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COMMENTARY | INSIGHT entanglement Vlatko Vedral Recent advances in quantum reveal the deep connections between entanglement and , many-body theory, quantum and its link to macroscopicity.

uantum physics started with the outcome is completely uncertain. But In fact, entanglement also serves as an ’s ‘act of desperation’, consider that tossing two coins becomes less order parameter characterizing quantum Qin which he assumed that uncertain. Indeed, quantum mechanically, phase transitions7, and there is growing energy is quantized in order to explain the state of two coins could be completely evidence that quantum topological phase the intensity profile of the black-body known, whereas the state of each of the coins transitions can only be understood in radiation. Some twenty-five years is still maximally uncertain. terms of entanglement8. A quantum phase later, , , In theory, this transition is a macroscopic change driven , Erwin Schrödinger and leads to negative conditional . by a variation in the of a many- wrote down the complete laws When it comes to quantum coins, as we body system at zero temperature9. But, of . A pertinent question know the outcome, two predictable tosses in contrast to an ordinary phase, no local then immediately came up — and was have zero . However, if we only order parameter can distinguish between subsequently hotly debated by the founding toss one coin, the outcome is completely the ordered and the disordered topological fathers of quantum physics: what features uncertain and therefore has one unit of phases10. For instance, because the change of quantum theory make it different entropy. If we were to quantify the entropy from non-magnetic to magnetic behaviour from classical ? Is it Planck’s of the second toss, given that the first has constitutes an ordinary phase transition, quantization, Bohr’s complementarity, been conducted, we would come up with we can check whether an ordinary phase Heisenberg’s or the one negative bit — that is, the entropy is magnetic by measuring the state of just superposition principle1? of two tosses minus the entropy of one one . However, a topological phase Schrödinger felt that the answer was toss: 0 – 1 = –1 bit. transition cannot be characterized by a local none of the above. In some sense, each of It is precisely because of such parameter — it requires an understanding of these features can also be either present or peculiarities that the pioneers of quantum the global entanglement of the whole state. mimicked within : energy physics considered entanglement weird and This is good news for stable encoding can be coarse-grained classically — by counterintuitive. However, after around of quantum information. The idea is to use brute force if nothing else; waves can be twenty years of intense research in this area, topological phases as quantum memories11. superposed; and complementarity and we are now accustomed to entanglement This is precisely because topological states uncertainty can be found in the trade-off and, moreover, as we learn more about it are gapped (that is, the energy gap between between the knowledge of the wavelength we discover that entanglement emerges in the ground and excited states is finite) and and position of the wave. But the one unexpected places. no local noise can kick the topological state effect Schrödinger thought had no classical Negative entropies have a physical out of the protected subspace. The ground counterpart whatsoever — the characteristic meaning in thermodynamics. My colleagues states are also degenerate, meaning that trait of quantum physics — is entanglement2. and I have shown3 that negative entropy there are different states with the same The reason entanglement is so refers to the situation where we can erase level of robustness that can be used to counterintuitive and presents a radical the state of the system, but at the same encode information. departure from classical physics can be time obtain some useful work from it. In Quantum information theory has nicely explained in terms of modern classical physics we need to invest work also expanded our understanding of quantum information theory mixed with in order to erase information — a process entanglement in other areas. Exciting some of Schrödinger’s jargon. The states known as Landauer’s erasure4, but quantum recent work focuses on ways of quantifying of quantum systems are described by mechanically we can have it both ways. This entanglement. The most fruitful general idea what Schrödinger called ‘catalogues of is possible because the system erasing the is to quantify entanglement by measuring information’ (psi-wavefunctions). These information could be entangled with the how different quantum states are from catalogues contain the probabilities for all system that is having its information erased. their best possible classical approximations. possible outcomes of the In that case, the total state could have zero But, there are many non-equivalent ways we can make on the system. Schrödinger entropy, so it can be reset without doing of capturing this difference, which leads thought it odd that when we have two work. Moreover, the eraser now also results to a great deal of ongoing research12. For entangled physical systems, their joint in a zero-entropy state and so it can be used instance, nonlocality13, which, strictly catalogue of information can be better to obtain one unit of work. speaking, means that no local realistic model specified than the catalogue of each Furthermore, we realized that can be found to explain the outcomes of individual system. In other words, the whole entanglement can exist in many-body measurements performed on entangled can be less uncertain than either of its parts! systems (with arbitrarily large numbers of systems, is not the same as inseparability. This is impossible, classically speaking. ) as well as at finite temperature5. This is because separable states are still Imagine that someone asks you to predict Entanglement can be witnessed using quantum states, whereas local hidden the toss of a single (fair) coin. Most likely macroscopic , such as the heat variables can be drawn from more general you would not bet too much on it because capacity (see ref. 6 for recent experiments). probabilistic theories. Moreover, quantum

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Box 1 | Charting the set of many-body physical states.

