International Journal of Quantum Information

In majority of protocols of secure quantum communication (such as, BB84, B92, etc.), the unconditional security of the protocols are obtained by using conjugate coding (two or more mutually unbiased bases (MUBs)). Initially, all the conjugate-coding-based protocols of secure quantum communication were restricted to quantum key distribution (QKD), but later on they were extended to other cryptographic tasks (such as, secure direct quantum communication and quantum key agreement). In contrast to the conjugate-coding-based protocols, a few completely orthogonal-state-based protocols of unconditionally secure QKD (such as, Goldenberg–Vaidman and N09) were also proposed. However, till the recent past, orthogonal-state-based protocols were only a theoretical concept and were limited to QKD. Only recently, orthogonal-state-based protocols of QKD are experimentally realized and extended to cryptographic tasks beyond QKD. This paper aims to briefly review the orthogonal-state-based protocols of secure quantum communication that are recently introduced by our group and other researchers.

We discuss some applications of various versions of uncertainty relations for both discrete and continuous variables in the context of quantum information theory. The Heisenberg uncertainty relation enables demonstration of the Einstein, Podolsky and Rosen (EPR) paradox. Entropic uncertainty relations (EURs) are used to reveal quantum steering for non-Gaussian continuous variable states. EURs for discrete variables are studied in the context of quantum memory where fine-graining yields the optimum lower bound of uncertainty. The fine-grained uncertainty relation is used to obtain connections between uncertainty and the nonlocality of retrieval games for bipartite and tripartite systems. The Robertson-Schrödinger (RS) uncertainty relation is applied for distinguishing pure and mixed states of discrete variables.

Quantum systems which interact with an unknown environment cannot be described in terms of a unitary evolution on the system alone. For such evolution one can use a map from one density operator to another and use any other known information to model the system. Such maps are required to be positive (at least on their domain) — they take positive density operators to positive density operators — so as to be physically reasonable. A map $\mathrm{\Phi }$ which is positive but not completely positive (CP) is one which takes positive operators to positive operators, but when extended by the identity $I_n$, i.e. $I_n\otimes \mathrm{\Phi }$ does not give a positive map for some $I_n$. The map is CP if and only if the extension is positive for all $n$. Recently some effort has been put forth to try to understand if, and/or, under what circumstances one might utilize a non-CP map. Here, after a tutorial-type introduction to maps from one density operator to another, some examples which do not fit the standard prescription (SP) for deriving a CP map are given. In cases where the physical system is constrained by some knowledge of the interaction, it is possible that the SP cannot be used directly. Our example is robust to changes in the initial and final conditions.

Entanglement is one of the most useful resources in quantum information processing. It is effectively the quantum correlation between different subsystems of a composite system. Mathematically, one of the most hard tasks in quantum mechanics is to quantify entanglement. However, progress in this field is remarkable but not complete yet. There are many things to do with quantification of entanglement. In this review, we will discuss some of the important measures of bipartite entanglement.

Quantum theory as formulated in conventional framework using statevectors in Hilbert spaces misses the statistical nature of the underlying quantum physics. Formulation using operators ${{𝒞}}^{\ast }$ algebra and density matrices appropriately captures this feature in addition leading to the correct formulation of particle identity. In this framework, Hilbert space is an emergent concept. Problems related to anomalies and quantum epistemology are discussed.

In 1935, Albert Einstein and two colleagues, Boris Podolsky and Nathan Rosen (EPR) developed a thought experiment to demonstrate what they felt was a lack of completeness in quantum mechanics (QM). EPR also postulated the existence of more fundamental theory where physical reality of any system would be completely described by the variables/states of that fundamental theory. This variable is commonly called hidden variable and the theory is called hidden variable theory (HVT). In 1964, John Bell proposed an empirically verifiable criterion to test for the existence of these HVTs. He derived an inequality, which must be satisfied by any theory that fulfill the conditions of locality and reality. He also showed that QM, as it violates this inequality, is incompatible with any local-realistic theory. Later it has been shown that Bell’s inequality (BI) can be derived from different set of assumptions and it also find applications in useful information theoretic protocols. In this review, we will discuss various foundational as well as information theoretic implications of BI. We will also discuss about some restricted nonlocal feature of quantum nonlocality and elaborate the role of Uncertainty principle and Complementarity principle in explaining this feature.

In this paper, we discuss the issue of distinguishing a pair of quantum operation in general. We use Krause theorem for representing the operations in unitary form. This supports the existence of pair of quantum operations that are not locally distinguishable, but distinguishable in asymptotic sense in some higher dimensional system. The process can even be successful without any use of the entangled initial state.

