## Abstract

We present an experimental illustration on the quantum sensitivity of decision making machinery. In the decision making process, we consider the role of available information, say hint, whether it influences the optimal choices. To the end, we consider a machinery method of decision making in a probabilistic way. Our main result shows that in decision making process our quantum machine is more highly sensitive than its classical counterpart to the hints we categorize into “good” and “poor”. This quantum feature originates from the quantum superposition involved in the decision making process. We also show that the quantum sensitivity persists before the quantum superposition is completely destroyed.

## Introduction

We live in a chain of decisions everyday. We make a decision whether to take an umbrella as assessing the chance of raining. Decisions are made by accounting for available information, e.g., the dark clouds through a window and/or the 30% chance of raining that the weather forecast announces. Yet we often make wrong decisions due to inadequate or noisy information. The relations of decisions with given information were studied in the theory of decision making (DM)^{1}. However, it is not easy that DM processes are consistently analyzed^{2,3}. This is mainly because each decision maker has the different degree of “sensitivity” to a given available information; ones are more biased with the given information than others^{1,4}. This is an intrinsic trait of decision makers^{5}. In this work we focus on the sensitivity to the available information which we categorize as “good” and “poor” hints, qualitatively.

Our DM study is presented in a framework of game theory^{6}. Game theory deals with the strategies by which players (decision makers in this paper) maximize their own rewards. Nowadays quantum science has extended game theory to the quantum domain, revealing distinctive quantum features and opening a new avenue of applications^{7,8,9}. As in quantum game theory, we are to investigate a quantum trait in decision makers, which originates from quantum properties^{10,11}, i.e. the quantum sensitivity to the available information during a DM process. This is intimate to an issue of quantum game theory, whether any quantum effects are revealed when no quantum strategies are involved. This has been regarded to be negative^{12,13}. To this end, we consider machines which play (or simulate) rational decision makers, equipped with a simple and reasonable DM algorithm. We then compare the two types of decision making machines, classical and quantum. Here, game elements including strategies are assumed to be classical, except the decision processes, in which the quantum machine is allowed to exploit a quantum algorithm^{14}. Our main result shows that the quantum decision maker is more highly sensitive than its classical counterpart to given available information, categorized to good and poor hints. This is attributed to the quantum coherence involved in the quantum DM process. We also show that the quantum sensitivity persists before the quantum coherence is completely destroyed. These results will be applicable to reinforcement learning and preference updating^{15,16,17,18}; they expect a risk-averse machine to learn more slowly.

## Results

### Secret-bit guessing game

We suggest a simple game, called the “secret-bit guessing game” (see Fig. 1a)^{19}. In this game, one player (say Alice) has a couple of cards *C*_{κ} (*κ* = 0, 1), on each of which her secret-bit number *x*_{κ} is written. The other player (say Bob) should make a guess *y*_{κ} (or “strategy” in the language of game theory) at her secret-bit *x*_{κ}. By a successful guess (i.e., *y*_{κ} = *x*_{κ}), Bob receives a positive score of *ξ*/2; however, by a wrong guess (i.e., \({y}_{\kappa }\ne {x}_{\kappa }\)), Bob receives a penalty, i.e., a negative score of −*ξ*/2 (see Fig. 1b). After the two guesses, Bob will get a score among {−*ξ*, 0, *ξ*}. Then, Bob wins (loses) with a score of *ξ* (−*ξ*). The game ends in a draw if Bob has a score of zero. Here, we raise a question whether some (additional) hints can help Bob to increase his winning probability or score. In particular, we explore how Bob’s winning probability depends on a DM algorithm, considering the two types of DM processes which work classically and quantum-mechanically, respectively. Our results suggest that some quantum features play roles in the DM process with no use of quantum strategies.

