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The Measurement Postulates of Quantum Mechanics Are Operationally Redundant

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The Measurement Postulates of Quantum Mechanics Are Operationally Redundant ARTICLE https://doi.org/10.1038/s41467-019-09348-x OPEN The measurement postulates of quantum mechanics are operationally redundant Lluís Masanes1, Thomas D. Galley 1,2 & Markus P. Müller2,3 Understanding the core content of quantum mechanics requires us to disentangle the hidden logical relationships between the postulates of this theory. Here we show that the mathe- matical structure of quantum measurements, the formula for assigning outcome probabilities ’ 1234567890():,; (Born s rule) and the post-measurement state-update rule, can be deduced from the other quantum postulates, often referred to as “unitary quantum mechanics”, and the assumption that ensembles on finite-dimensional Hilbert spaces are characterized by finitely many parameters. This is achieved by taking an operational approach to physical theories, and using the fact that the manner in which a physical system is partitioned into subsystems is a subjective choice of the observer, and hence should not affect the predictions of the theory. In contrast to other approaches, our result does not assume that measurements are related to operators or bases, it does not rely on the universality of quantum mechanics, and it is independent of the interpretation of probability. 1 Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK. 2 Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada. 3 Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria. Correspondence and requests for materials should be addressed to T.D.G. (email: [email protected]) NATURE COMMUNICATIONS | (2019) 10:1361 | https://doi.org/10.1038/s41467-019-09348-x | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-09348-x hat sometimes is postulated as a fundamental law of Postulate (composite systems). The joint pure states of systems Wphysics is later on understood as a consequence of Ca and Cb are the rays of the tensor-product Hilbert space more fundamental principles. An example of this Ca Cb. historical pattern is the rebranding of the symmetrization pos- Postulate (measurement). Each measurement outcome of tulate as the spin-statistics theorem1. Another example, according system Cd is represented by a linear operator Q on Cd satisfying to some authors, is the Born rule, the formula that assigns 0 ≤ Q ≤ I, where I is the identity. The probability of outcome Q on probabilities to quantum measurements. The Born rule has been ψ Cd derived within the framework of quantum logic2–5, taking an state 2 is – – operational approach6 9, and using other methods10 14. But all PðQjψÞ¼hψjQjψi : ð1Þ these derivations assume, among other things, the mathematical structure of quantum measurements, that is, the correspondence A (full) measurement is represented by the operators correspond- ; :::; between measurements and orthonormal bases, or more gen- ing to its outcomes Q1 Qn, which must satisfy the normal- Pn erally, positive-operator valued measures15,16. I ization condition Qi ¼ . Taking one step further, the structure of measurements toge- i¼1 ther with the Born rule can be jointly derived within the many- The more traditional formulation of the measurement worlds interpretation of quantum mechanics (QM)17,18 and the postulate in terms of (not necessarily positive) Hermitian – framework of entanglement-assisted invariance19 22. But these operators is equivalent to the above. But we have chosen the derivations involve controversial uses of probability in determi- above form because it is closer to the formalism used in the nistic multiverse scenarios, which have been criticized by a presentation of our results. – number of authors21 31. Also, these frameworks require the Postulate (post-measurement state-update). Each outcome is universality of QM, meaning that the measurement apparatus represented by a completely-positive linear map Λ related to the and/or the observer has to be included in the quantum descrip- operator Q via tion of the measuring process. While this is a meaningful Λ ψ ψ ψ ψ ; assumption, it is interesting to see that it is not necessary, as tr ðj ih jÞ ¼ h jQj i ð2Þ proven in the present article. for all ψ. The post-measurement state after outcome Λ is In this work we take an operational approach, with the ΛðjψihψjÞ notions of measurement and outcome probability being pri- ρ ¼ : ð3Þ mitive elements of the theory, but without imposing any par- trΛðjψihψjÞ ticular structure on them. We use the fact that the subjective A (full) measurement is represented by the maps corresponding choices in the description of a physical setup in terms of Pn Λ … Λ Λ operational primitives must not affect the predictions of the to its outcomes 1, , n whose sum i is trace-preserving. theory. For example, deciding to describe a tripartite