aa r X i v : . [ qu a n t - ph ] A ug Is independent choice possible?
Hitoshi Inamori
Soci´et´e G´en´eraleBoulevard Franck Kupka, 92800 Puteaux, FrancePrevious academic affiliation:Centre for Quantum Computation, Clarendon Laboratory, Oxford University
August 31, 2016
Abstract
This paper questions the generally accepted assumption that one canmake a random choice that is independent of the rest of the universe.We give a general description of any setup that could be conceived togenerate random numbers. Based on the fact that the initial state ofsuch setup together with its environment cannot be known, we show thatthe independence of its generated output cannot be guaranteed. Someconsequences of this theoretical limitation are discussed.
Keywords:
Random number generation, Independence, Entangle-ment
We are ultimately able to make choices, independently of the world that sur-rounds us. This is free will , a belief which is so deeply rooted in our culture,that we generally take it for granted without even noticing. The assumptionthat one can make an independent decision is however crucial in many logicalreasonings and algorithms.Making a decision is a physical process. Being able to make a random choiceimplies that there is a physical setup that provides random outcomes, which canbe represented – without loss of generality – as numbers. Independent randomchoice implies the existence of a physical setup that provides random numbersthat are independent of the rest of the universe.It is already known that one can expand an initial random number thanks toquantum algorithms known as Private Randomness Expansion [1, 2, 3, 4], whoseaim is to increase (in entropy) randomness using untrusted devices. PrivateRandomness Expansion does however not deal with the generation of the initialrandom number, as it is proved [1] that one cannot generate random number1rom scratch with untrusted devices, but can only expand an initial randomnumber.The scope of this paper is different: we study the feasibility of actual gen-eration of random numbers using trusted devices, which does not assume thepresence of initial random numbers serving as a seed. Based on the law of quan-tum mechanics, and the fact that the state of the overall system including thesetup and its environment can never be known, we prove that such generationcannot be guaranteed to be truly independent of the rest of the universe.The paper is organized as follows: we start by considering a simple setupthat is usually believed to generate independent random numbers. We showhowever that such setup can be entangled with its environment prior to theexperiment, and as such, the random numbers returned by the setup can neverbe guaranteed in theory to be independent of the environment.We show that this result can be generalized to any physical setup that couldbe conceived to generate random numbers. Independence of random numbergeneration cannot be guaranteed, and we discuss the theoretical consequencesof such limitation.
Throughout this paper, our question will be the following: can we conceive asource of random binary number that is guaranteed to be independent of therest of the universe?The usual setup that comes to mind when one wants a random source of bi-nary number is the following. Consider a qubit and let {| i , | i} be its canonicalbasis. If we had a certified way of preparing the qubit in the state | + i where |±i = √ ( | i ± | i ), then we could measure such qubit in the basis {| i , | i} .If the state is observed in the state | i then we say that the setup generatedthe random number 0. If the state is observed in the state | i then the ran-dom number is 1. The process is guaranteed to generate a random bit that isindependent of the rest of the universe.But how do we prepare the state | + i in practice? The setup that is usuallyemployed works as follows: The measured qubit is prepared in some state,and is first observed in the {| + i M , |−i M } basis. We know that the measuredqubit is now in either the | + i M state or |−i M state, information which mustbe explicitly kept (otherwise the effect of the measurement can be cancelled, asshown in the Quantum Eraser experiment [5]). This state is then measured inthe {| i M , | i M } basis.The whole setup can be represented as in Figure 1: The qubit P representsthe outcome of the first measurement ( P stands for “projection”), whereby thefirst qubit is measured in the | + i M state or the |−i M state. We look at the valuereturned from the measurement of M , given the outcome of the measurementof P .Note that although we find it natural to describe the measurement of P asbeing prior to the measurement of M , it actually does not matter whether M ✈ H ❥ MP measurementmeasurement Figure 1: An usual setup for random bit generationis measured before or after P .With such a setup, we generally assume that measurement of M leads toa random binary outcome, that is completely independent of the rest of theuniverse. This belief is natural if one accepts that the initial state of the setup M ⊗ P is known, or at least, that the initial state of the setup is not entangledwith the rest of the universe.However, in theory, the initial quantum state of a given experimental setupis never known. One can choose arbitrarily the basis for the Hilbert spacedescribing the system under study, but one cannot guarantee that the systemunder study is not entangled with the rest of the universe. Said differently,there is no theoretical guarantee that a physical system under study had notbeen entangled at an earlier point with another part of the universe.Suppose for instance that the setup above was entangled initially with athird qubit, that is part of the rest of the universe and is not known to us. Let’sdenote this third qubit E (Figure 2). H ✈ H ❥ MPE measurementmeasurementmeasurement
