Making Mistakes Saves the Single World of the Extended Wigner s Friend Experiment. Szymon Łukaszyk 1. Abstract

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1 Making Mistakes Saves the Single World of the Extended Wigner s Friend Experiment Szymon Łukaszyk bstract n Extended Wigner s Friend gedankenexperiment devised by Daniela Frauchiger and Renato Renner consisting of a quantum system containing an agent who draws conclusion, upon observing the outcome of a measurement of a quantum system prepared by another agent lead them to conclude that no single-world interpretation of quantum mechanics can be self-consistent. Jeffrey ub in turn proposed to consider this experiment in a context of many-body entangled quantum system. In this paper I show that the logical paradoxes of such an experimental setup stem from the very assumption of perfection of the actions based on inherently imperfect measurements. In a remarkable [FR6] publication Daniela Frauchiger Renato Renner proposed an Extended Wigner s Friend gedankenexperiment that leads to a logical paradox, which forced them to give up the view that there is one single reality. Since the many-worlds interpretation of quantum mechanics has paradoxical features of its own ([EV93]) and is not only counterfactually indefinite but factually indefinite as well ([G]), such a conclusion has been questioned. uthors of [HW6] for example argue that the contradiction disclosed in [FR6] expresses the incompatibility of collapse and unitarity resulting in different formal descriptions of a measurement. They argue that it is the formal description of the subjective collapse that leads to contradicting predictions for Wigner s-friend experiments and the conflict becomes manifest only if encapsulated observers can communicate. uthor of [S7] goes even further arguing that this incompatibility makes quantum theory self-contradictory by itself. ut the proposed Extended Wigner Experiment is not just a brain teaser. In a recent update ([FR7]) authors of [FR6] provide a game-theoretic viewpoint of their experiment, arguing that if it was realised as a game between a gambler and a casino, both parties would likely end up in a dispute, putting forward contradicting arguments based on quantum-mechanical reasoning that should have been accepted as two alternative facts about what the outcome of the coin toss was. Recently ([J7]) it has been proposed to consider the whole experimental system of [FR6] as big many-body quantum system that has evolved unitarily to a combined entangled state, which I briefly describe below with lice, ob and Charlie taking the role of a single super-observer. szymon@patent.pl

2 . Extended Wigner s Friend Experiment with Global Super-Observer In her perfectly isolated lab lice prepares a first qubit in a state h t () 3 3 and nondestructively measures it in an orthonormal basis h, t to find lice prepares then a second qubit h or t. iff iff h t () lice hands the second qubit to ob residing in another perfectly isolated lab 3. ob nondestructively measures the received second qubit in an orthonormal basis, to find or. super-observer Charlie measures the first qubit of lice in a first Hadamard basis ok h t, fail h t and the second qubit of ob in a second Hadamard basis ok, (3) fail (4) The qubits of lice and ob might be two photons that have been linearly polarised while being measured by lice in her basis h, t and ob in his basis,, the first one currently passing through another polariser (3) rotated at an angle 4 h, t and the second one relative to currently passing through another polariser (4) rotated at an angle 4,. with respect to ccording to [J7] from Charlie s perspective lice and ob, their measuring instruments and all the systems in their laboratories that become entangled with the measuring instruments in registering and recording the outcomes of the first qubit measurement and the second qubit measurement, including the entangled environment, are just two big many-body quantum systems that have evolved unitarily to a combined entangled state that may be expressed in terms of the original bases of lice h, t and ob, as Gruesomely describing the process of the preparation of the final state of the second qubit as a black box one might say that lice puts a cat into a Schrödinger s box provided with all the necessary equipment (a Geiger counter, a flask of hydrocyanic acid, etc.) so that it becomes a superposed living-dead creature if she measures t, while if she measures h she kills the cat before putting its dead body into the box. She then delivers the box to ob who will open it to find whether the cat is alive or dead. 3 This is the only action that violates otherwise perfect isolation of lice s and ob s labs.

