Quantum entanglement: A modern perspective

The discovery of quantum teleportation by IBM researcher Charles Bennett and five collaborators in 1993 marks the starting point of the modern view. In quantum teleportation (see the article by Charles Bennett in Physics Today, October 1995, page 24), an experimentalist, Alice, wishes to send an unknown state $|s⟩=α|0⟩+β|1⟩$ of a two-level quantum system to another experimentalist, Bob, in a distant laboratory. The two-level system could refer, for example, to the polarization of a single photon, the electronic excitation of an effective two-level atom, or the nuclear magnetic spin of a hydrogen atom. Alice and Bob do not have the means of directly transmitting the quantum system from one place to another (for photons, this could be the case when using a high-loss optical fiber), but let us imagine that they do share an entangled state. Consider the case in which Alice and Bob each have one spin of a shared singlet state of two spin-½ particles $| Ψ − ⟩ = 1 / 2 ( | ↑ , ↓ ⟩ − | ↓ , ↑ ⟩ ) ,$ ⁠, also called an EPR pair. Alice can transmit her spin $|s⟩$ to Bob by performing a certain joint measurement on her spin state $|s⟩$ and her half of the EPR pair. She tells Bob the result of her measurement and, depending on her information, Bob rotates his half of the EPR pair to obtain the state $|s⟩$⁠. The teleportation protocol demonstrates that the resources of classical communication and the sharing of prior EPR entanglement are sufficient to transmit an unknown spin state $|s⟩$⁠. (For the experimental realization, see Physics Today, February 1998, page 18.)

The spin-singlet EPR state that Alice and Bob share in quantum teleportation is called a maximally entangled state. Even though the two spins together constitute a definite pure state, each spin state is maximally undetermined or mixed when considered separately. In mathematical terms, Alice’s local density matrix—obtained by tracing over Bob’s spin degrees of freedom, $Tr B ( | Ψ − ⟩ ⟨ Ψ − | )$—has equal probability for spin up and spin down. In keeping with Schrödinger’s understanding of entanglement, one measures the amount of entanglement in a general pure state $φ$ in terms of the lack of information about its local parts. The von Neumann entropy $S(ρ)=−Tr(ρlogρ)$ is used as a measure of that information. In other words, the entropy of entanglement $E$ of the pure state $φ$ is equal to the von Neumann entropy of, say, Alice’s density matrix $ρ= Tr B |φ⟩⟨φ|$⁠.

In the quantum teleportation scenario, we imagined, unrealistically, that Alice and Bob shared an EPR pair free of noise or decoherence. More generally, Alice and Bob have quantum systems that interact directly or through another mediating quantum system—like Rydberg atoms in a laser cavity that interact via photons, or two ions in an ion trap that interact through phonon modes of the trap.⁵ A related example of interest in quantum computation is an array of interconnected ion traps, each holding a small number of ions that are coupled by traveling photons or by ions that are moved between the traps.⁶ The interaction, or “quantum link,” between a pair of systems is subject to noise or decoherence through photon loss or heating of the phonons, for instance. For simplicity, assume that Alice and Bob’s local operations on the quantum systems—operations on the ions in a single trap, say—are perfect, and their exchange of classical information is also perfectly noise free. That idealization enables one to measure the strength of the quantum link between the systems.

An essential question is, Given unavoidable noise levels, is it possible to establish a strong quantum link—a set of pure EPR pairs, in other words—between two systems? If it is, then the noise is weak enough to permit the error-free exchange of quantum information between the systems, since the teleportation through the generated EPR pairs will be error free. That capability may come at a certain cost, determined by the amount of noisy interaction required to generate an EPR pair. If it is not possible to generate EPR pairs, that decoherence in the system imposes a fundamental limitation on our ability to perform quantum information processing.