system i¼1 A·B·C as either the bipartite system AB·C or as A·BC must not If the measurement is repeatable and minimally disturbing33,34 ; :::; modify the outcome probabilities. Using these constraints we then Q1 Qn are projectors and the above maps are of the form Λ ρ = ρ “ characterize all possible alternatives to the mathematical i( ) Qi Qi, which is the standard textbook projection structure of quantum measurements and the Born rule, and we postulate”. Below we prove that the “measurement” and “post- prove that there is no such alternative to the standard mea- measurement state-update” postulates are a consequence of the surement postulates. This theorem has simple and precise first three postulates. premises, it does not require unconventional uses of probability theory, and it is independent of the interpretation of prob- The structure of mixed states. Mixed states are not mentioned in ability. A further interesting consequence of this theorem is that the standard postulates of QM, but their structure follows the post-measurement state-update rule must necessarily be straightaway from the measurement postulate (1). Recall that a that of QM. mixed state is an equivalence class of indistinguishable ensembles, ψ and an ensemble ( r, pr) is a probability distribution over pure states. Note that the notion of distinguishability depends on what Results the measurements are. For the particular case of quantum mea- The standard postulates of QM. Before presenting the main surements (1), the probability of outcome Q when a source prepares state ψ with probability p is result we prepare the stage appropriately. This involves reviewing r X r some of the postulates of QM, reconstructing the structure of PðQjðψ ; p ÞÞ ¼ p PðQjψ Þ¼trðQρÞ ; mixed states from them, and introducing a general character- r r r r ð4Þ r ization of measurements that is independent of their mathema- where we define the density matrix tical structure. X Postulate (states). To every physical system there corresponds a ρ ψ ψ : Cd ¼ prj rih rj ð5Þ complex and separable Hilbert space , and the pure states of r the system are the rays ψ 2 PCd. It will be convenient to use the notation Cd both for Hilbert This matrix contains all the statistical information of the ensemble. Therefore, two ensembles with the same density matrix spaces of finite dimension d, and also for countably infinite- dimensional Hilbert spaces which we denote by C1. This are indistinguishable. Theimportantmessagefromtheaboveisthatadifferent notation is justified, since all countably infinite-dimensional Hilbert spaces are isomorphic32. Analogously we use U(∞)to measurement postulate would give different equivalence classes 1 of ensembles, and hence, a different set of mixed states. Thus, in denote the unitary transformations of C . In this document we ψ Cd proving our main result, we will not assume that mixed states represent states (rays) by normalized vectors 2 . are of the form (5). An example of mixed states for a non- Postulate (transformations). The reversible transformations (for quantum measurement postulate is described in the section Cd example, possible time evolutions) of pure states of are the “Non-quantum measurement postulate violating associativ- ψ ψ ∈ unitary transformations 7!U with U U(d). ity”. A full classification of the sets of mixed states for non- 2 NATURE COMMUNICATIONS | (2019) 10:1361 | https://doi.org/10.1038/s41467-019-09348-x | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-09348-x ARTICLE quantum measurement postulates is given in Supplementary which represents the same measurement outcome Note 1. ψ ψ φ ; fφ;g ð Þ¼gð Þ ð9Þ Formalism for any alternative measurement postulate . Before ψ 2 PCd : proving that the only possible measurement postulate is that of QM, we have to articulate what “a measurement postulate” is in Next we consider local measurements in multipartite systems. general. In order to do so, we introduce a theory-independent In order to do so, it is useful to recall that the observer always has characterization of measurements for single and multipartite the option of describing a systems Ca as part of a larger system systems. This is based on the concept of outcome probability Ca Cb, without this affecting the predictions of the theory. In function (OPF), introduced in35 and defined next. order to do so, the observer needs to know how to represent the fi De nition (OPF). Each measurement outcome that can be OPFs of the small system F a as OPFs of the larger system F ab. observed on system Cd is represented by the function f : PCd ! This information is contained in the star product, defined in what ½0; 1 being its corresponding probability f ðψÞ¼Pðf jψÞ for each follows. d fi ⋆ pure state ψ 2 PC ; and we denote by F the complete set of De nition ( -product). Any pair of local OPFs, f 2Fa and d ? Cd fi g 2Fb, is represented as a global OPF ðf gÞ2Fab via the star OPFs of system .
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