Quantifying quantum correlations: how to measure quantum weirdness

To design powerful new technologies such as quantum computers, and to assess the role of quantum effects in biology and physiology, we need to understand a mysterious feature of quantum theory called quantum correlations (QCs). These are a kind of connection or relation, unknown in our everyday world, that may exist between microscopic particles as a result of their combined quantum state. Objects sharing QCs can cause one another’s states to change without any communication. Finding reliable ways to quantify the strength and character of these correlations in systems of interest is crucial for understanding them. Several such measures have been proposed, but many questions remain unresolved.

Diagram of classification of quantum states and correlations.
Figure 1. Classification of quantum states and quantification of correlations. The pink area (T) represents the set of all quantum states. The blue area (S) represents the non-entangled states, also known as separable states. The yellow region (C) represents classical states, states with no quantum correlations. The gray area (P) represents the set of classical product states, states with no correlations at all. The black dot represents the state we are interested in. To find out how much of each of the different correlations this state has, one measures the shortest distance between the dot and a state in the region that lacks the correlation. For example, to measure entanglement, one measures the distance between the dot and the closest separable state, represented by the red dashed line. (Source: Asma Al-Qasimi)

Quantum mechanics is not a very intuitive subject. Unlike everyday (classical) objects, which have definite states (e.g., your cup of tea is either in your hands or on the table, but not both), a quantum object can be in multiple states simultaneously, including multiple different positions. If you attempt to find where it is, you will change its configuration. There is no guarantee that the position after measurement is the same as that before. With two (or more) objects related by a quantum correlation (QC), things become even more bizarre: operations on one object can alter the other’s configuration even if they are isolated from each other. Albert Einstein famously called this “spooky action at a distance.” The challenge for theorists is to understand the nature of QCs and to quantify them.

Besides being of innate interest because of their mysterious nature, QCs have very important applications, most notably in the field of quantum information. Using quantum systems, such as those made up of atoms, ions, or photons, will allow us to build quantum computers. Among many applications, these computers have the ability to factor tremendously large numbers much more efficiently than seems to be possible with conventional computers. The factoring of large numbers allows you to break certain security codes. The security of RSA encryption (named after the researchers Ronald Rivest, Adi Shamir, and Leonard Adleman) depends on the near impossibility of factoring large numbers, but, with quantum computers, this security will evaporate. Conversely, quantum systems also can create unbreakable codes. The major goal of quantum cryptography is to guarantee ways of communicating with absolute secrecy. In addition, since many biological and physiological processes occur on the molecular level (and very small molecules can exhibit quantum behavior), it may be that QCs play a role in these processes. Until we have a better understanding of the nature of QCs and how to accurately quantify them, however, it is premature to make strong claims about their role in biology.

Progress in exploiting quantum systems for computation and communication is not as fast as we would like. Thus, a goal for researchers studying QCs is to spell out how experimentalists and technologists might use them for such applications. For example, we need to specify just how much and what kind of QC is needed to achieve efficient factoring. With this in mind, theorists have developed a variety of measures of QCs, each with different properties or characteristics.

Talking about quantifying QCs could lead us to discuss many types of quantum systems made up of various numbers of particles. Here I will focus on the simple case of two objects called qubits (pronounced like “Q-bits”), which suffices to illustrate the different approaches to finding quantities or functions that capture quantum correlations. A qubit is a two-state quantum system, the quantum analogue of a classical bit, or binary digit. A bit can be in two possible states, usually labeled zero and one. A qubit can be a zero or a one, but it can also be in a linear combination of these two states. We need two qubits because that is the smallest size system to exhibit a QC.

For years, one of the most popular approaches to quantifying QCs was an information-based approach.[1] I will describe it in the context of the correlation known as quantum entanglement (QE). Many people think of QE as the inability to express the state of a system as a product of the states of the smaller components of the system. In the case of two entangled qubits, A and B, you cannot say that qubit A is in one quantum state and B is in another. Instead, the system must be described as some linear combination of such joint states. A state with zero QE is called separable because it can be separated into a product of simple states.

Quantum systems can have different levels of QE. Basically, the more the system can exhibit the ‘spooky action at a distance’ behavior, the more entangled it is. Certain states have the maximum possible entanglement. Examples are the Bell states, named for John Stewart Bell, one of the founders of this field.[2] These states are then used as a reference to compare to the entanglement of other states of interest, revealing the strength of the latter. In the case of our two qubits in some specific entangled state, the idea is to create many realizations of the pair, with different people holding the A’s and the B’s. Then, by performing certain permitted operations on the qubits and by sending classical information between the holders of the qubits, one tries to create as many Bell states as possible. The number of Bell states that can be extracted from a large number of realizations gives us information about how much entanglement our state of interest has. For instance, if we can create 100 Bell states from 300 instances of our pair, this suggests the pair’s state has one third the entanglement of a Bell state. The very popular quantification of QCs known as concurrence is based on these ideas.[3][4]