### Classical & quantum decision making algorithm

To proceed, we adopt a DM algorithm, which is assumed to work in Bob’s brain. The DM algorithm is modeled as a machinery process (see Fig. 2), which runs with two channels: an input channel of a single bit for Alice’s card number \(\kappa \in \{0,1\}\), and the other is an ancillary channel for processing the input with an output which is used for Bob’s guess. The ancillary channel consists of two *probabilistic* operations *u*_{j} (*j* = 0, 1), each supposed to be either the identity \(1\) (doing nothing) or the logical-not *X* (flipping the signal). Here, applying *u*_{1} is conditioned on the input *κ*: i.e., *u*_{1} is applied only if *κ* = 1. The algorithm commences with receiving an input *κ* from Alice. The two probabilistic operations *u*_{j} in the ancillary channel are carried out with respect to the probabilities \(P({u}_{j}\to 1)\) and \(P({u}_{j}\to X)=1-P({u}_{j}\to 1)\). Here, \(P({u}_{j}\to 1)\) and *P*(*u*_{j} → *X*) are the probabilities that *u*_{j} is to be \(1\) and *X*, respectively. The ancillary input is prepared to a fiducial bit *α* in the classical case or state |*α*〉 in the quantum case. It is flipped or unchanged as successively passing through *u*_{0} and *u*_{1}. The output is measured with an outcome \({m}_{k}\in \{0,1\}\). Then, Bob’s guess *y*_{κ} at Alice’s secret numbers *x*_{κ} is made such that \({y}_{\kappa }={m}_{\kappa }\oplus \alpha \) for each input *κ*. Note that this DM algorithm is universal in the sense that it realizes all possible guesses *y*_{κ} of Bob (for more details, see Table in Fig. 2 and/or Sec. S1-A of the Supplementary Material).

Here the probabilities \(P({u}_{j}\to \mathrm{1)}\) and *P*(*u*_{j} → *X*) (*j* = 0, 1) refer to the DM preferences^{6}. For example, if \(P({u}_{j}\to \mathrm{1)}\) is larger than \(1/2\), Bob (or his brain) prefers setting \({u}_{j}\to 1\) to *u*_{j} → *X*. We can represent these probabilities as (for *j* = 0, 1)

where hint \({h}_{j}\in [\,-\,1/2,1/2]\). Note that the hints are not always informative^{20}; for instance, a decision maker may acquire some hint fabricated with malicious, which we say poor. We thus need to characterize the quality of given hints, which we represent by a hint vector **h** = (*h*_{0}, *h*_{1})^{T}. We categorize hint vectors into “good” and “poor.” A hint vector **h** is categorized to good if, by using it, Bob can improve his winning probability. Otherwise, it is to poor.

We consider and compare the machinery DM processes of two types, classical and quantum. The classical DM (cDM) is defined using the classical elements for the ancillary channel: the input *α* is a classical bit number and *u*_{j} (*j* = 0, 1) is applied in a classical probabilistic way, namely, either to be \(1\) or to be *X* based on Eq. (1). In this case, the probabilistic application of *u*_{j} is represented by a stochastic evolution matrix,

On the other hand, the quantum DM (qDM) runs with the quantum state |*α*〉 and the application of *u*_{j} is represented by a unitary matrix,

Here we note that the additional degree of freedom, i.e., the quantum phase *ϕ*_{j}, is introduced in the unitary operation. The qDM utilizes these phases with the directional condition **h** = (*h*_{0}, *h*_{1})^{T} in addition to the individual components of **h**, according to the following rules:

where Δ = |*ϕ*_{1} − *ϕ*_{0}| is defined as the absolute difference of the quantum phases *ϕ*_{j}. These rules were built based on the postulate of “rational” game player (Bob, here) who can find the best algorithm by utilizing all available resources—which is often referred to as the theory of rationality^{6}. Actually, the rules in Eq. (4) optimizes Bob’s DM algorithm and thus maximizes his winning probability (see Sec. S1-B of the Supplementary Material). It is worth noting that we run the DM process quantum-mechanically, even though we keep the game strategies classical, such as Alice’s secret numbers and Bob’s guesses.

### Quantum sensitivity to additional hints

In such settings, we investigate quantum sensitivity to the given hints. First, we indicate that qDM allows Bob to enjoy much higher winnings with good hints. More specifically, by analyzing Bob’s average score Ξ (often-called the average payoff function — a term from game theory)^{6}, we arrive at

where the indices *C* and *Q* denote classical and quantum, respectively. Bob’s quantum score differentiates from the classical by the amount of Γ. We set *α* = 0 and *ξ* = 1 for a sake of simplicity. As in Eq. (S15), the Supplementary Materials, the differential

and clearly this leads to an advantage for qDM since Γ ≥ 0. If the hints are poor, on the other hand, qDM makes it more difficult to make the correct guesses. In the worst case [see Eq. (S16) in the Supplementary Materials],