The possibility of generating shared EPR entanglement in noisy environments is not only of interest in entanglement theory, but is crucial for the realization of long-distance quantum communication⁷ and possibly large-scale quantum computation. For example, it was recently shown⁸ that fault-tolerant quantum computation can be achieved in the presence of very high noise levels in the interaction link—a link can have an error rate of two-thirds—between quantum systems that are “small” in a particular sense, if one assumes that local quantum processing on each end is (almost) error free.

Pure quantum states have their entanglement quantified fairly intuitively by considering the degree of local “mixedness” or entropy. However, mixtures of entangled and unentangled states are murkier: Recognizing which mixtures are still entangled may be difficult. So, just what physical systems can we call “entangled”? An operational description—expressing entanglement in terms of its negation—is helpful. Suppose that Alice and Bob, working in their distant labs, each receive the same random number over the phone. Depending on the random number, each of them locally prepares a certain quantum state. The physical state of their whole system, expressed as a density matrix, typically exhibits correlations between the two systems. However, those correlations would be classical, since they arise from classical random numbers. A quantum state that can be prepared in this way over the phone is called “unentangled” or separable, and such a state can be mathematically expressed as a mixture of unentangled pure states (see figure 2). Conversely, a state is “entangled” if it cannot be prepared over the phone, but requires coherent interaction between the two systems or the transmission of superpositions of quantum states.

For mixed states, it is harder to establish a good measure of entanglement, since such a measure has to distinguish between entropy arising from classical correlations in the state—a state of thermal equilibrium, for example—and local entropy due to purely quantum correlations. Two measures of entanglement that have explicit physical meaning in the processing of quantum information have emerged from the quantum-link notion just described: the entanglement cost $E(ρ)$ of a quantum state and the distillable entanglement $D(ρ)$ of a quantum state, first defined in reference 9.

Assume that Alice and Bob have created, using their noisy link, many (⁠$n$⁠) shared copies of an entangled quantum state $ρ$⁠; we denote such a collection as $ρ ⊗ n$⁠. To distill some EPR pairs from those copies, Alice and Bob perform several rounds of local, error-free operations to their parts of the copies and communicate their measurements (or other classical data) to each other. Such a protocol is called entanglement distillation; figure 3 illustrates one round of such a scheme. The aim is to produce fewer states that are, however, more entangled than the initial ones. Ideally, the protocol produces nearly perfect maximally entangled EPR pairs in the limit of a large number of input states $ρ ⊗ n$ with $n$ → ∞. The distillable entanglement $D(ρ)$ is then the number of such EPR pairs that can be extracted per copy of $ρ$ in this asymptotic limit.

The reverse process also has physical meaning. What is the smallest number $k$ of EPR pairs that Alice and Bob initially need to create a set of $n$ copies of $ρ$ for $n$ → ∞ by local error-free operations? This asymptotic ratio $k/n$ is the second measure of entanglement, the entanglement cost $E(ρ)$⁠.

Attentive readers may have noticed a quirk in our notation: The formalism uses the same symbol $E$ to denote both the entanglement cost for general states and the entropy of entanglement for pure states. The notation coincidence is harmless since the creation cost of a pure state equals the local entropy of entanglement $E$⁠. Furthermore, for a pure state $φ$⁠, it turns out that $E(φ)=D(φ)$ (see box 1). Physically, this means that the process of entanglement dilution—converting EPR pairs into lesser entangled pure states $φ$—can be reversed without loss of entanglement. The reverse process is called entanglement concentration and it produces $D(φ)n=E(φ)n$ EPR pairs from an initial supply of $n$ states $φ$⁠.

For mixed states, $D$ is believed to be generically less than $E$⁠, which implies that the preparation of mixed states from EPR pairs is a process involving an irreversible loss of entanglement. Curiously, the $D<E$ conjecture has only been proven for some special classes of mixed states.¹⁰ 

In 1998, the Horodecki family of Gdańsk, Poland (father Ryszard and sons Paweł and Michał), identified a class of entangled states that exhibit an extreme form of irreversibility. They proved that no entanglement can be distilled (⁠$D$ = 0) from these “bound entangled states.”¹¹ And for a large set of states from that class, irreversibility was established by proving that entanglement is required to prepare the states $E$ > 0.

