Experimental Test of Tracking the King Problem

In quantum theory, the retrodiction problem is not as clear as its classical counterpart because of the uncertainty principle of quantum mechanics. In classical physics, the measurement outcomes of the present state can be used directly for predicting the future events and inferring the past events which is known as retrodiction. However, as a probabilistic theory, quantum-mechanical retrodiction is a nontrivial problem that has been investigated for a long time, of which the Mean King Problem is one of the most extensively studied issues. Here, we present the first experimental test of a variant of the Mean King Problem, which has a more stringent regulation and is termed “Tracking the King.” We demonstrate that Alice, by harnessing the shared entanglement and controlled-not gate, can successfully retrodict the choice of King's measurement without knowing any measurement outcome. Our results also provide a counterintuitive quantum communication to deliver information hidden in the choice of measurement.


Introduction
The thought experiment, named the Mean King Problem (MKP), is a quantum-mechanical retrodiction problem, which originates from the paper in 1987 by Vaidman et al. [1]. The original research dealt with spin-half particles, i.e., the case in 2-dimension Hilbert space, and was generalized to prime dimensionality [2] and power of prime dimensionality [3,4]. Actually, the MKP and its variants are continuously developed in the general frame of quantum theory [5][6][7][8].
In the original version described by Vaidman et al., a measurement chosen from fσ x , σ y , σ z g is performed on a spin − ð1/2Þ particle at a given time. By means of measurements carried out both before and after the time in question, they came up with a method to ascertain the result of the spin measurement, even though they do not know in which direction the spin was measured and no matter what those measured results may happen to be. The solution of this problem is intriguing because it is by no means obvious to make definite inferences about σ x , σ y , σ z .
In the Mean King Problem, a physicist named Alice prepares a two prime-dimensional particle state and avails one of the particles to the King who then measures his particle by freely choosing one of the mutually unbiased bases (MUB) [4,[9][10][11][12]. The King challenges Alice to perform a control measurement of her liking and then state the outcome of his measurement correctly with the knowledge of his choice of measurement.
It seems that in the MKP the physicist can assign the state of the system in the past. However, with the knowledge of the King's choice of measurement, this state retrodiction is actually conditional. What if the King does not tell Alice his choice? This question leads to another extension of the MKP termed "Tracking the King Problem" (TKP). In this new problem, Alice is not informed by the King's choice (nor the outcome). Inversely, her control measurement is designed to track the basis that the King used. The interesting thing is, in TKP, the King's choice of measurement (without recording the outcome) can be viewed as a novel quantum communication [13,14] signal, viz., a message sent to Alice [15].
The counterintuitive part of this problem is that in classical physics, measurements without outcomes being recorded carry no information and hence cannot be used for communication. However, such a measurement in quantum mechanics causes traceable disturbance to the measured system so that Alice's control measurement can retrieve the King's choice. Until recently, this thought experiment has not been realized yet.
In this work, we experimentally demonstrate the TKP of the 2-dimension case in an optical system. By sharing maximally entangled photon pairs and employing a controllednot gate (C-NOT) as Alice's control measurement, we show that it is possible for Alice to retrodict the choice of King's measurement without knowing any measurement outcome. The realization of this thought experiment unravels the intrinsic characteristics of quantum measurement and deepens our understanding of quantum theory. Furthermore, the choices of measurements can also be seen as the delivered messages, which put it forward to realize a counterintuitive quantum communication.
The scheme of the TKP is shown in Figure 1. Alice starts with preparing one of the maximally entangled bipartite states: where d = 2; ω = i; c, r ; s = 0, 1, ⋯, d − 1; and jni = jn + di for any n. Superscript s labels a basis for Hilbert space of the two qubits, while c, r label the d 2 orthonormal states within the basis. For example, s = c = r = 0 represents the state of jϕ + i = 1/ ffiffi ffi 2 p ðj1ij1i + j0ij0iÞ.

Experimental Setup
The schematic view of the experimental setup is shown in Figure 2. In Alice's station, polarization-entangled photon pairs are prepared via type-II spontaneous parametric downconversion. The 390 nm pulsed laser is generated in the LiB 3 O 5 (LBO) crystal by frequency doubling and is focused on the 2 mm thick type-II degenerate noncollinear cut betabarium borate (BBO) crystal as a pump light [16]. The combination of a half-wave plate (HWP) and a 1 mm thick BBO crystal is used in each arm to compensate the spatial and temporal walk-off of the generated photons. Two extra HWPs are used here to create four Bell states as different initial states: where HðVÞ represents horizontal (vertical) polarization of the qubit.
One of the entangled photons (P1) is stored with a fixed time delay by being coupled into a single-mode fiber. The other one (P2) is sent to the King's station, where a standard single-qubit analyzer consisting of two quarter-wave plates (QWPs) and a polarizer (POL) serves as the required nonselective measurement (see the black box in Figure 2). The collapsed photon state is then coupled into a single-mode fiber and sent back into the C-NOT gate together with the P1 in Alice's station. We measure a truth table in the computational basis ZZ and obtain an average fidelity of the C-NOT gate up to 0.827. More details about the implementation and optimization of C-NOT gate can be found in Methods and Figure 3. With single-qubit polarization projection for each photon and coincidence measurement, we are able to project P1 and P2 onto four Bell states. To test such an ability for distinguishing Bell states, we inject all the four maximally entangled states in equation (5) and measure their coincidence, in the basis fD, Ag for the control arm and in the basis fH, Vg for the target arm. The results shown in Figure 3(c) show a good ability of transforming the maximally entangled states to corresponding product states, which is directly related to the performance of the required control measurement. Step1: Alice prepares a bipartite state | c , r ; s 〉 1,2 ., and sends one qubit to the King.
Step2: King measures the qubit he holds in one of the MUBs labeled by b = 0 ,0,1... , d -1, then returns it back to Alice.
Step3: Alice performs a measurement in the basis: Thereby she can almost always retrieve the message labeled by b.