This implies that the differential Γ becomes disadvantageous with the minus sign. Here, the most surprising fact is that, in qDM, Bob’s score exhibits an abrupt transition near the boundary between good and poor hints. For example, when the amounts of hints are small but non-zero, approximately Bob’s scores \({{\rm{\Xi }}}_{Q}\simeq +\,{\rm{\Gamma }}\) and \({{\rm{\Xi }}}_{Q}\simeq -\,{\rm{\Gamma }}\) for the good and poor hints, respectively, if the hints are symmetric, i.e., |*h*_{0}| = |*h*_{1}| = |*h*|, where the symmetric hints were taken into account as hints are usually dependent and correlated. As the symmetric hint comes to zero, more explicitly, Bob’s quantum score

where we used Ξ_{C} → 0 as |*h*^{(G,P)}| → 0. Here, *h*^{(G)} and *h*^{(P)} respectively stand for the good and poor symmetric hints. This abrupt score-transition (which resembles quantum phase transition)^{21} is a representative of the quanum sensitivity. Without any hints, i.e., |**h**| = 0, however, there is no gain or loss from the quantum assumption (for detailed calculations and theoretical analyses, see Sec. S1-B of the Supplementary Material).

### Experimental demonstration

Now, we design linear-optical settings for the proof-of-principle experiments, as drawn in Fig. 3. To simulate the qDM algorithm, we use single-photon light as the ancillary system input^{22}. Horizontal and vertical polarizations of the photon represent the qubit signal, such that \(|H\rangle \leftrightarrow |0\rangle \) and \(|V\rangle \leftrightarrow |1\rangle \). The unitary operations *u*_{j} (*j* = 0, 1) can be realized as combinations of half-wave-plate (HWP) and quarter-wave-plate (QWP). More specifically, *u*_{0} is composed of HWP(*ϑ*_{0})-QWP(*φ*_{0})-QWP(*χ*), and *u*_{1} is realized by one HWP(*ϑ*_{1}). Here, *ϑ*_{0}, *φ*_{0}, and *ϑ*_{1} are controllable rotation angles of the wave plates. The angle *χ* is fixed to be \(\pi /4\). Such a setting for qDM can generate all possible outputs for Bob’s guesses by controlling the wave plate angles, according to the following rules:

We then also simulate the cDM algorithm for comparison. For cDM, we prepare the thermal state of light as the ancilla input, leaving no room for unexpected quantum effects on the cDM. The signal bits are also represented by the light polarization, i.e., \(H\leftrightarrow 0\) and \(V\leftrightarrow 1\). However, in such a cDM, application of the given hint **h** is limited without the ability to fully exploit the quantum superposition; i.e., the directional information of **h** cannot be encoded. The classical operations *u*_{j} (*j* = 0, 1) can thus be implemented with only HWPs placed at either *ϑ*_{j} = 0 (for \({u}_{j}\to 1\)) or \({\theta }_{j}=\pi /4\) (for *u*_{j} → *X*), probabilistically, based on Eq. (1) (see Fig. 3b).

The experiments are carried out for all of Alice’s possible strategies, i.e., her choices of the secret bits *x*_{0} and *x*_{1}. In the experiments, we evaluate Bob’s average scores Ξ_{C} and Ξ_{Q} by repeating 10^{4} games for a given **h** = (*h*_{0}, *h*_{1})^{T}. We perform such evaluations by varying *h*_{0} and *h*_{1} from −0.5 to 0.5 at 0.01 increments. Thus a given hint **h** is good or poor for the secret bits *x*_{κ}, which holds for both in cDM and qDM. We represent the experimental results of Ξ_{C} and Ξ_{Q} as density-plots in the space of *h*_{0} and *h*_{1} (see Fig. 4). The average scores Ξ_{C} and Ξ_{Q} are undifferentiated at each corner point, whereas they differentiate, if far from the corners, maximally near to the origin, i.e., when the hints are very small. At the origin, i.e., *h*_{0} = *h*_{1} = 0, the average scores are to be zero in both DMs. Here, note that in the qDM, Bob’s average score Ξ_{Q} is discontinuous as crossing the axes, while Ξ_{C} is continuous everywhere in the cDM. Meanwhile, Ξ_{Q} is always higher (lower) than Ξ_{C} for good (poor) hints. To see these features conspicuously, we also perform experiments for the symmetric hints, i.e., |*h*_{j}| = |*h*|, along the blue and red dashed lines in Fig. 4a,b. These lines, which are toward the best and worst hints from the origin, are represented by *h* whose sign is positive (negative) when its quality is good (poor). The result clearly shows the abrupt score-change between the quantum advantage Γ and disadvantage −Γ (see Fig. 5). All these results indicate that qDM exhibits higher sensitivity between the boundary for good and poor hints, as described in Eq. (8).