Consider the metaphor illustrated in figure 4. If EPR pairs were nodes connected by lines or strands that represent quantum correlations between particles, then one could think of mixed entanglement as entanglement in which the strands are simply mixed up. The mixing may make it hard to reconstruct which particle of Alice is entangled with which particle of Bob. Cutting a few strands reduces the clutter, but every line cut represents an EPR pair lost (compare this process with the distillation protocol in figure 3). Bound entangled states are those mixtures that are so thoroughly mixed up that every single line has to be cut to remove the noise or clutter from the system. But, when every line is cut, no entanglement remains to be distilled.

Because the modern theory of entanglement treats quantum states as physical resources for processing information, one might consider them hierarchically. A simple and ideal world would have only two classes of quantum states: unentangled, classically correlated states that are useless as a resource in quantum teleportation and don’t violate any Bell inequalities, and entangled states whose distillation rate $D$ measures their usefulness in quantum teleportation. If the distillation rate $D$ is nonzero, one can distill from such states some EPR pairs, known to violate Bell inequalities.

Bound entanglement tells us that life is not so simple. Bound entangled states are costly (⁠$E$ > 0), but useless in various quantum-information-processing protocols like teleportation. Furthermore, there is evidence that bound entangled states do not violate any Bell inequalities.

In those two senses, bound entangled states are the “black holes” of quantum information theory. Entanglement goes in but is impossible to recover. And like black holes in the theory of gravitation, bound entangled states test the limits of our understanding and puzzle us by their intrinsic irreversibility.

In what sense are bound states so thoroughly mixed up that no entanglement at all can be extracted? Bound entangled states behave intrinsically differently from every other entangled state: They remain physical under the unphysical operation of partial transposition.

Researchers realized that they could characterize entanglement in terms of how states behave under certain unphysical operations.¹² In 1996, Asher Peres at the Technion-Israel Institute of Technology in Haifa, Israel, noted that matrix transposition is just such an unphysical operation when applied to entangled states. Taking the transpose of a system’s density matrix produces another density matrix—a physically valid result. And taking the transpose of, say, Bob’s part of an unentangled state $ψ A ⊗ ψ B$ yields another physically valid quantum state, since each part of the quantum state can transform separately; $ψ A$ is not changed, and the density matrix of $ψ B$ is transposed. But when applied to part of a pure entangled state, matrix transposition produces an unphysical result. (For details, see box 2.)

Peres conjectured that partial transposition was the defining criterion for entanglement. In other words, all entangled states—pure or mixed—should map onto unphysical states by partial matrix transposition, and all unentangled states will remain physical under the same operation.

Remarkably, the truth of that conjecture depends on the dimension of the underlying Hilbert spaces or phase spaces. If one considers the state of two spin-½ particles, the polarization degrees of freedom of two laser beams, or two modes of a light field having a Gaussian Wigner function, then, indeed, all entangled states map onto unphysical states by partial transposition. However, for two spin-one (or higher-dimensional system) particles or a Gaussian light field with at least two modes for both Alice and Bob, that is no longer true in general; there exist entangled mixed states that pass the “partial transpose” test and have therefore lost an essential property of entanglement.

The loss of that property is precisely what the Horodecki family showed would lead to a zero distillation rate $D$⁠. Entangled states that pass the partial transpose test are the bound entangled states in which the entanglement is forever locked or “bound” inside.

Given that entanglement can be such a subtle property of quantum states, just how can one distinguish between entangled and unentangled states? A violation of a Bell inequality has been the traditional telltale sign of entanglement in a quantum system. Examples of such experiments³ used pairs of entangled photons created from nonlinear optical processes, especially parametric down-conversion; the polarization degrees of freedom of the emitted photons carried entanglement. Alice and Bob checked for a Bell inequality violation by using local analyzers to measure the polarization of the photons along various angles.

