Experimental information complementarity of two-qubit states

The concept of information offers a more complete description of complementarity than the traditional approach based on observables. We present the first experimental test of information complementarity for two-qubit pure states, achieving close agreement with theory. We also explore the distribution of information in a comprehensive range of mixed states. Our results highlight the strange and subtle properties of even the simplest quantum systems; for example, entanglement can be increased by reducing the correlations between two subsystems.

. The three families of quantum states we use to explore information complementarity: (i) pure states ρ pure ; (ii) highly entangled mixed states ρ werner and ρ mems (where |ψ − , |φ + are two of the four Bell states); and (iii) separable states ρ as , ρ s . I 2 and I 4 are the one-and two-qubit identity matrices. (i) ρ pure = |ψ ψ|; |ψ = cos α|00 + sin α|11 α ∈ [0, π/4] 1] This knowledge deficit for a single particle in a mixed state can be attributed to entanglement with a second particle. In addition to the local observables V and P, one therefore includes the non-local observable C (which is the concurrence [9], a measure of entanglement) and finds that [10,11] (V 2 + P 2 ) local + (C 2 ) corr 1. ( The complementarity is now between the local properties of an individual subsystem and its correlations with another subsystem. For pure two-particle states, relation (2) has been experimentally tested to some extent in [12,13]-albeit in its early form of interferometric complementarity [14]. In [15], equation (2) has been measured for a 'system' qubit coupled to an 'environment' qubit via an amplitude damping channel. But again, in the case of mixed states, (2) does not saturate its bound. One could of course explain this as due to entanglement to yet another particle and so on, ad infinitum 4 . It would, however, be preferable to precisely identify the quantities involved in complementarity without resorting to virtual higher-dimensional Hilbert spaces.
Here we experimentally study a version of complementarity between two quantum bits that does not require infinite regression. It is phrased in terms of information and is symmetric and exact, i.e. all subsystems are treated on an equal footing and the relations are saturated also for mixed states. We experimentally test the complementarity for pure two-qubit states and explore the distribution of information in many interesting mixed states; see table 1. Finally, we show why traditional complementarity relations such as equation (2) cannot be exact for mixed states. They do not take into account correlations that are not captured by entanglement.
Information complementarity goes back to the insight that information is physical [18] and that the amount of information in a two-level quantum system (or qubit) is limited to 1 bit [19,20]. Complementarity may now arise because there is insufficient information to simultaneously specify the results of different measurements that can be performed on a 3 quantum system. The single-particle complementarity relation (1) reflects this by being bounded to 1 (bit), a value that is saturated for a pure qubit. For a mixed state, the left-hand side of (1) is equal to a suitable measure of the reduced information content of the system [20].
This provides an alternative interpretation of the two-particle complementarity, equation (2). The total information content of two particles, I total , can be split into information stored in subsystems, I local , and a remaining part that we call the correlation information, I corr , giving rise to the following relation [21] I local + I corr = I total , which defines I corr . The natural way to quantify the information of a given quantum state is to use its entropy. The information measure originally proposed in [20] is linked to the so-called linear entropy and, as we show in appendix A, equation (3) is equivalent to (2) for pure states. Note that (3) has the elegant feature of being symmetric with respect to subsystems, i.e. I local = I a + I b , whereas the local observables V 2 and P 2 in (2) capture only one of the two subsystems. Their equivalence for pure states is a consequence of Schmidt decomposition according to which both subsystems are described by the same density operator. Unfortunately, for certain mixed two-particle states, the linear entropy measure leads to a negative I corr ; see appendix A. We therefore adopt the von Neumann information I (ρ) = log 2 d − S(ρ), where S(ρ) = −Tr(ρ log 2 ρ) is the von Neumann entropy of a d-dimensional system described by the density matrix ρ, and we choose bits as the units of information content (N qubits carry at most N bits). The correlation information is then the quantum mutual information, , which is non-negative for all physical states and is a measure for the total correlations present in a quantum state [22,23]. Unlike traditional wave-particle duality, the information approach (3) has not been explored in any experimental system to date. We now investigate information complementarity for a range of two-photon quantum states. For pure two-qubit states, we can test complementarity relation (3), with its right-hand side bounded to I total = 2; for mixed states, it will allow us to highlight the different types of correlations present in these systems. Our states can be grouped into three families (table 1): (i) 'pure' states, i.e. our best experimental approximation to pure; (ii) two classes of highly entangled mixed states-the Werner and maximally entangled mixed states [17,24] (MEMS); and (iii) two classes of separable states. These states were chosen because-except for our particular choice of MEMS 5 -they represent the boundaries of the physical parameter space for two-qubit states in the tangle-entropy plane, figure 1.