Figure 2: The same setup taking into account the environmentSuppose now that the initial state of the resulting system is: | α i = 1 √ (cid:0) | i M ⊗ | i P ⊗ | i E + | i M ⊗ | i P ⊗ | i E | i M ⊗ | i P ⊗ | i E + | i M ⊗ | i P ⊗ | i E (cid:1) , (1)then, calculation shows that the final state of the system is, after going throughthe experimental setup: | α i = 1 √ (cid:0) | i M ⊗ | i P ⊗ | i E + | i M ⊗ | i P ⊗ | i E + | i M ⊗ | i P ⊗ | i E + | i M ⊗ | i P ⊗ | i E (cid:1) , (2)which is identical to the initial state and which can be rewritten as: | α i = 1 √ (cid:0) | i M ⊗ | i E + | i M ⊗ | i E (cid:1) ⊗ | i P + 1 √ (cid:0) | i M ⊗ | i E + | i M ⊗ | i E (cid:1) ⊗ | i P . (3)In other words, whichever result is observed for the projection qubit P ,the measured qubit M and the environment qubit E are perfectly entangled.Measurement outcome at M and E will be perfectly correlated if we use thesame measurement basis for M and E .We see that the proposed setup M ⊗ P which is usually used as a randomand independent source of random number, cannot actually be guaranteed tobe independent of the rest of the universe. We saw in the section above that a setup that was believed to generate inde-pendent random choices could actually not be guaranteed to be independent.Our next question is: can any setup in general produce random numbersthat are guaranteed to be independent of the rest of the universe?However complex such a setup can be, it can be described as follows: thesetup is a physical system, made of two subcomponents M and P , that interactand evolve possibly in a most general and complex manner. The subcomponents M and P are then observed. We denote by U the unitary map describing theevolution of M and P . The outcome of these observations are denoted m and p respectively. The generation of the random number is deemed successful if p is part of a pre-agreed set S corresponding to “successful” outcomes. In such acase, the random number is given by a pre-agreed deterministic function of m and p , f ( m, p ). The setup is represented in Figure 3.Now, whatever the size of the setup, we assume that it is sufficiently smallcompared to the entire universe. Consequently we can define a quantum system4..... U ...... MP measurement m measurement p f ( m, p )if p ∈ S Figure 3: General setup for random number generation E that is part of the rest of the universe, described by a Hilbert space of thesame dimensionality as M . We denote by V = U ⊗ I E the unitary operator thatapplies U on the system M ⊗ P and leaves E unchanged (Figure 4).Consider the following state for the combined physical system made of M , P and E : | Ψ i = N X k | k i M ⊗ | φ i P ⊗ | k i E . (4)where | φ i P is a state for P that leads to a successful fixed outcome p ∈ S , and {| k i M } and {| k i E } are bases for M and E , respectively. The number N is anormalisation factor.By construction, if the initial state of the entire system M ⊗ P ⊗ E is inthe state | Ψ i = V − | Ψ i , then the experiment returns the state | Ψ i in which M and E are perfectly entangled, and with which the measurement at P leadsto a successful result p ∈ S . Measurement of E leads to the same outcome asthe measurement of M , and the outcome f ( m, p ) can be completely deducedfrom the measurement outcome of E . Therefore, there exists at least one initialstate | Ψ i with which the proposed setup does not return independent randomnumbers.As such no physical setup can guarantee generation of independent randomnumbers. We have demonstrated the following: Property 1
No generation of random choices can be guaranteed to be indepen-dent of the remaining part of the universe.
We have proven that – at least in theory – no random choice can be guaranteedto be independent of the rest of the universe. This is mainly due to the fact that,if we consider the state of any physical setup together with its environment, thensuch state is unknown. One could argue that, statistically, no relevant impact5..... U ......... MPE | Ψ i V − | Ψ i Figure 4: General setup taking into account environmentcan be expected from these entanglements that are only theoretical possibilities.One could also argue that the likelyhood that the overall state of the universeis such that it precisely introduces dependence between the generated randomnumber and its environment is dim. However, we cannot rely on statisticalargument when independence of random choices is itself under question.The fact that we cannot guarantee independence of random choices has manyconsequences, few of them are discussed below.
Positive-Operator Valued Measure [6] is considered to be the most general for-mulation of a measurement in quantum mechanics. It allows the introductionof probability weights between different projective measurements, as if it waspossible to choose randomly between these projective measurements. However,we have just seen that the random choice between these different projectivemeasurements cannot be guaranteed to be independent of the rest of the uni-verse. Therefore, the mathematical description using POVM is misleading, inthe sense that it introduces the false belief that independent random choice ispossible. Our view in this paper is that the most fundamental description ofmeasurement remains the law based on projection of quantum states onto themeasurement basis.
The experimental violation of Bell’s inequality [7] is generally considered as aconfirmation that laws of nature cannot be described by local hidden variables.However, existence of local hidden variable is not the only assumption which isnecessary in the derivation of Bells inequality [8, 9]. In particular, Bell assumes6hat random and independent decisions can be taken at the two distant loca-tions. We suggest in this paper that, what could be forbidden by the violationof the Bell’s inequality is actually not the existence of local hidden variable,but rather the true independence between the observed system and the waythat system is observed. In other words, violation of Bell’s inequality could beinterpreted as the demonstration that one cannot make decisions that are fullyindependent of the environment.
Many algorithms, such as the Monte Carlo integration, error-correction andcryptography to name a few, rely on the ability to generate random numbersthat are independent. The theoretical possibility that some remaining part ofthe universe may be correlated to these random numbers may, or may not berelevant. In any case, for theoretical completeness, one should not take forgranted that sources of random choices are truly independent of the rest of theuniverse. Impact of potential dependence of such sources with the rest of theuniverse should be analyzed.
We generally assume that one can take a random decision (using if necessary adevice as described in the case study above), and that such decision can be takenindependently of all the rest of the universe. This is free-will , this commonlyshared belief that if one wants, one is able to decide by oneself, independentlyof its environment.We have shown in this paper that no random choice can be guaranteed to beindependent of the rest of the universe. For practical purposes, the theoreticalpossibility that one part of the universe may be correlated with our experimentalsetup may have little importance. This said, if we believe that our currentuniverse expanded from a single common state, then we cannot dismiss thepossibility that any experimental setup under study is entangled with anotherobservable part of the universe.
The author thanks Norbert L¨utkenhaus for his comments on the initial versionof this paper and for pointing to the existing work on Private RandomnessExpansion. 7 eferenceseferences