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What really happens upon quantum measurement?[n eeds revision] References are more fully listed in my Phys Rev A paperPhys Rev A paper Art Hobson Prof Emeritus of Physics University of Arkansas

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ABSTRACT Measurement causes the measured quantum system to entangle non-locally with the measurement apparatus. Quantum theory, and non-local two-photon interferometry experiments, show that the locally observed states of both subsystems are mixtures of their eigenstates, while the unitarily-evolving composite global state, which can be accessed only by comparing after-the-fact records from the two local subsystems, evolves as a coherent superposition of correlations rather than of subsystem eigenstates. Thus, nature violates the eigenvalue-eigenstate link: Entanglement causes the subsystems to exhibit mixed state outcomes (such as a dead-or-alive Schroedinger’s cat, rather than dead-and-alive), but the subsystems do not actually collapse into the corresponding eigenstates. Instead, the subsystems remain entangled.

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OUTLINE I.What’s the problem? II.An enlightening experiment. III.A proposed solution

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I. WHAT’S THE PROBLEM? Consider a 2-state (qubit) system S, Hilbert space H S spanned by |s1>, |s2>. Superposition |> S = α|s1> + β|s2>. Ex: 2-slit exp, observed at the slits. |s1> & |s2> represent an electron passing through slit 1 or 2. Apparatus A “measures” S, obtaining a “definite outcome” (eigenvalue) s1 or s2. Measurement postulate: Upon measurement, S instantly “collapses” into either |s1> or |s2>. The problem: If the world obeys Q physics, we should be able to derive this collapse from the Schrodinger eq. for A & S. How to do this?

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Let’s set up the problem: (See e.g. Schlosshauer 2007) Assume A has states |ready>, |a1>, |a2> ∊ H A such that: |ready>|s1> → |a1>|s1> and |ready>|s2> → |a2>|s2> --“ideal non-disturbing meas.” Linearity of the Schrodinger evolution implies |ready>|> S → α|a1>|s1> + β|a2>|s2> Note that the meas process entangles S and A. Thus the correlations between S and A are non-local (Gisin 1991). This turns out to be crucial.

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Apparent contradictions: |> = α|a1>|s1> + β|a2>|s2> (measurement state, MS) appears to describe a macro superposition of SA with superposed states |a1>|s1> & |a2>|s2>. Ex: If A is Schrodinger’s cat, and S is a radioactive nucleus that kills the cat if it decays, then MS appears to be a superposition of dead & alive. But that’s absurd. Where’s the collapse, to |a1>|s1> or |a2>|s2>? Such a collapse would be non-linear, contradicting the linear Sch. eq. Most experts think the problem is solvable only by altering the quantum fundamentals.

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When SA is in the MS, neither S nor A is superposed. E.g. S cannot be described by a state of the form γ|s1>+δ|s2>. Proof: see Phys Rev A paper.Phys Rev A paper So cat is not predicted to be dead & alive. This mistake results from neglect of the non-local connection between S and A. Entangled states are nonlocal (Gisin 1991). Exactly what is superposed in the MS? What interferes? One thing immediately wrong with these conclusions:

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II. AN ENLIGHTENING EXPERIMENT The Rarity-Tapster & Ou (RTO, 1990) exp. As we’ll see, there is a problem in the foundations. RTO demonstrates that non-locality is the key to understanding the meas prob. We must distinguish between two kinds of states of entangled systems: local states and the global state. RTO exp shows precisely what is superposed in the MS --It’s not your ordinary superposition!

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Exp of Rarity/Tapster, & Ou et al (RTO) 1990. Two entangled photons, A & S. The exp, which used beam splitters, variable phase shifters, and photon detectors, is equivalent to the following 2-photon double slit exp: A source sends entangled photon pairs through two Mach-Zehnder interferometers:

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Source of two entangled photons photon A path a1 path a2 x mirror y photon S path s2 path s1 With no entanglement, this would be two 2-slit exps: states (|a1>+|a2>)/√2 and (|s1>+|s2>)/√2. Interference fringes at both screens. With entanglement, the photons are in the MS |> = (|a1>|s1> + |a2>|s2> )/√2. Each photon “measures” the other! The RTO exp is a probe of the MS …with variable phases!