The correlations between a pair of spins in a many-spin state encoded in the diagonal and off-diagonal spin–spin correlation functions. These spin functions determine the quantum correlation measures, like pair-wise concurrence, quantum discord and other measures of quantum information. We show that for isotropic and translationally invariant states, the quantum correlations depend only on the diagonal spin correlation function. The pair concurrence shows a strict short-ranged behavior. The distribution of concurrence for a random W-like state exhibits a long tail for both time-reversal invariant states and for states that break the time reversal. The quantum discord can be related to the diagonal spin correlation function. As the spin function is long range close to a critical point, analogously the quantum discord exhibits a long range behavior. For the isotropic state, the conditional entropy distribution is a Dirac delta function, whereas it has a twin-peak structure for the anisotropic model.

An overview is given of the nature of the quantum mechanical wave function.

The two-slit experiment with quantum particles provides many insights into the behavior of quantum mechanics, including Bohr’s complementarity principle. Here, we analyze Einstein’s recoiling slit version of the experiment and show how the inevitable entanglement between the particle and the recoiling slit as a which-way detector is responsible for complementarity. We derive the Englert–Greenberger–Yasin duality from this entanglement, which can also be thought of as a consequence of sum-uncertainty relations between certain complementary observables of the recoiling slit. Thus, entanglement is an integral part of the which-way detection process, and so is uncertainty, though in a completely different way from that envisaged by Bohr and Einstein.

Using classical laser beams, we generate a general complex superposition state, cebit, of orbital angular momentum (OAM) of light. We use a nonseparable beam of polarization and OAM generated by a modified Sagnac interferometer for the generation of OAM cebits which can be represented as points on the OAM Poincaré sphere. The general cebit state is represented as a function of the rotation angle of the wave plates so that one can easily generate the required state.

Completely positive trace preserving (CPTP) maps are essential for the formulation of the second law of thermodynamics. The dynamics of quantum systems initially correlated with their environments are in general not described by such maps. We explore, how this issue can be fixed by describing the classical analogue of this problem. We consider initially correlated probability distributions, whose subsequent system dynamics is ill-described by stochastic maps, and prescribe the correct way to describe the dynamics. We use this prescription to discuss the classical version of the second law, valid for initially correlated probability distributions.

It has been suggested that there may exist quantum correlations that go beyond entanglement. The existence of such correlations can be revealed by information theoretic quantities such as quantum discord, but not by the conventional measures of entanglement. We argue that a state displays quantumness, that can be of local and nonlocal origin. Information theoretic measures not only characterize the nonlocal quantumness, but also the local quantumness, such as the “local superposition”. This can be a reason, why such measures are nonzero, when there is no entanglement. We consider a generalized version of the Werner state to demonstrate the interplay of local quantumness, nonlocal quantumness and classical mixedness of a state.

Certain predictions of quantum theory are not compatible with the notion of local-realism. This was the content of Bell’s famous theorem of the year 1964. Bell proved this with the help of an inequality, famously known as Bell’s inequality. The alternative proofs of Bell’s theorem without using Bell’s inequality are known as “nonlocality without inequality (NLWI)” proofs. We review one such proof namely the Hardy’s proof which due to its simplicity and generality has been considered the best version of Bell’s theorem.

The ghost interference observed for entangled photons is theoretically analyzed using wave-packet dynamics. It is shown that ghost interference is a combined effect of virtual double-slit creation due to entanglement, and quantum erasure of which-path information for the interfering photon. For the case where the two photons are of different color, it is shown that fringe width of the interfering photon depends not only on its own wavelength, but also on the wavelength of the other photon which it is entangled with.

Gibbs paradox in the context of statistical mechanics addresses the issue of additivity of entropy of mixing gases. The usual discussion attributes the paradoxical situation to classical distinguishability of identical particles and credits quantum theory for enabling indistinguishability of identical particles to solve the problem. We argue that indistinguishability of identical particles is already a feature in classical mechanics and this is clearly brought out when the problem is treated in the language of information and associated entropy. We pinpoint the physical criteria for indistinguishability that is crucial for the treatment of the Gibbs’ problem and the consistency of its solution with conventional thermodynamics. Quantum mechanics provides a quantitative criterion, not possible in the classical picture, for the degree of indistinguishability in terms of visibility of quantum interference, or overlap of the states as pointed out by von Neumann, thereby endowing the entropy expression with mathematical continuity and physical reasonableness.