Analyzing further, we consider the decoherence effects, which cause degradation of the quantum superposition, during the process of qDM. Here, without loss of the generality, the signals transmitted in the ancillary system in qDM are assumed to be decohered (mathematically, a decay of off-diagonal elements of the density matrix of the signal state \(\hat{\rho }\))^{23} at a rate of 1 − *γ* ≤ 1. Then, it is predicted that the decoherence effectively results in a smaller hint-sensitivity with

With this prediction, the experiments are carried out for symmetric hints |*h*| = |*h*_{0}| = |*h*_{1}|. Here, the hints are assumed to be good. The experiments are repeated for 10^{4} games to evaluate the average score Ξ_{Q}. The experimental results clearly confirm the prediction: the quantum advantages become smaller with increasing decoherence rate *γ* (see Fig. 6). However, note that even in this case, qDM still has more advantages than cDM, unless the quantum superposition is completely washed out. This result is also quite remarkable, since quantum properties usually disappear rapidly with very small decoherence.

## Discussion

We performed the study of quantum decision making, adopting a two-player game where one player (Bob) tries to guess the secret bit numbers chosen by the other player (Alice). In this game, we focused on Bob’s decision process in terms of his guesses. Primarily, we attempted to investigate novel quantum features, assuming that Bob (i.e., the decision maker) uses a pre-programmed algorithm by which favorable quantum properties can be exploited. As the main result, we demonstrated both theoretically and experimentally that the quantum aspects make the choosing tendency stronger in the quantum, establishing the high sensitivity at the boundary of opposite hint quality. This quantum feature originates from the fact that quantum DM is able to find additional way of using the quality (i.e., the directional condition) of the given hint **h**, while the classical DM uses only the amount (i.e., the size). Through the further experiments and analyses, we also demonstrated that the high hint-sensitivity persists before the quantum coherence is completely destroyed. Our study is expected to provide the insight to understand some DM processes at the quantum level.

This work is also intimate to the issue whether novel quantum features exist in a classical game. The issue has been regarded to be negative, while quantum features in quantum games have been discussed mostly by considering quantum strategies^{12,13}. To attack the issue, on the other hand, we proposed to employ the machinery that plays (or simulates) the decision processes made by the rational players. We hope that the present work would accelerate the studies on potential applications, including quantum cryptography^{24,25} and quantum machine learning^{26}.

## Methods

### Preparation of the ancillary input

In the qDM experiments, we prepared a heralded single-photon state (*H*-polarized) as the ancillary input. Photon pairs are produced in type-II spontaneous parametric down conversion (SPDC) using a periodically poled KTiOPO_{4} crystal (length, 10 mm) and a continuous wave pump laser (wavelength, 401.5 nm). The vertically polarized photons reflected by a PBS are used as trigger photons, and the transmitted horizontally polarized photons are used as signal photons. Signal photons were counted only when the trigger photons were detected. Here, if this post-selection is not applied, the signals toward the gate operations are the thermal state with supper-Poissonian photon statistics. In the cDM experiments, the thermal state of light was employed as the ancillary input, which does not possess the quantum coherence (see Fig. 3).

### Experimental simulation of decoherence

Effectively, the decoherence can be simulated in the experiments by setting the relative phases of the states either as 0 or as *π* (a phase flip) randomly with a ratio of 1 − *γ*/2 to *γ*/2. Then, statistically, the state *ρ* can be described as^{23}

## References

Zsambok, C. E. & Klein, G. (Eds)

*Naturalistic decision making*. (Psychology Press, 2014).Tversky, A. & Kahneman, D. Judgment under uncertainty: Heuristics and biases.

*Science***185**, 1124–1131 (1974).Tversky, A. & Shafir, E. The disjunction effect in choice under uncertainty.

*Psychol. Sci.***3**, 305–310 (1992).Blackhart, G. C. & Kline, J. P. Individual differences in anterior EEG asymmetry between high and low defensive individuals during a rumination/distraction task.

*Pers. Individ. Dif.***39**, 427–437 (2005).Resulaj, A., Kiani, R., Wolpert, D. M. & Shadlen, M. N. Changes of mind in decision-making.

*Nature***461**, 263 (2009).González-Díaz, J., García-Jurado, I. & Fiestras-Janeiro, M. G.

*An Introductory Course on Mathematical Game Theory*, vol. 115 of Graduate Studies in Mathematics (American Mathematical Society, 2010).Meyer, D. A. Quantum Strategies.

*Phys. Rev. Lett.***82**, 1052–1055 (1999).Eisert, J., Wilkens, M. & Lewenstein, M. Quantum Games and Quantum Strategies.

