Operational Resource Theory of Imaginarity

Wave-particle duality is one of the basic features of quantum mechanics, giving rise to the use of complex numbers in describing states of quantum systems and their dynamics and interaction. Since the inception of quantum theory, it has been debated whether complex numbers are essential or whether an alternative consistent formulation is possible using real numbers only. Here, we attack this long-standing problem theoretically and experimentally, using the powerful tools of quantum resource theories. We show that, under reasonable assumptions, quantum states are easier to create and manipulate if they only have real elements. This gives an operational meaning to the resource theory of imaginarity. We identify and answer several important questions, which include the state-conversion problem for all qubit states and all pure states of any dimension and the approximate imaginarity distillation for all quantum states. As an application, we show that imaginarity plays a crucial role in state discrimination, that is, there exist real quantum states that can be perfectly distinguished via local operations and classical communication but that cannot be distinguished with any nonzero probability if one of the parties has no access to imaginarity. We confirm this phenomenon experimentally with linear optics, discriminating different two-photon quantum states by local projective measurements. Our results prove that complex numbers are an indispensable part of quantum mechanics.

Figure 1

Experimental setup for local state discrimination. The experiments are carried out using linear optics. We prepare entangled photon sources and perform local projective measurements on each photon, identifying the successful guessing probability by classical communications. The optical elements are HWP, QWP, $\beta \text{−}\mathrm{Ba}{\mathrm{B}}_2{\mathrm{O}}_4$ (BBO), interference filter (IF), adjustable aperture (AA), beam splitter (BS), mirror (M), quartz plate (QP), and fiber coupler (FC). We refer to Ref. [41] for more details.

Figure 2

Experimental results for local state discrimination. All discrimination tasks concern output states after some local free operations, when the initial state is prepared as a Bell state. (a),(b) present the probabilities for the extreme examples in Eq. (7). (b) shows the experimentally measured guessing probabilities under local measurements. The horizontal axes of (a) represent the measurement results $\{++,+-,-+,--\}$, where $+,-$ corresponds to the result $\pm 1$ of local observable $A\in \{X,Y,Z\}$. (c)–(f) present results for experimental discrimination of mixed states via local projective measurements and classical communication. The families of states are prepared as follows: (c) $|{\varphi }^{+}⟩⟨{\varphi }^{+}|$ and $[u|{\varphi }^{+}⟩⟨{\varphi }^{+}|+(1-u)|\psi ⟩⟨\psi |+|{\varphi }^{-}⟩⟨{\varphi }^{-}|]/2$, where $|\psi ⟩$ is a nontrivial entangled pure state; (d) $p|{\varphi }^{+}⟩⟨{\varphi }^{+}|+(1-p)\mathbb{1}/4$ and $(|{\varphi }^{+}⟩⟨{\varphi }^{+}|+|\psi ⟩⟨\psi |+2|{\varphi }^{-}⟩⟨{\varphi }^{-}|)/4$; (e) $(|{\varphi }^{-}⟩⟨{\varphi }^{-}|+|{\psi }^{+}⟩⟨{\psi }^{+}|)/2$ and $p|{\varphi }^{+}⟩⟨{\varphi }^{+}|+(1-p)\mathbb{1}/4$; (f) $(|{\varphi }^{-}⟩⟨{\varphi }^{-}|+|{\psi }^{+}⟩⟨{\psi }^{+}|)/2$ and $s|{\varphi }^{+}⟩⟨{\varphi }^{+}|+(1-s)|{\varphi }^{-}⟩⟨{\varphi }^{-}|$, with purity $a=1–2s+2s^2$. Dashed and dotted lines represent bounds for guessing probabilities using general projective measurements and real projective measurements, respectively. The shaded areas represent the advantage of using complex measurements over real ones. In (d), general projective measurements do not provide any advantage compared to real projective measurements in the range $p<0.173$. See [41] for more details.