2\vert \eta _3\vert ^2\sin 2c_+\cos \left( 4c_3+2\chi \right) \right) ^2\right] ^{1/2} . \end{aligned}$$(73)
Conditions for maximally entangled states in case C5
\(\phi \) | \(\omega \) | j | c |
---|---|---|---|
0 | \(\pi /4\) | Even | \(3\pi /4+\pi n-\chi \) |
Odd | \(\pi /4+\pi n-\chi \) | ||
\(3\pi /4\) | Even | \(\pi /4+\pi n-\chi \) | |
Odd | \(3\pi /4+\pi n-\chi \) | ||
\(\pi /2\) | – | Even | \(1/2\left[ (2n+1)\pi -2\chi \right] \) |
Odd | \(\pi n-\chi \) | ||
\(\pi \) | \(\pi /4\) | Even | \(\pi /4+\pi n-\chi \) |
Odd | \(3\pi /4+\pi n-\chi \) | ||
\(3\pi /4\) | Even | \(3\pi /4+\pi n-\chi \) | |
Odd | \(\pi /4+\pi n-\chi \) | ||
\(3\pi /2\) | – | Even | \(\pi n -\chi \) |
Odd | \(1/2\left[ (2n+1)\pi -2\chi \right] \) |
Conditions for maximally entangled states in the case C6
\(\phi \) | l | c | \(c_+\) |
---|---|---|---|
0 | Even | \(\pi /4+\pi n-\chi \) | \(\pi /4+\pi n\) |
\(3\pi /4+\pi n-\chi \) | \(3\pi /4+\pi n\) | ||
Odd | \(\pi /4+\pi n-\chi \) | \(3\pi /4+\pi n\) | |
\(3\pi /4+\pi n-\chi \) | \(\pi /4+\pi n\) | ||
\(\pi \) | Even | \(\pi /4+\pi n-\chi \) | \(3\pi /4+\pi n\) |
\(3\pi /4+\pi n-\chi \) | \(\pi /4+\pi n\) | ||
\(\pi /4+\pi n-\chi \) | \(\pi /4+\pi n\) | ||
\(3\pi /4+\pi n-\chi \) | \(3\pi /4+\pi n\) |
Conditions for maximally entangled states in case C7
\(\phi \) | \(\omega \) | \(c_+\) | \(c_3\) |
---|---|---|---|
0 | \(\pi /4\) | \(\pi /4+\pi n\) | \(1/4\left[ (2n+1)\pi -2\chi \right] \) |
\(3\pi /4+\pi n\) | \(1/2\left[ \pi n-\chi \right] \) | ||
\(3\pi /4\) | \(\pi /4+\pi n\) | \(1/2\left[ \pi n-\chi \right] \) | |
\(3\pi /4+\pi n\) | \(1/4\left[ (2n+1)\pi -2\chi \right] \) | ||
\(\pi /2\) | – | \(\pi /4+\pi n\) | \(1/4\left[ \pi /2+2\pi n -2\chi \right] \) |
\(3\pi /4+\pi n\) | \(1/4\left[ 3\pi /2+2\pi n -2\chi \right] \) | ||
\(\pi \) | \(\pi /4\) | \(\pi /4+\pi n\) | \(1/2\left[ \pi n-\chi \right] \) |
\(3\pi /4+\pi n\) | \(1/4\left[ (2n+1)\pi -2\chi \right] \) | ||
\(3\pi /4\) | \(\pi /4+\pi n\) | \(1/4\left[ (2n+1)\pi -2\chi \right] \) | |
\(3\pi /4+\pi n\) | \(1/2\left[ \pi n-\chi \right] \) | ||
\(3\pi /2\) | – | \(\pi /4+\pi n\) | \(1/4\left[ 3\pi /2+2\pi n -2\chi \right] \) |
\(3\pi /4+\pi n\) | \(1/4\left[ \pi /2+2\pi n -2\chi \right] \) |
It is worth noting that in all above cases, one can achieve full range of values of the concurrence. This effect indicates that it is not the geometry and dimensionality of the particular state manifold that is decisive, rather it is the location of such manifold within the \(S^7\) and its common points with the torus \(S^2\times S^2\) of separable pure state manifold. For states with vanishing concurrence for special choices of the relevant parameters, we can talk about a ’parametric death’ of entanglement—paraphrasing the ’sudden death’ of entanglement discussed in the literature for specific evolutions. However, having in mind the entangling power of the considered families of Hamiltonians we have collected conditions for preparation of maximally entangled states for all cases under consideration.
6 Conclusions
The geometric characterization of the state manifold of quantum system is of great value, but for compound systems such task becomes very complex when addressed in the general setting.
In the present work, we have studied quantum state manifolds for two-qubit system obtained by means of the unitary evolution defined by large family of physically interesting Hamiltonians. Despite the knowledge of the whole set of the two-qubit quantum states, it is important to know what manifolds lying inside this set can be reached using the evolution governed by the realistic Hamiltonians. The geometry of such obtained compact quantum state spaces is of Riemannian type defined by the Fubini–Study metrics depending on initial conditions and parameters entering the definition of the families of Hamiltonians. We have given the classification of possible state manifolds and thoroughly discussed the explicit description of two-qubit unitary orbits generated by physically relevant Hamiltonians. The relevant Fubini–Study metrics were obtained with the use of the explicit parametrizations.
It is worth noting that we have also studied the question of how obtained geometries are modified by the noncommutative linear perturbation term included into the original Hamiltonian. We describe its influence on the scalar curvature of the relevant state spaces. In some cases, the answer turns out to be nontrivial.
As an important physical characterization of the considered systems, we have studied the degree of entanglement of states for all obtained quantum state spaces and we have provided conditions for producing maximally entangled states in each case, where the concurrence was used as an entanglement monotone. Such knowledge seems to be of great importance for the quantum information and quantum computation applications.
Notes
Acknowledgements
One of the authors (A.K.) wishes to thank the Institute of Theoretical Physics at the University of Wroclaw for hospitality and financial support as well as he acknowledges that the work was supported by Project FF-30F (No. 0116U001539) from the Ministry of Education and Science of Ukraine.
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