The experimental scheme is depicted in figure 2. A source based on a polarization Sagnac loop [26,27] produces two-photon states close to the ideal form |ψ ab = cos α|H a H b + sin α|V a V b , where the logical qubit states '0' and '1' are encoded into the horizontal (H) and vertical (V) polarizations of the photons a and b. The degree of entanglement is determined by α, which is set by the pump laser polarization [26]. The resulting photons pass through two channels, E 1 and E 2 (figure 2). We perform both full single-qubit state tomography [28] on the individual photons a and b-from which we reconstruct ρ a and ρ b -and, separately, two-qubit tomography [28] on the two-photon state, which yields ρ ab . These density matrices contain all information that one can possibly have about the underlying quantum states, and we can from them readily compute properties such as the tangle T -the concurrence C squared-or the entropy, which are otherwise not directly accessible experimentally. The 48 created states are shown in figure 1: for each, we calculate I a = I (ρ a ) and I b = I (ρ b ) to obtain I local = I a + I b and I total = 2 − S(ρ ab ), respectively. For the correlation information I corr , we subtract I local from I total .
Information complementarity for our experimental states ρ pure is shown in figure 3. When T vanishes, the total information is stored exclusively in the individual subsystems. As the entanglement increases, so does I corr , and as T approaches 1, I local goes to zero. This phenomenon is strictly quantum-a pure classical state cannot contain any mutual information. More importantly, for pure quantum states, I corr corresponds to the entanglement of formation [29]. This means that information complementarity reduces to a sum of entanglement and local information, each of which can be measured independently, and thus (3) has the same form as (2), the only difference being that (3) is symmetric with respect to the subsystems, i.e. it accounts for the local properties of both subsystems a and b. Because any pure entangled two-qubit state can be obtained from the Schmidt form ρ pure via local operations, the data in figure 3 represent a conclusive test of (3) and (2) for all pure two-qubit states (see also the appendix).
For mixed states, I corr is no longer exclusively identified with entanglement [30,31]. The Werner states, ρ werner , for example, are a statistical mixture of a maximally entangled state and white noise. The individual qubits in this state are always fully mixed, I local = 0, and their entire information content is stored in I corr , as one can see in figure 4(a). As is well known, Werner states are separable for a high noise admixture (T = 0 for S(ρ werner ) > 1.8, figure 1) but their I corr does not vanish correspondingly, emphasizing that it does not measure entanglement in this case. The transmitted beam polarization is rotated by 90 • and the reflected beam is polarized at a polarizing beam splitter (PBS) before the beams are recombined incoherently. Two steering mirrors control the splitting ratio between the two beams and thus parameter p in ρ mems (table 1). This technique allows the creation of ρ mems in the range of 0 S(ρ) 0.92. The remaining states can be covered by dephasing the second photon once p = 2/3 is reached. In practice, the initial tangle in our experiment was too low to create states with significantly higher entanglement than the corresponding ρ werner . (d) Dephasing channel for the creation of ρ as . Jamin-Lebedeff interferometers introduce optical path delays between the orthogonal polarization components of incoming photons in a given basis. Two interferometers are used to first individually decohere photon 1 from zero to fully mixed and then photon 2.
