The entangled quantum world
§23.10 23.10
indeed, is the only connecting link between us that is capable of supporting the amount of ‘information’ that is required. The trouble with it, of course, is that it contains a stretch that extends 5 years into the past!
23.10 Quanglement I must make it very clear that I am not trying to give support to the idea that ordinary information can be propagated backwards in time (nor can EPR effects be used to send classical information faster than light; see later). That kind of thing would lead to all sorts of paradoxes that we should have absolutely no truck with (I shall return to this kind of issue in §30.6). Information, in the ordinary sense, cannot travel backwards in time. I am talking about something quite diVerent that is sometimes referred to as quantum information. Now there is a diYculty about this term, namely the appearance of the word ‘information’. In my view, the preWx ‘quantum’ does not do enough to soften the association with ordinary information, so I am proposing that we adopt a new15 term for it: QUANGLEMENT At least in this book, I shall refer to what is commonly called ‘quantum information’ as quanglement. The term suggests ‘quantum mechanics’ and it suggests ‘entanglement’. This is very appropriate. This is what quanglement is all about. Quanglement also does have something very much to do with information, but it is not information. There is no way to send an ordinary signal by means of quanglement alone. This much is made clear from the fact that past-directed channels of quanglement can be used just as well as future-directed channels. If quanglement were transmittable information, then it would be possible to send messages into the past, which it isn’t. But quanglement can be used in conjunction with ordinary information channels, to enable these to achieve things that ordinary signalling alone cannot achieve. It is a very subtle thing. In a sense, quantum computing and quantum cryptography, and certainly quantum teleportation, depend crucially on the properties of quanglement and its interrelation with ordinary information. As far as I can make out, quanglement links are always constrained by the light cones, just as are ordinary information links, but quanglement links have the novel feature that they can zig-zag backwards and forwards in time,16 so as to achieve an eVective ‘spacelike propagation’. Since quanglement is not information, this does not allow actual signals to be sent faster than light. There is also an association between quanglement and ordinary spatial geometry (via the connections between the Riemann sphere and spin, as pictured in Figs. 22.10, 22.14, 22.16), this association 603
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Non-linear crystal
Fig. 23.8 Parametric down-conversion. A photon, emerging from a laser, impinging upon a suitable ‘non-linear crystal’, produces a pair of entangled photons. This entanglement manifests itself in the EPR nature of the correlated polarization states of the secondary photons, but also in the fact that their 3-momentum states must sum to that of the incident photon.
being spatially reXected at a reversal of time direction, with interesting implications.17 It would take us too far aWeld to explore these in detail. One of the most direct uses of the idea of quanglement is in certain experiments where a pair of entangled photons is produced according to the process referred to as parametric down-conversion (see Fig. 23.8). This occurs when a photon, produced by a laser, enters a particular type of (‘non-linear’) crystal which converts it into a pair of photons. These emitted photons are entangled in various ways. Their momenta must add up to the momentum of the incident photon, and their polarizations are also related to one another in an EPR way, like the examples given earlier, above. In one particularly striking experiment, one of the photons (photon A) passes through hole of a particular shape as it speeds towards its detector DA . The other photon (photon B) passes through a lens that is positioned so as to focus it, appropriately, at its detector DB . The position of detector DB is moved around slightly as each photon pair is emitted. The situation is illustrated schematically in Fig. 23.9a. Whenever DA registers reception of photon A and DB also registers reception of B, the position of DB is noted. This is repeated many times, and gradually an image is built up by the detector DB , where only the positions of B are counted when simultaneously DA registers. The shape of the hole that A encounters is gradually built up at DB , even though photon B never directly encounters the hole at all! It is as though DB ‘sees’ the shape of the hole by looking backwards in time to the emission point C at the crystal, and then forwards in time in the guise of photon A. It can do this because the ‘seeing’ process in this situation is achieved by quanglement. This Xitting back and forth in time is precisely the kind of thing that quanglement is allowed to do. Even the strength and positioning of the lens can be understood in terms of quanglement. To obtain the lens location, think of a mirror placed at the emission point C. The lens (a positive lens) is placed so that the image of the hole, as reXected in this mirror at C, is focused at the detector DB . Of course there is no actual mirror at C, but the quanglement links act as though reXected at a mirror, but they are reXected in time as well as space.[23.11] [23.11] See if you can give a fuller explanation of this, using quanglement ideas or otherwise.