*Phys. Rev. Lett.***83**, 3077–3080 (1999).Lee, C. F. & Johnson, N. F. Efficiency and formalism of quantum games.

*Phys. Rev. A***67**, 022311 (2003).Deutsch, D. Quantum theory of probability and decisions.

*Proc. R. Soc. A***455**, 3129 (1999).Pothos, E. M. & Busemeyer, J. R. A quantum probability explanation for violations of ‘rational’ decision making.

*Proc. R. Soc. B***276**, 2171 (2009).van Enk, S. J. & Pike, R. Classical rules in quantum games.

*Phys. Rev. A***66**, 024306 (2002).Aharon, N. & Vaidman, L. Quantum advantages in classically defined tasks.

*Phys. Rev. A***77**, 052310 (2008).Bang, J., Ryu, J., Pawłowski, M., Ham, B. S. & Lee, J. Quantum-mechanical machinery for rational decision-making in classical guessing game.

*Sci. Rep.***6**, 21424 (2016).Mihatsch, O. & Neuneier, R. Risk-sensitive reinforcement learning.

*Mach. Learn.***49**, 267–290 (2002).Lee, D. Game theory and neural basis of social decision making.

*Nat. Neurosci.***11**, 404 (2008).Molleman, L., Van den Berg, P. & Weissing, F. J. Consistent individual differences in human social learning strategies.

*Nat. Commun.***5**, 3570 (2014).Ghahramani, Z. Probabilistic machine learning and artificial intelligence.

*Nature***521**, 452 (2015).Lungo, A. D., Louchard, G., Marini, C. & Montagna, F. The Guessing Secrets problem: a probabilistic approach.

*J. Algorithm.***55**, 142–176 (2005).Lehner, P. E., Mullin, T. M. & Cohen, M. S. When Should a Decision Maker Ignore the Advice of a Decision Aid?

*arXiv preprint arXiv*:*1304*.*1515*(2013).Park, C. Y.

*et al*. Quantum macroscopicity measure for arbitrary spin systems and its application to quantum phase transitions.*Phy. Rev. A***94**, 052105 (2016).Naruse, M.

*et al*. Single-photon decision maker.*Sci. Rep.***5**, 13253 (2015).Audretsch, J.

*Entangled Systems*:*New Directions in Quantum Physics*. (John Wiley & Sons, 2008).Werner, A. H., Franz, T. & Werner, R. F. Quantum cryptography as a retrodiction problem.

*Phys. Rev. Lett.***103**, 220504 (2009).Kaniewski, J. & Wehner, S. Device-independent two-party cryptography secure against sequential attacks.

*New J. Phys.***18**, 055004 (2016).Clausen, J. & Briegel, H. J. Quantum machine learning with glow for episodic tasks and decision games.

*Phys. Rev. A***97**, 022303 (2018).

## Acknowledgements

The authors thank Jaewan Kim and Byoung Seung Ham for valuable discussions. J.B. thanks Marcin Wieśniak, Wiesław Laskowski, Marcin Pawłowski. This research was supported through the National Research Foundation of Korea (NRF) grant (No. 2014R1A2A1A10050117 and No. 2016R1A2B4014370) and the Institute for Information and communications Technology Promotion (IITP-2018-2015-0-00385), funded by the Korea government(MSIT), Korea. This research was also implemented as a research project on quantum machine learning (No. 2018-104) by the ETRI affiliated research institute. J.B. acknowledge the support of the R&D Convergence program of NST (National Research Council of Science and Technology) of Republic of Korea (No. CAP-18-08-KRISS).

## Author information

### Authors and Affiliations

### Contributions

J.B., J.L. and K.G.L. developed the theoretical idea. J.S.L. and K.G.L. performed the experiments, and J.B. and J.L. provided theoretical support. J.B. and J.S.L. wrote the manuscript. All the authors contributed to analysis of the results. The first two authors (J.S.L. and J.B.) contributed equally to this work and can be regarded as the main authors.

### Corresponding authors

## Ethics declarations

### Competing Interests

The authors declare no competing interests.

## Additional information

**Publisher’s note:** Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Supplementary information

## Rights and permissions

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

## About this article

### Cite this article

Lee, JS., Bang, J., Lee, J. *et al.* Experimental Demonstration on Quantum Sensitivity to Available Information in Decision Making.
*Sci Rep* **9**, 681 (2019). https://doi.org/10.1038/s41598-018-36945-5

Received:

Accepted:

Published:

DOI: https://doi.org/10.1038/s41598-018-36945-5

## Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.