Results
We test the TKP in two different initial conditions, in which Alice prepares the singlet state jψ − i and the triplet state jϕ + i, respectively. After receiving the qubit sent by Alice, the King performs the nonselective measurement chosen from fσ x , σ y , σ z g by changing the angles of the combination of the POL and the QWPs. Then he sends the qubit back to Alice. The reunited two qubits go through the device consisting of two QWPs and a HWP in order to compensate the polarization rotation induced by the fibers. Afterwards, Alice performs the control measurement in the Bell basis fjϕ + i, jϕ − i, jψ + i, ψ − g on the two qubits to retrieve the King's choice of measurement b. The expected probabilities are listed in Table 1 as a theoretical truth table. The experimental results obtained from Alice's control measurement are shown in Figure 4, from which we do observe a good agreement with the theoretical truth table. Both the expected and unexpected probabilities are much far away from 0.25, a uniform probability distribution. It should be noticed that the King's measurement outcomes are not used at all in the control measurement in our experiment, but are only borrowed to check whether our results are consistent with the theoretical truth table. The reliability of identifying the King's choice, the number of successful events divided by all trial times, is found to be up to 0.813 on average, going well beyond the rate of 0.5 from wild guessing.
The demonstrated abilities of retrodicting the King's choice of measurement apparently can be considered a form of communication protocol. We ask two volunteer students to act as Alice and the King, who execute the   whole process like a game. In the case that Alice prepares the initial state jϕ + i, the King randomly chooses a series of nonselective measurements b out of f € 0, 0, 1g. Alice then can "guess" the King's choice from f € 0, 0, 1g relying on her measurements and the truth table. We show a fraction of trial events in Figure 5. We can see that, while the reliability is not a unit, an appropriate statistics can help Alice unambiguously reveal the King's choice and win the game. In summary, by experimentally testing a variant of MKP, i.e., the TKP, we exemplify how tasks that seem impossible by classical reasoning can be experimentally achieved within quantum mechanics frameworks. By harnessing the shared entanglement and controlled-not gate, we demonstrate that we can successfully retrodict the choice of King's measurement without knowing any measurement outcome. Our results provide a strong distinction for the features between classical and quantum systems. While performing nonselective measurements on classical systems, no matter how correlated they are, they cannot carry or manipulate information [17,18]; the trackability of nonselective measurements on quantum systems inevitably introduces distinguishable disturbance [15].
The realization of such a thought experiment also provides a counterintuitive quantum communication to deliver information hidden in the choice of measurement: the King sends a message that he does not want to send. We may also conceive an anti-eavesdropping scheme to reveal all eavesdroppers' actions when they try to hack into an entanglement-distributed quantum network, tracking the King.

Experimental
Details. The C-NOT gate, as a key part of the control measurement, is realized by an essential partial polarization beam splitter (PPBS-I) and two supplemental PPBS-IIs [19,20]. The PPBS-I reflects vertically polarized light perfectly and reflects (transmits) 1/3 (2/3) of horizontally polarized light, which performs as a quantum phase gate while attenuating the H components by a factor of 1/ ffiffi ffi 3 p . The two PPBS-IIs are inserted to each of the interferometer paths as local polarization compensators, which transmits H components perfectly and transmits (reflects) 1/3 (2/3) V components. The two incident photons are ensured to perfectly interfere on the PPBS-I to erase their which-way information. We optimize their spatial and temporal overlap by observing the Hong-Ou-Mandel interference with an injected identical polarization of H. Figure 3(a) shows the measured Hong-Ou-Mandel dip, with which we obtain a visibility of 66.3% by fitting with the Gaussian curve. In light of the ideal value of 80% determined by the specification of the PPBS-I, the achieved visibility represents a good mode match in both space and time. As is shown in Figure 5(b), we measure a truth table in the computational basis ZZ and obtain an average fidelity of the C-NOT gate up to 0.827. Take it specifically, if we inject the V into the control arm, the target bit will keep its original state. Otherwise, if we inject H into the control arm, the target bit will flip, which we denote using red bars in Figure 3(b). With this basic function, we can use the C-NOT gate to discriminate four Bell states. For example, the injection of state jϕ + i will come out with the result D in the control arm and V in the target arm (see Table 2).