I mentioned earlier that measurements typically disturb quantum systems in ways that do not occur with classical systems. This key difference can be exploited to distinguish between quantum and classical correlations. This approach led Harold Ollivier and Wojciech Zurek to discover quantum discord (QD).[5] Their idea, for a two-qubit system, was to imagine performing a measurement on one of the two qubits and assess whether the measurement would produce a disturbance. How does one see whether there is a disturbance? By looking at a function of both qubits. Ollivier and Zurek looked at a function known as the mutual information function, which tells us about the information shared between the two qubits. The quantum discord is the amount by which the mutual information has changed after the measurement. People became excited about QD because, even when entanglement disappears from a quantum system, there can still be discord. Entanglement is a very fragile property in experimental quantum systems, and loss of entanglement is a really big problem for those who are building quantum computers. The research into QD revived the hope of alternative approaches to using entanglement in technological applications. For a two-qubit system, the general relationship between QE and QD is not simple,[6] but there are some important observations. In particular, although QD can exist in a system with zero QE, QE cannot exist in a system with zero QD. We do not yet fully understand the nature of QCs to explain this observation.

Another interesting way to capture correlations (including QCs) of a state of interest is to measure the ‘distance’ between this state and the closest state that lacks the correlation that we are trying to measure.[7][8] To illustrate this concept, consider Figure 1. Here, we are looking at the space of all quantum states (technically speaking, density matrices) and classifying subclasses based on the lack of correlations, whether classical or quantum. (For two qubits, this state space actually has 15 dimensions, so I have taken a certain amount of artistic license!) Our state of interest is represented by the black dot. If we wanted to know how much entanglement is in the system, we measure the distance between the dot and the closest separable state (i.e., the closest state that has zero entanglement). A quantity called dissonance is defined as the distance between this separable state and the closest classical state (classical states have no quantum correlations). Kavan Modi and his colleagues argue[8] that discord, QD, is the quantity that encompasses all possible QCs. They interpret QE and dissonance as distinct subclasses of QD. Others (including me) think that perhaps QE and QD are just two different approaches to quantify or capture the same set of correlations.[6]

You might be wondering what I mean by measuring distance in this context. There are many possible candidate functions that satisfy certain general mathematical properties that are essential for them to be acceptable as measures of QCs.[7][8]

The idea of measuring distances is neat and has a nice geometric representation, and a schematic diagram (such as the figure) makes it appear straightforward. In practice, however, such distances are not easy to calculate. The states do not necessarily occupy nice, known, simply shaped regions as in the diagram. To find the shortest distance, one must consider all the possible states with the requisite property in the multidimensional space of all states. That can be a very time consuming numerical problem. Moreover, how do we know for sure that discord is the embodiment of all classes of QC? How do we know there are actually different classes? If so, how can we tell how many we have? Not only for this case, but also with other approaches, we are expecting the mathematics of quantum theory to tell us about the ‘spooky action at a distance’ behavior.

Although the various approaches to measuring QCs are interesting, and they definitely are helpful in studying the evolution of these correlations, we do not have a good feel for the nature of QCs. We do know that they can give us power in building useful technologies, but where is this power coming from? Which type of QC—QE, QD, or something else—is essential? Considering quantum mechanics is not a very intuitive subject and that we do not yet have very satisfying answers, I think that it is crucial that theorists who want to come up with quantifications of QCs do so with experimentalists’ results in mind. This keeps them in touch with reality. After all, physics is an experimental science.

References

  1. C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, Concentrating partial entanglement by local operations, Phys. Rev. A 53, pp. 2046–2052, 1996.
  2. J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, 2nd ed., Cambridge University Press, Cambridge, 2004.
  3. S. Hill and W. K. Wootters, Entanglement of a pair of quantum bits, Phys. Rev. Lett. 78, pp. 5022–5025, 1997.
  4. W. K. Wootters, Entanglement of formation of an arbitrary state of two qubits, Phys. Rev. Lett. 80, pp. 2245–2248, 1998.
  5. H. Ollivier and W. H. Zurek, Quantum discord: a measure of the quantumness of correlations, Phys. Rev. Lett. 88, p. 017901, 2001.
  6. A. Al-Qasimi and D. F. V. James, Comparison of the attempts of quantum discord and quantum entanglement to capture quantum correlations, Phys. Rev. A 83, p. 032101, 2011.
  7. V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Quantifying entanglement, Phys. Rev. Lett. 78, pp. 2275–2279, 1997.
  8. K. Modi, T. Paterek, W. Son, V. Vedral, and M. Williamson, Unified view of quantum and classical correlations, Phys. Rev. Lett. 104, p. 080501, 2010.

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