3 h t t (5) This state does not contain the h pure state and by simple algebraic substitutions may also be expressed in terms of Charlie s bases ok, fail and ok, fail as 9 fail fail fail ok ok fail ok ok (6) S 4 fail fail ok fail t (7) S h fail h ok t fail h t fail (8) S Therefore the state h (5), the state ok (7), the state t ok and S S ok ok is superposed in the state (6) with a probability amplitude of and thus possible S to be measured. ll these four conditions contradict each other, as the state ok is entangled (is inseparable) with the state by virtue of equation (7), the state ok is entangled with the state h by virtue of equation (8), the state h must not be measured together with the state by virtue of equation (5) and yet both states virtue of equation (6). ok and ok may be observed with frequentist probability by The super-observer methodology proposed in [J7] enables to immediately observe the contradiction in the [FR6] remarkable gedankenexperiment. ut this is based on the assumption that from Charlie s perspective, lice s and ob s laboratories have evolved unitarily to a composite entangled state of (5) from two single qubits, after lice has handed the second qubit to ob. ut this assumption is not entirely correct.. Unitary evolution of the system The state may be expressed in a computational basis as (5a) where h and t. Obviously represents an entangled state as it may not be presented as a tensor product of its components, that is the first and the second qubit (regardless of the version of the second qubit prepared by lice). Nonetheless from Charlie s super-observer perspective, quantum register containing the first qubit and the second qubit or after it has 3 3 3

4 been prepared by lice would initially contain either two separable pure states measured h ), (if she (9a) or four separable pure states t,,, (if she measured or t ) 6 6 (9b) In order to effect the unitary evolution on such quantum registers in order to bring both of them to the entangled state that would be the same regardless of the initial state or lice must use some unitary operations. To this end, in a practical embodiment she might obviously employ quantum gates. t If she measures h she may use first the following unitary matrix (a) while if she measures t she may use first the following unitary matrix t (b) 4

5 Then she may use the following unitary matrix R () to receive R from or. h t t There are obviously infinitely many possibilities of unitary transformations that would bring or to, as groups of unitary 4x4 matrices acts transitively on the unit vectors in 4. The t above forms are therefore just exemplary. It should be noted however that even if these forms were the only transformations of this kind they fulfil the conditions of the [FR6] experiment and correctly model the unitary evolution of the composite system, provided they are correctly used. Indeed, such an approach assumes that in order to arrive at lice must firstly define the outcome of her measurement of the first qubit, secondly record the outcome with the basis of this definition to be either h or t and eventually act according to this definition in some predefined manner, in this case by applying an appropriate unitary matrix transformation. ut what if lice does not define the reality she perceives and does not act according to perceived definitions? Or what if lice makes a mistake in her measurement of the first qubit and recognises it as h even though it should have been truly measured as t or vice versa and due to this mistake she applies a wrong transformation or t? R or R to a given input quantum register t We might assume for example that there is no agent (no lice) performing the actual measurement of the first qubit in lice s isolated lab but just some mechanism applying randomly (with uniform probability of ½) 4 any of the two composite matrix transformations R or R to t the initial separable state or. The following table lists the results of such a mechanism t operation. initial state initial state probability applied transformation and resultant state R 3 R t th R t 3 t ht R applied transformation probability resultant probability t t 6 4 Or even alternately. 5

6 In this scenario state would be obtained with frequentist probability of, state frequentist probability and state with probability. The wrong states 6 ht 3 (expressed in terms of the original bases of lice and ob) are ht with th and th 3 h 3 t t th (5a).847 h.537 t.36 t and h h t t ht.684 h.357 h.684 t.36 t ut substituting h fail t from (3) the states Sth and th 5 fail t t fail.3334 t.36 t become ht (5b) (7a).96 fail fail.996 t (7b) Sht while substituting fail from (4) the states and become th ht Sth.847 h.6498 t.95 fail t (8a).96 h.865 t.3333 h fail.95 t fail (8b) Sht and neither of these is a proper quantum state, as it does not normalise to unity 5. Therefore though the combined entangled quantum state that lice exposes to ob may be the same regardless of the outcome of her measurement of the first qubit, it may not be a quantum state at all, if she is only allowed to make a mistake while affecting the unitary evolution of this state. In this setup the logical paradoxes of [FR6] stem from the very assumption of lice s infallibility. 3. Discussion Very roughly speaking authors of [FR6] argue to have arrived at the contradiction by letting the initial state of the second qubit that ob receives from lice depend on a random value (the first qubit) generated by lice. ut the contradiction is not a result of letting the initial state to depend on something that was generated in some physical process (understood as letting the initial state of the second qubit to unitarily evolve) but the result of creating the second qubit by lice. nd there is no universal law of physics that would force her to create this second qubit this way or the other. 5 Sth , etc. 6