By comparing the Werner to MEMS states, we show that I corr is not even a monotonic function of entanglement. Out of the several different existing classes [17] of MEMS, we chose to create rank-2 MEMS, ρ mems , using the method from [25], illustrated in figure 2(b). As one can see in figure 4(b), the local information contents of these states are nonzero. For any value of S, even though they are more entangled (cf figure 1), I corr for ρ mems is lower than that for the corresponding ρ werner . In particular, we find that for I total = 1.4 bits the Werner state has more mutual information I corr = 1.387 ± 0.001 and less tangle T = 0.647 ± 0.004 than the corresponding MEMS, where I corr = 1.315 ± 0.018 and T = 0.667 ± 0.007. The higher entanglement of MEMS states in comparison to Werner states coincides with a relative decrease of correlations. In our experiment, this difference is small, due to the difficulty of producing    high-quality MEMS. There are, however, states for which the effect is far more pronounced, as we show in appendix B. Such states indicate that complementarity relations such as (2) miss some quantities as both local information and entanglement can grow from one state to another.
Our third example shows that aspects of complementarity between local information and correlations are already present in classical states. We consider two classes of mixed separable states, ρ as and ρ s (table 1). The former, ρ as , represents two individually dephased photons. The latter, ρ s , consists of a product state with a white noise admixture; see figure 2(d) for the experimental details.
In figure 5(a), one can clearly see that product states with individually added noise, ρ as , do not contain any correlations, I corr = 0. In contrast, states with globally added white noise, ρ s , contain I corr = 0.094 ± 0.023 bits of mutual information at I total = 1 bit, whereas I local is reduced accordingly ( figure 5(b)).
In conclusion, we performed a test of information complementarity and traditional complementarity for pure two-qubit states and showed that the two approaches can be reconciled in this case. We measured the information distribution in mixed quantum systems, which allowed us to demonstrate that correlations are not a monotonic function of entanglement. This suggests what is missing in traditional complementarity relations such as (2). These relations are not exact, because they do not take into account any other correlations than those due to entanglement. It remains an open question whether a traditional complementarity relation-based on directly observable quantities-can be formulated which includes classical correlations and quantum discord [30,31], dissonance [32] or other recently proposed correlation measures [33,34]. An interesting alternative might be to tie wave-particle duality to the Bell inequality violations, as shown in [35] for Werner states.  Theory plot for states ρ d (γ ) with 0 < γ < 1 in comparison with the Werner states ρ werner (red line) and MEMS ρ mems (green line). The blue dashed lines represent states dephased in two conjugate bases, with different weightings γ and γ α . The information content of dephased states is the same as that for ρ werner and is larger than that for the MEMS states even for entanglement arbitrarily close to zero.
traditional complementarity relation. Note that information complementarity based on linear entropy has been discussed, for example, in [36]. Figure A.1(b) shows that the information measure based on the linear entropy is nonadditive in the sense that I corr is negative for the product state ρ as , defined in the main text. This is the reason why we to stick to von Neumann entropy instead of linear entropy in the main text.

Appendix B. Doubly dephased states
Here we present a class of weakly entangled states of two 2-level systems, for which I corr is consistently higher than that for the MEMS states. Consider the states which can be viewed as a result of dephasing of the Bell singlet state |ψ − in the local bases of Pauli operators σ z and σ x . The dephasing in the z-basis has strength γ and that in the x-basis has strength γ α . Just like the Werner states, ρ d (γ ) has information only in correlations, i.e. I local = 0, cf figure 4(a). We plot the entanglement (tangle) of these states for various α in figure A.2. Since MEMS states with the same amount of total information have nonvanishing I local , they contain less information in correlations than does ρ d (γ ), even at S(ρ) > 1 and large α, in which case the entanglement of the dephased states is close to zero [37]. In the most extreme case, at S(ρ) = 1, the state ρ d has zero tangle but retains 1 bit of (classical) correlations I corr . The rank-3 MEMS (the boundary of the shaded area in figure 2), in comparison, has a tangle of T ∼ 0.48 and I corr ∼ 0.94 bit.