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Results: In a series of trials, neither screen shows any sign of interference or phase dependence: x & y are distributed randomly; no sign of coherence or superposition. The reason: Each photon acts as a “which path” detector, decohering both, causing each to come through only 1 slit. Locally (i.e. at S’s screen & A’s screen), each photon is described by an incoherent mixture with density operators ρ S = (|s1>

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Where does the coherence go? --It can’t vanish, because the Schrodinger evolution is “unitary.” Ans: the correlations become coherent. When coincidences of entangled pairs are detected at A’s and S’s screens: coincidence rate ∝ cos( A - S ) = cos(y-x). Photon A strikes it’s screen randomly at x, and photon S then strikes its screen in an interference pattern around x! And vice- versa. This certainly seems non-local, and in fact the results violate Bell’s inequality. y-x is proportional to the difference of the two phases A - S.

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What are “coherent correlations”? If the phase difference is A - S = 0, 2π, …, A and S are correlated: ai occurs iff si occurs. When A - S =π, 3π, 5π,..., A and S are anti- corr: ai occurs iff si does not occur. When A - S =π/2, 3π/2, 5π/2, …, A and S are not at all correlated. This is an interference of correlations between states, rather than the usual single-photon interference of states. A - S =0, 2π, 4π M-Z interferometer:

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Bell’s theorem implies…. …that the results are truly non-local: Cannot be explained by “prior causes” or by “causal communication.” If S’s phase shifter changes, the outcomes on A’s (and S’s) screen are instantly (i.e. faster than light) altered. Aspect (1982) tested these predictions (but with photon polarizations): The results confirmed violation of Bell’s ≠ and the observed changes showed up at the distant station sooner than a lightbeam could have gotten there.

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III. PROPOSED RESOLUTION The RTO exp shows that, when A & S are in the MS, the local results for A alone and for S alone are correctly described by the incoherent mixtures ρ S = (|s1>

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In fact, all local observations at S and A must show no sign of S-A correlations. Here’s why: S and A are non-locally connected. Thus all info about S-A correlations must be “camouflaged from” local observers of S & A. The reason: If an observer could, by observing S alone, detect any change when A is changed, A could send an instant signal to S across an arbitrary distance. Special relativity (Einstein causality) does not allow this.

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In fact Ballentine 1987 and Eberhard 1989 show that quantum probabilities do just what’s needed: When A alters φ A, only the correlations change—the statistics of the outcomes at S don’t change. RTO exp confirms this. This protection of Einstein causality is a remarkable and delicate feature of quantum entanglement. It is key to the measurement problem. Thus, to a local observer of S, all correlations with A must be “invisible.” Such correlations can be made known to S only by comparing outcomes at S and A— i.e. only via the global state. Thus all physical facts at S or at A must be fully described by the local states.

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In other words … ρ S = (|s1>

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Solution to “problem of outcomes:” The standard quantum axioms imply that, when 2 systems are entangled in the MS, they exhibit definite values s1 or s2 and a1 or a2. However, S and A are not actually in the corresponding eigenstates. They are in the MS. That is, S and A appear to be in the corresponding eigenstates, but they retain non-local (and thus unobservable locally) correlations between them. Thus the postulated collapse, into the corresponding eigenstates, doesn’t occur at the time when S and A become entangled in the MS.

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Returning to the RTO exp: RTO separates the problem of outcomes from the problem of irreversibiity. The MS describes S & A after they are emitted from the source but before either photon impacts its screen. The MS is still reversible, in principle. Upon impact, the coherence of the MS transfers to the environment (the screens) as described by Zurek 1981. This leaves S & A in mixtures ρ S & ρ A --definite outcomes that are now irreversible i.e. macroscopic & permanent. This locks in specific outcomes. A x y S

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Physics 2170 – Spring 20091 Some interesting aspects of quantum mechanics The last homework is due at 12:50pm.

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