Entanglement can be understood as an extraordinary degree of correlation between states of quantum systems – a correlation that cannot be given an explanation in terms of something like a common cause. Entanglement can occur between two or more quantum systems, and the most interesting case is when these correlations occur between systems that are space-like separated, meaning that changes made to one system are immediately correlated with changes in a distant system even though there is no time for a signal to travel between them. In this case one says that quantum entanglement leads to non-local correlations, or non-locality. More precisely, entanglement can be defined in the following way. Consider two particles, A and B, whose (pure) states can be represented by the state vectors ψA and ψB. Instead of representing the state of each particle individually, one can represent the composite two-particle system by another wavefunction, Ψ AB. If the two particles are unentangled , then the composite state is just the tensor product of the states of the components: Ψ AB = ψ A ⊗ ψ B; the state is then said to be factorable (or separable). If the particles are entangled, however, then the state of the composite system cannot be written as such a product of a definite state for A and a definite state for B. This is how an entangled state is defined for pure states: a state is entangled if and only if it cannot be factored: Ψ AB ≠ ψ A ⊗ ψ B.
Entanglement can be understood as an extraordinary degree of correlation between states of quantum systems – a correlation that cannot be given an explanation in terms of something like a common cause. Entanglement can occur between two or more quantum systems, and the most interesting case is when these correlations occur between systems that are space-like separated, meaning that changes made to one system are immediately correlated with changes in a distant system even though there is no time for a signal to travel between them. In this case one says that quantum entanglement leads to non-local correlations, or non-locality. More precisely, entanglement can be defined in the following way. Consider two particles, A and B, whose (pure) states can be represented by the state vectors ψA and ψB. Instead of representing the state of each particle individually, one can represent the composite two-particle system by another wavefunction, Ψ AB. If the two particles are unentangled , then the composite state is just the tensor product of the states of the components: Ψ AB = ψ A ⊗ ψ B; the state is then said to be factorable (or separable). If the particles are entangled, however, then the state of the composite system cannot be written as such a product of a definite state for A and a definite state for B. This is how an entangled state is defined for pure states: a state is entangled if and only if it cannot be factored: Ψ AB ≠ ψ A ⊗ ψ B.
In classical mechanics, two (or more) systems that do not interact – that do not exert any kind of force on each other – are completely separated (see Sec. 1.1), i.e. experiments performed locally on one of them cannot in any way influence experiments performed locally on the other. Quantum mechanics, on the other hand, admits that there can be a form of interdependence even in absence of physical interaction. This type of correlation goes under the name of entanglement (see Def. 5.1: p. 183). We already know that entanglement means non-factorizability of the state vector describing a system with at least two different degrees of freedom. In this chapter, we shall see that this may be the case also for systems that are very far away (space-like distant) from each other and we shall learn many interesting consequences of this state of affairs. Recent developments show that one may entangle even particles that have never interacted with each other. The starting point of the following analysis is necessarily represented by the pioneering paper of Einstein, Podolsky, and Rosen, who inaugurated an incredible series of theoretical and experimental investigations right up to the present day that have confirmed the predictions of quantum mechanics. The “irony” of this history is that Einstein, Podolsky, and Rosen, EPR for short, aimed to present a definitive proof of the inconsistency of quantum mechanics relative to classical physical theory. In fact, they initiated one of the most prolific fields of modern science.
Introduction We will now turn to quantum entanglements and their contemporary, possibly practical, interest. Entanglements, first discussed in the E.P.R. paradox (Einstein, Podolsky, Rosen, 1935) are a perplexing nonlocal feature of quantum mechanics. This was immediately and succinctly discussed by Schrödinger (1935). The long history of this apparent paradox is outlined in the wonderful book of Jammer (1974). We will not focus on the central issue of “hidden variables” and their resolution by the Bell inequalities (Bell, 1964) and the test, nor the failure, of these in experiment (see Fry, 1998). The distinctive quantum nature of entanglements has led to two quantum effects which we will discuss: quantum information teleportation (Zeilenger, 1998) and quantum computation by means of entangled states. A nice, recent, elementary introduction to the latter is in the Los Alamos reports of James and Kwiat (2002). The quantum correlations or entanglements are sensitive to environmental destruction. This was pointed out early by Zurek, who termed this “decoherence.” The loss of coherence may occur on a short time scale. Recent discussions have been given by Zurek (2002, 2003). We have already given a theoretical example early in Chapter 2. It is pertinent to discuss this here, since it is a property of opensystem quantum master equations that is a central part of our study in this book. In a sense, the decoherence of correlations has turned out to be a “practical” application of these theoretical notions. Are there remedies for unwanted decoherence? Quantum error correction is a possibility. This will be mentioned also. Entanglements: foundations Following the reading of the E.P.R. paper, Schrödinger (1935b) quickly repeated the argument from quantum theory, but more generally. He introduced the term “entanglement” to describe what he seemed to agree was a curious, if not unacceptable, property. Let us follow his point of view. Given a state ψ ( x, y ) of a composite system of particles x and y formed in their mutual interaction. ψ ( x , y ) is not a product state.
