Quantum information theory: entanglement and coherence

Coherence has long been one of central concepts of quantum physics and hence its detection and quantification is a fundamental task.  The distinction between classical (coherence in the absence of quantum fluctuations e.g. bright coherent light) and quantum coherence is has been made traditionally using phase space distributions and higher order correlation functions.  Though these methods give a distinction between the classical and quantum forms of coherence they do not give us any procedure to measure the amount of coherence in the system. One of the new entrants in the field of quantum correlations is quantum coherence.   A scheme for measuring coherence using the framework of quantum information theory was proposed by Baumgratz, Cramer, and Plenio [1] recently.   In this work, the conceptual steps were taken and the definition of incoherent states, incoherent operations and maximally coherent states were defined.  Further the list of properties a measure should satisfy in order to be classified as a coherence measure were also proposed.  Developments have been made towards understanding quantum coherence and using it as resource in information theory.

This method of Baumgratz, Cramer, and Plenio was very similar to the one used in entanglement where the distance of the given state to the closest separable state is used.  For measuring coherence we need to find the distance to the closest incoherent state which in turn requires a definition of a completely incoherent state.  Also under these contexts it is important to introduce incoherent operations which are the set of operations which do not generate coherence from incoherent states.  Using these two quantities namely the incoherent states and operations a set of axioms were proposed.  Any measure of coherence should satisfy these axioms to be considered as a proper measure of coherence.  Based on these axioms several measures of coherence are being proposed, with their properties and applications being investigated.    This is one of the major directions of current research on the topic of quantifying coherence.

 

coherence

Fig 1:  A four particle multipartite system in which the intra-qubit coherence is denoted by CL and the inter-qubit coherence is given by CI and the total-coherence is given by C. 

In our recent work [2] we have shown how to decompose coherence further, and precisely locate in what way and where the coherence is located in the quantum system.  This can be done because quantum coherence is a manifestation of the superposition which can occur either between the qubits or between the levels of the qubit.  An illustration of the two different kinds of coherence is given in Fig. 2 where we show a simple four particle system with each qubit shown by a black colored filled circle.  One kind of coherence arises due to the correlations between the qubits which is shown through the red colored arrow between the qubits represented by the black dots.  This coherence is the inter-qubit or Intrinsic coherence. An example of this kind of coherence is the one in the state  .  The second form of coherence arises purely due to the superposition between the levels of the quantum system.  In Fig.2 this is shown by magnifying the qubit to exhibit the two levels which are superposed.  Since this intra-qubit coherence is localized within the qubit we refer to it as local coherence.  The quantum state        is an example of this form of coherence.  These two forms of coherence are complementary to each other and we proposed a scheme to isolate and estimate these different forms of coherence.

 

References

[1] T. Baumgratz, M. Cramer and M.B. Plenio, Phys. Rev. Lett. 113, 140401 (2014).
[2] R. Chandrashekar, P. Manikandan, J. Segar, Tim Byrnes, Phys. Rev. Lett. 116, 150504 (2016)