7 nd it seems therefore that quantum theory may not be applied to model an experimenter who herself uses (understood as acts upon the measurements) quantum theory. The experimenter who uses quantum theory in this manner faces inherent limitations as all experiments are burdened with some random error caused, for example, by the change of experiment conditions, tolerance of experimental tools, etc. [SL3]. nd no oolean macrolevel providing a suitable structure of alternative possibilities enabling to talk about an event as definitely occurring or not occurring [J7] is required but just the level of (de)finite (including counterfactually (de)finite) precision 6 of this event occurrence and some predefined action taken even if this event occurred (or not occurred) outside of this precision boundaries. Without such actions quantum mechanics, though reveals its weirdness yet remains self-consistent 7. The contradiction in the [FR6] experiment manifests itself not due to the mere registration of the measurement outcome but due to acting upon this registration. I am not advocating here for a consciousness causes collapse theories, as human beings are not essential to the measuring process and measurements may be carried out by automated equipment, while their results may be stored in a computer memory. To perform the gedanken experiment of [FR6] one might consider employing a quantum computer (as authors of [FR7] propose themselves) or some automated experimental setup based on single-photon sources, polarisers and photon detectors. Yet all these tools are prone to errors 8 and practical implementation of this experiment requires further research. Surely an experimenter with a modicum of free will is the most fit to assign labels to states of quantum particles, which must have their own share of this valuable commodity [CK6]. ut the logical contradictions of the quantum mechanics manifest if we assume that these labels universally and unmistakably describe the world that evolves unitarily around us all the time. Erare humanum est. cknowledgements I would like to thank nuj Dawar for his illuminating explanations concerning unitary transformations and Renato Renner for patiently answering my poorly, to say the least, posed questions. References [S7] nthony Sudbery, Single-world theory of the Extended Wigner s Friend Experiment. arxiv: v3 [quant-ph] 4 Mar 7. [HW6] Veronika aumann, rne Hansen, and Stefan Wolf, The measurement problem is the measurement problem is the measurement problem. arxiv:6.v [quant-ph] 3 Nov 6 6 uthors of [HW6] define a story as anything that can be stated which is finite and can represent, for instance, real numbers only up to some finite precision. ut the same holds true for every definition. The word define itself denotes drawing a boundary around some term thus making it finite at least in the context of a statement of the meaning of this term. nd definition of a phenomenon is always prone to errors and evolves itself (take Wikipedia for example). esides the labels that we apply to conglomerates of quantum particles are definite and real while the quantum particles themselves reside in infinite and complex Hilbert space. 7 cf. No-communication theorem for example % polarizing efficiency is typical. 7

8 [CK6] [EV93] [FR6] [FR7] [G] [J7] [SL3] John Conway, Simon Kochen, The Free Will Theorem. arxiv:quant-ph/6479v pr 6 vshalom C. Elitzur(a) and Lev Vaidman, Quantum mechanical interaction-free measurements. arxiv:hep-th/935v 5 May 993 Daniela Frauchiger Renato Renner, Single-world interpretations of quantum theory cannot be self-consistent. arxiv:64.74v [quant-ph] 5 pr 6 Daniela Frauchiger Renato Renner, Can quantum mechanics be considered consistent?, Guy laylock, "The EPR paradox, ell s inequality, and the question of locality". merican Journal of Physics. merican ssociation of Physics Teachers (PT). 78 ():. ISSN doi:.9/ (). Jeffrey ub, Why ohr Was (Mostly) Right. arxiv:7.64v [quant-ph] 5 Nov 7 Szymon Łukaszyk, New concept of probability metric and its applications in approximation of scattered data sets. Computational Mechanics, March 4, Volume 33, Issue 4, pp