The set of many-body (in this case, many-) states is broadly divided into |00...0〉 |GHZ 〉 |W 〉 entangled and separable (or disentangled) states. Among separable states there is a –1 (|000...〉 〈000...| + |111...〉 〈111...|) tiny, nowhere-dense subset of zero discord, 2 which is called a classically correlated state. Classical states can be written as mixtures 1 〉 〈 of states |a a … a 〉, |b b … b 〉, where –2 (|00...0 00...0| 1 2 N 1 2 N +|++...+〉 〈++...+|) {|a1〉, |b1〉} form an orthogonal for qubit 1, {|a2〉, |b2〉} form an orthogonal basis for qubit 2 and so on for all up to N. The rest of the separable states contain non-zero discord. The geometry of this set |+++... + 〉 is not well understood (other than perhaps for the special case of two qubits). The set of all states is convex and so is the set of separable states. The global entanglement of a given state can be measured, for instance, by the relative , which is defined as the relative entropy BACKGROUND IMAGE: © ALEXEY PAVLUTS / ALAMY © ALEXEY PAVLUTS IMAGE: BACKGROUND S(σ || ρ) = tr(σ log σ – σ log ρ), between the state (σ) and the closest |001〉 + |010〉 + |100〉. With respect to global |1〉 are orthogonal). Mixtures of the states to it, ρ (ref. 5). Separable states are those entanglement, W states are more entangled |0000〉 and |++++〉, on the other hand, obtained by mixing product states of the than the GHZ states, and entanglement are not only classically correlated but

form ρ1 ⊗ ρ2 ⊗… ⊗ ρN. scales with the logarithm of the number also contain , although Some important classes of states are of qubits. they are not entangled. States like this are shown in the figure. The (many-qubit) Cluster states are even more entangled, thought to still be useful for some quantum GHZ states, |00…0〉 + |11…1〉, are close to where entanglement scales as N/2, with computations, but their exact power is not separable states because they always have N being the number of qubits. The most fully understood at present. one unit of entanglement independent entangled states are typical states, which The scaling of global entanglement of the number of qubits. In terms of are simply random states of N qubits. Their with the number of qubits does not global entanglement they are close to, for entanglement scales as N. The maximally necessarily reflect other measures instance, the product states |00…0〉, which mixed state (in which all possible qubit of entanglement, such as quantum are disentangled. Another example of a states are mixed with equal probability) is macroscopicity. This notion is designed product state is the state |++++〉, where completely disentangled and contains no to capture the state in Schrödinger cat every qubit is in a superposition of the basis correlations of any type. experiments, namely a superposition states |+〉 = |0〉 + |1〉. The ‘dephased’ GHZ states are mixtures of two, or more, macroscopically The is a symmetric of the states |00…0〉 and |11…1〉 and distinct quantum states (as in the superposition of states containing a are classically correlated, but have no GHZ state, which has a high degree fixed ratio of zeroes and ones, such as quantum discord (since the states |0〉 and of macroscopicity). nonlocality is just one possible way of product states — that has revolutionized cryptography and protocols such as violating Bell’s inequalities and one can some aspects of solid-state physics17. The quantum and super-dense always imagine more nonlocal theories idea is simple: given as few as twenty half- coding, then being able to quantify (see also the Review by Popescu in this spins (or qubits), we would need 220 bits to entanglement is crucial for characterizing issue14). In addition, there is the notion of store their — this is already the efficiency of such protocols. non-contextuality — the fact that different practically intractable for today’s classical Entanglement was initially thought quantum measurements do not necessarily computers. However, if we know that no two necessary for facilitating the speed-up in commute15 — and on top of that there groups of qubits are entangled by more than quantum computation. More precisely, if are many different types of entanglement one unit of entanglement, the size of the our quantum computer never contains more (bipartite, multipartite and global) and they approximation can be drastically reduced. than a certain finite number of entangled can all be quantified in different ways16,17. Take ten qubits versus the other ten qubits — qubits, then it can never be universal. This Why is the question of quantifying in principle we need 210 states to describe the is true for computers with registers that are entanglement important? First, if we can entanglement between the two subsystems, always pure. It is simple to understand why: estimate how close classical states are to but given that we know that they contain a universal computer should be capable quantum ones, we can tell how easy it would only one unit of entanglement, only two of preparing any physical state, but if be to simulate quantum states of many-body states for each subsystem will suffice. entanglement must always be bounded, then systems. This is the logic behind a very Second, if we think of quantum those states with more entanglement cannot powerful numerical method — called matrix entanglement as a resource in quantum be reached18. However, when it comes to