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The entangled quantum world
§23.10 23.10
Non-linear crystal C
A
DA
B DB (a)
Beam splitter
C A B (b)
Fig. 23.9 Transmission of an image via quantum effects. (a) Entangled photons A, B are produced by parametric down-conversion at C. Photon A has to pass through a hole of some special shape to reach detector DA , while B passes through a lens, positioned so as to focus it at detector DB . Detector positions are gradually moved, appropriately in conjunction, and when they both register, the position of DB is noted. Repeated many times, an image of the hole shape is gradually built up by DB , where only those positions of B are counted when DA also registers. (This is schematically illustrated here by having, instead, DB as a fixed photographic plate that is only activated when DA registers.) Quanglement is illustrated by the lens positioning being determined as though C were a ‘mirror’ that reflects the photon backwards in time as well as in direction. (b) An alternative scheme using an adaption of the Elitzur– Vaidman bomb test of Fig. 22.6 (which is to be reflected in a horizontal line). The photographic plate at B receives the photon only when the photon ‘would have been stopped’ by the template at C, but actually took the lower route!
In case the reader Wnds this experiment far-fetched, I should make clear that this is a real eVect. It has been successfully conWrmed in experiments18 performed at the University of Baltimore, Maryland. Various other related experiments involving parametric down-conversion, which can be best understood in terms of quanglement, have also been carried out.19 On the other hand, the general type of situation illustrated in Fig. 23.9a might be regarded as not being ‘essentially quantum mechanical’. For one could envisage a device at C which simply ejects classical particles pairwise in the appropriate directions and, apart from the lensing, similar results could be obtained. We can remedy this by using a modification of the Elitzur– Vaidman set-up illustrated in Fig. 22.6 (reflected horizontally); see Fig. 23.9b. Now there is only one photon at a time. It can register at the 605
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photographic plate B only if the interference is destroyed when the alternative route for the photon would miss the hole at C. Now, let us look again at an ordinary EPR eVect, like the Stapp and Hardy examples considered earlier. In the ordinary application of the quantum R process, one imagines a particular reference frame in which there is a time coordinate t providing parallel time slices, each corresponding to a constant t value, through the spacetime. The normal procedure is to adopt the (nonrelativistic) viewpoint that, when one member of an EPR pair is measured, the state of the other is simultaneously reduced, so that a later measurement looks at a reduced (unentangled) state rather than an entangled state. This kind of description can be used, for example, in my speciWc EPR examples. Let us suppose that, from the point of view of a reference frame stationary with respect to the Sun, it is my colleague on Titan whose measurement takes place Wrst, some 15 minutes before my own measurement here on Earth. So, in this picture of things, it is my colleague’s measurement that reduces the state, and I subsequently perform a measurement on a particle with an unentangled state. But we might imagine that, instead, the whole situation is described from the perspective of some observer O passing by at great speed (say 23c) in the general direction from my colleague on Titan to me. From O’s viewpoint, I was the one who Wrst made the measurement on the EPR pair, thereby reducing the state, and it was my colleague who measured the reduced unentangled state (Fig. 23.10) (see §18.3, Fig. 18.5b). The joint probabilities come out the same either way, but O has a diVerent picture of ‘reality’ from the one that I and my colleague had before. If we think of R as a real process, then we seem to be in conXict with the principle of special relativity, because there are two incompatible views as to which of us Earth
Titan
O B⬘⬘ A⬘ A B B⬘ A⬘⬘
EPR source
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Fig. 23.10 Conflict between relativity and the objectivity of state reduction? Spacetime diagram of an EPR situation, with detectors on Earth and Titan and source closer to Titan than Earth. From the perspective of an inertial frame, stationary with respect to the Sun, the detector on Titan registers first (at B) and this reduces the state simultaneously (at B0 ) on Earth. Only later does detection on Earth take place (at A) of a state now unentangled (simultaneous with A0 on Titan). However, to an observer O, travelling towards Earth from Titan with very great speed, detection takes 00 place first on Earth (at A, simultaneous with A on Titan, according to the ‘sloping’ simultaneity lines of O) and Titan receives the reduced 00 unentangled state (at B, simultaneous with B on Earth).