In the case of mixed states ρAB of the composite system A ⊗ B, the previous relation generalizes to ρAB = ∑ k pkρ A k ⊗ρk , where pk are probabilities and ρk ’s (ρk ’s) are generic density matrix of the system A (B). The entanglement derives from “subtle” nonlocal correlations between the parts of a quantum system and combines three basic structural elements of quantum theory: the superposition principle, the quantum non-separability property and the exponential scaling of the state space with the number of partitions. This unique resource can be used to perform computational and cryptographic tasks otherwise impossible with classical systems. An entangled state shared by two or more separated parties is a valuable resource for fundamental quantum protocols, such as quantum teleportation [2], quantum computing [3] and quantum cryptography. By using entangled states we can deeply investigate the nonlocal properties of the quantum world. In fact, an entangled system exhibits correlations between two (or more) parties, which cannot be reproduced by a classical theory or, in general, by a Local Hidden Variables Theory (LHVT). Quantum optics represents an excellent experimental test bench for various novel concepts introduced within the framework of quantum information theory. Quantum states of photons, generally produced by a probabilistic process, namely spontaneous parametric down conversion (SPDC), may be easily and accurately manipulated using linear and nonlinear optical devices and measured by efficient single-photon detectors. Photonic qubits represent the typical carrier of quantum information and can be distributed over long distances, either in free-space or in low-loss optical fibers. Two-particle experiments correspond to operate with qubits belonging to a (2 × 2) Hilbert space. Many quantum information tasks and fundamental tests of quantum mechanics, such as the simulation of properties of quantum systems, the realization of quantum algorithms of increasing complexity and the investigation of the quantum world at a mesoscopic level, deal with a large number of qubits. In order to take full advantage of the possibilities offered by quantum mechanics, more qubits must be added to a quantum state. Two approaches may be followed to increase the number of qubits. The first one consists of increasing the number of entangled particles. In this way, multi-qubit entangled states are created by distributing the qubits between the particles so that each particle carries one qubit. A second strategy consists of encoding more than one qubit in each particle, by exploiting different degrees of freedom (DOFs) of the photon such as the polarization, the path, the emission time, the orbital angular momentum. This has been used to create two-photon fourand six-qubit states and up to five-photon ten-qubit graph states. The entanglement of two particles in different DOFs corresponds to socalled hyperentanglement. It is worth noting that multi-qubit hyperentanglement allows to achieve higher
Up to now we have limited ourselves to states of a single particle. In the present chapter we shall introduce the description of two-particle states. Once this case is understood, it will be easy to generalize to any number of particles. States of two (or more) particles lead to very rich configurations called entangled states. A remarkable feature is that two entangled quantum particles, even at arbitrarily large spatial separations, continue to form a single entity and no classical probabilistic model is able to reproduce the correlation between particles. In the first section we shall present the essential mathematical formalism, that of the tensor product. This will permit us in Section 6.2 to describe quantum mixtures using the state operator formalism. Section 6.3 is devoted to the study of important physical consequences like the Bell inequalities and interference experiments involving entangled states, which will lead us to a deeper understanding of quantum physics. Finally, in the last section we shall briefly review applications to measurement theory and quantum information theory. The latter is undergoing rapid development at present and has applications to quantum computing, cryptography, and teleportation. The tensor product of two vector spaces Definition and properties of the tensor product We wish to construct the space of states of two physical systems which we assume initially to be completely independent. Let ℌ N 1 and ℌ M 2 be the spaces of states of the two systems, of dimension N and M , respectively.
Introduction The first realization that the validity of the quantum superposition principle in the Hilbert space describing a composite quantum system may give rise to fundamentally new correlations between the constituent subsystems came in the landmark 1935 paper by Einstein, Podolsky, and Rosen (EPR), where it was shown how the measurement statistics of observables in certain quantum states could not be reproduced by assigning definite wavefunctions to individual subsystems. It was in response to the EPR paper that Schrödinger, in the same year, coined the term entanglement ( Verschränkung ) to acknowledge the failure of classical intuition in describing the relationship between the “parts” and the “whole” in the quantum world: Whenever one has a complete expectation catalog – a maximum total knowledge – a ψ function – for two completely separated bodies, …then one obviously has it also for the two bodies together. But the converse is not true. The best possible knowledge of a total system does not necessarily include total knowledge of all its parts, not even when these are fully separated from each other and at the moment are not influencing each other at all. While Bell's strengthening of the original EPR-paradox setting and the subsequent experimental verification of Bell inequalities irreversibly changed the perception of entanglement from a property of counterintuitive “spookiness” to (beyond reasonable doubt) an experimental reality, the concept and implications of entanglement continue to be associated with a host of physical, mathematical, and philosophical challenges. In particular, investigation of entanglement in both its qualitative and quantitative aspects has intensified under the impetus of quantum information science (QIS).
Abstract Two or more particles which have interacted in the past, may in general maintain nontrivial quantum mechanical correlations, described by non-factorized wave functions. These correlations, under optimal conditions, can manifest themselves no matter how far they travel away from each other and even when the separate parts cannot dynamically influence each other, leading to apparently paradoxical situations. These correlations are known as quantum entanglement. However counter-intuitive and surprising it may sometimes look, it is rather one of the more beautiful and experimentally well established predictions of quantum mechanics.