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mixed states (Box 1), there are examples entangled state can be (Box 1). The global be fully reconciled with quantum of computations that, despite requiring entanglement (measured, for instance, by entanglement21, and how far into the macro a small amount of entanglement (never the relative entropy between this state and domain do quantum effects really need to more than a single entangled bit), can still the closest separable state) is always 1 for be taken into account? This brings up a achieve an exponential speed-up relative GHZ states, no matter how many particles whole new set of exciting questions ranging to their classical counterparts. It has been are involved. from whether living could also suggested that these computers exploit a GHZ states are examples of states that exploit entanglement22 to whether quantum more general type of quantum correlation, are difficult to prepare in practice20, but are effects can ever have an impact in the known as quantum discord (see Box 1 and very easy to simulate classically. States that gravitational domain. ❐ ref. 12 for a review). Unfortunately, even the are difficult to simulate are, in general, the amount of discord is bounded during these ones where entanglement scales with the Vlatko Vedral is at the Clarendon Laboratory, computations, so it is hard to see how it number of particles in the system. This is University of Oxford, Oxford OX1 3PU, UK could make a difference. the case for typical many-body interacting and the Centre for Quantum Technologies, The third point is perhaps the most systems, as well as for cluster states used in National University of Singapore, intriguing as it touches on the issue of -based quantum computation. Singapore 117543, Singapore. ‘macroscopicity’ (see also the Review by On the other hand, cluster states do not e-mail: [email protected] Arndt and Hornberger in this issue19). usually exhibit quantum macroscopicity. Namely, is there a limit to how big a system Although it seems to be a problem, References 1. Kumar, M. Quantum: Einstein, Bohr and the Great Debate about can be and still exhibit sizable quantum the dichotomy between macroscopicity the Nature of (Icon Books, 2008). mechanical features? Here it seems and the amount of entanglement could 2. Schrödinger, E. Proc. Camb. Phil. 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Preprint at http://arxiv.org/ state, which is written as |00…0〉 + |11…1〉, Technologically, it is still not fully abs/1209.3745 (2012). where the state |0〉 signifies the atom in understood how far quantum computers 16. Horodecki, R. et al. Rev. Mod. Phys. 81, 865–942 (2009). 17. Eisert, J. et al. Rev. Mod. Phys. 82, 277–306 (2010). one location and |1〉 the atom in the other can be scaled up, nor can the full range of 18. Jozsa, R. & Linden, N. Proc. R. Soc. A 459, 2011–2032 (2003). location and there are 1018 zeroes and their applications be easily predicted. On 19. Arndt, M. & Hornberger, K. Nature Phys. 10, 271–277 (2014). ones. Based on the proximity of classical the fundamental side, the problem is how 20. Korsbakken, J. I., Wilhelm, F. K. & Whaley, K. B. Europhys. Lett. 89, 30003 (2010). states, it is not very entangled. In fact, it is to bridge the gap between the micro and 21. Dorner, R. et al. Phys. Rev. Lett. 109, 160601 (2012). just about as close to a classical state as an the macro domains. Can thermodynamics 22. Sarovar, M. et al. Nature Phys. 6, 462–467 (2010).

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Correction In the Commentary ‘’ (Nature Physics 10, 256–258; 2014) reference 6 contained a typographical error and should have read: Singh, H. et. al. New J. Phys. 15, 113001 (2013). This has now been corrected in the HTML and PDF versions after print 4 August 2014.

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