Unfortunately, many quantum states, including the set of bound entangled states, are not known to violate any Bell inequality. And considering the existing limitations on experimental control of quantum systems, experimentalists prefer to check for entanglement using the fewest possible local measurements. The theoretical framework of an entanglement witness, of which a Bell inequality is a particular example,¹³ addresses those two issues. The defining property of an entanglement witness $W$ is that its expectation value with respect to any unentangled state $ρ$ is always nonnegative, $Tr(Wρ)$ ≥ 0. At the same time, there exist entangled states $σ$ for which $Tr(Wσ)$ < 0. Measuring $W$ on a quantum state $σ$ and finding a negative expectation value thus establishes the entanglement of $σ$⁠. The good news is that there is an entanglement witness for every entangled state; given an experimental means, any entanglement, bound or otherwise, can be detected. The bad news is that entanglement witnesses are nonlocal observables. Nevertheless, one can measure the expectation value of $W$ by measuring the expectation value of a number of local observables $W i$⁠, such that $W = Σ i W i .$ ⁠. Research is under way to determine the minimal number of local measurements for a given witness.¹⁴ 

Given the framework of entanglement witnesses, what is special about Bell inequalities? Although they can be considered a type of entanglement witness, Bell inequalities do not, strictly speaking, test for entanglement but for a departure from local hidden-variable theories. Interpreted as such, Bell inequalities have taken on a whole new life in quantum-communication science. Researchers consider remote parties who have to carry out a certain task with minimal communication between them. One compares the amount of communication necessary if those parties are given shared random bits (that can be viewed as local hidden variables) or an entangled quantum state. Sharing entangled states leads to savings in communication precisely because the correlations in quantum states cannot always be adequately described by local hidden-variable theories¹⁵ (see the article by Andrew M. Steane and Wim van Dam, in Physics Today, February 2000, page 35).

The efforts of the quantum information theorists over the past eight years would come to little if the theory were not supplemented by an ability to create and manipulate entanglement in the lab. There is a rapidly growing list of physical systems—optical and atomic systems especially—in which it is possible to prepare various kinds of entangled states. As discussed previously, the use of photonic degrees of freedom, such as polarization or momentum, has been a long-time favorite way to create entanglement.³ Entangled states consisting of the quadrature observables of different modes of light have been prepared in optical parametric oscillators and optical fibers.¹⁶ Entanglement in the states of motion of the valence electrons⁵ of trapped ions or of Rydberg atoms in cavity quantum electrodynamics has involved up to four different atoms. Another promising avenue is the recently observed entanglement of large ensembles of atoms.¹⁷ 

This short review showcases just a few striking facets of the modern theory of entanglement. Most notably, entanglement shared between more than two subsystems is outside our scope here. The broader study of entanglement between many subsystems may lead the field to better understand the role of large-scale entanglement in quantum computation or quantum many-body systems.

We have focused on the role of entanglement in the transmission of quantum information. Entanglement also proves useful, however, when the goal is to transmit classical information as efficiently as possible. Researchers are studying many measures of mixed entanglement beyond the two most prominent measures discussed in this review. As for bound entanglement, there is some evidence that it may have a role to play as “helper” entanglement, useless by itself, but useful when combined with other sources of entanglement. For entanglement-theory overview articles that highlight the field, see volume 1 of Quantum Information and Computation (July 2001).

Barbara Terhal is a research staff member in the physical sciences department at the IBM Corp’s T. J. Watson Research Center in Yorktown Heights, New York. Michael Wolf is a physics PhD student at the Technical University at Brunswick in Brunswick, Germany. Andrew Doherty is a postdoctoral scholar in the quantum optics group at the California Institute of Technology.