The entangled quantum world
Notes
eVected the reduction of the state and which of us observed the reduced state after reduction. We may deduce from this that EPR effects, despite their seemingly acausal nature, cannot be directly used to transmit ordinary information acausally, which one might imagine could influence the behaviour of a receiver at spacelike separation from the transmitter. A reference frame can always be chosen in which it is the ‘reception event’ which occurs first, and the ‘transmitter’ then has only the reduced state to examine. It is ‘too late’, by then, for the entanglement to be used for a signal because it has already been destroyed by the state reduction. What is the quanglement perspective on these matters?20 See §30.3. On this picture, it is not correct to think of either measurement (mine or my colleague’s) as eVecting the reduction and the other (my colleague’s or mine) as measuring the reduced state. The two measurement events are on an equal footing with one another, and we think of the quanglement as providing a connection between these events which correlates the two. It makes no diVerence which event is viewed as being to the past of the other, for quanglement can equally be thought of as propagating into the past as propagating into the future. Not being capable directly of carrying information, quanglement does not respect the normal restrictions of relativistic causality. It merely eVects constraints on the joint probabilities of the results of diVerent measurements. Although quanglement is a useful idea in ‘making sense’ of this kind of puzzling quantum experiment, I am not sure how far these ideas can be carried, nor how precisely the eVects of quanglement can be delineated. The idea of quanglement certainly does not resolve the issue of quantum measurement, telling us little, if anything, about the circumstances under which R takes over from U. That issue will be addressed more fully in Chapters 29 and 30, especially in §30.12, but the precise role of quanglement in this in this is not yet very clear, to my mind. A more promising connection is with some of the ideas of twistor theory, and these will be examined brieXy in §33.2.
Notes Section 23.1 23.1. See Eddington(1929b); Mott(1929); Dirac (1932). Section 23.3 23.2. See Einstein et al. (1935); Schro¨dinger (1935b); also Afriat (1999). 23.3. In detail what it tells us about quantum computing is a subtle issue, however; see Jozsa (1998). 23.4. Perhaps the neatest and most widely quoted version of this inequality is that due to Clauser et al. (1969). It takes the form jE(A, BÞ E(A, D)j þ jE(C, B)
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þ E(C, D)j#2, where E(x, y) is the expectation value of agreement (E ¼ 1 for complete agreement and E ¼ 1 for complete disagreement) between the results of alternative measurements A, C for one component of the EPR pair and B, D for the other. 23.5. One of the most amazing is in Tittel et al. (1998). 23.6. See Stapp (1971, 1979); Hardy (1992, 1993). 23.7. See Nielsen and Chuang (2000) for a general discussion of such issues in entanglement. Section 23.4 23.8. See Note 23.5. Section 23.5 23.9. See Hardy (1992, 1993). 23.10. See Hardy (1993). Section 23.8 23.11. A scheme of this nature was eVectively adopted in my book Shadows of the Mind, §5.15, but without explicitly using ‘wedges’. Section 23.9 23.12. See Penrose (1989a). 23.13. Wooters and Zurek (1982). 23.14. See Jennewein et al. (2002). Section 23.10 23.15. See Penrose (2002a). 23.16. See Jozsa (1998). 23.17. See Penrose (1998). 23.18. See Shih et al. (1995). 23.19. See Gisin et al. (2003), for example, for a taste of this important area. 23.20. Compare with Aharonov and Vaidman (2001); Cramer (1988); Costa de Beauregard (1995); and Werbos and Dolmatova (2000).
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