The total information content of a composite system consisting of k qubits can either be completely encoded in a specific computational basis, or alternatively it can be partially encoded in a number of different bases. In that case the information encoded in a complete set of mutually complementary bases is again k bits. Using only two single-qubit gates and the controlled-NOT gate, one can implement coding and decoding in such a complete set. The total information content is then invariant under the particular choice of a complete set of mutually complementary bases. For maximally entangled states the k bits of information are not encoded into the k qubits separately but only into their joint properties.
The total information content of a composite system consisting of k qubits can either be completely encoded in a specific computational basis, or alternatively it can be partially encoded in a number of different bases. In that case the information encoded in a complete set of mutually complementary bases is again k bits. Using only two single-qubit gates and the controlled-NOT gate, one can implement coding and decoding in such a complete set. The total information content is then invariant under the particular choice of a complete set of mutually complementary bases. For maximally entangled states the k bits of information are not encoded into the k qubits separately but only into their joint properties.
We have seen how information can be encoded onto a quantum system by selecting the state in which it is prepared. Retrieving the information is achieved by performing a measurement, and the optimal measurement in any given situation is usually a generalized measurement. In between preparation and measurement, the information resides in the quantum state of the system, which evolves in a manner determined by the Hamiltonian. The associated unitary transformation may usefully be viewed as quantum information processing; if we can engineer an appropriate Hamiltonian then we can use the quantum evolution to assist in performing computational tasks. Our objective in quantum information processing is to implement a desired unitary transformation. Typically this will mean coupling together a number, perhaps a large number, of qubits and thereby generating highly entangled states. It is fortunate, although by no means obvious, that we can realize any desired multiqubit unitary transformation as a product of a small selection of simple transformations and, moreover, that each of these need only act on a single qubit or on a pair of qubits. The situation is reminiscent of digital electronics, in which logic operations are decomposed into actions on a small number of bits. If we can realize and control a very large number of such operations in a single device then we have a computer. Similar control of a large number of qubits will constitute a quantum computer. It is the revolutionary potential of quantum computers, more than any other single factor, that has fuelled the recent explosion of interest in our subject. We shall examine the remarkable properties of quantum computers in the next chapter. In digital electronics, we represent bit values by voltages: the logical value 1 is a high voltage (typically +5 V) and 0 is the ground voltage (0 V). The voltage bits are coupled and manipulated by transistor-based devices, or gates. The simplest gates act on only one bit or combine two bits to generate a single new bit, the value of which is determined by the two input bits. For a single bit, with value 0 or 1, the only possible operations are the identity (which does not require a gate) and the bit flip.
Abstract: Reduction of a state of a quantum system to a subsystem gives partial quantum information about the true state of the total system. Two subalgebras A1 and A2 of B(H) are called complementary if the traceless subspaces of A1 and A2 are orthogonal (with respect to the Hilbert-Schmidt inner product). When both subalgebras are maximal Abelian, then the concept reduces to complementary observables or mutually unbiased bases. In the paper several characterizations of complementary subalgebras are given in the general case and several examples are presented. For a 4-level quantum system, the structure of complementary subalgebras can be described very well, the Cartan decomposition of unitaries plays a role. It turns out that a measurement corresponding to the Bell basis is complementary to any local measurement of the two-qubit-system.
We now have all the basic tools we need to describe systems of one and two qubits. In this final chapter of Part I we look at some of the consequences of the peculiar properties of qubits when they are used to encode information. Many of these results can be traced to the properties of quantum measurement. Measuring a single qubit A key result in quantum information theory is that it is impossible to accurately characterize a single qubit. In other words, there is no experiment, or sequence of experiments, which allows us to find out the state of a single quantum bit. The reason for this problem is twofold. Firstly, we have to make some sort of decision about the basis we will use for the measurement. For example, when measuring a single qubit the most popular choice is to make a measurement in the computational basis. This is equivalent to asking the qubit whether it is in state |0⧽ or state |1⧽. If the qubit is indeed in one of the basis states the measurement process is simple, and we will get the obvious answer of 0 if it is in |0⧽ and 1 if it is in |1⧽. If, however, the qubit is in a superposition, such as α|0⧽+β|1⧽, then the situation is more difficult. Characterizing the state now means determining the values of the two complex numbers α and β, but the measurement can only return the answer 0 or 1, and for a superposition state one of these two answers will be returned at random, with probabilities |α| 2 and |β| 2 , respectively.
Classical information processing is performed using bits , which are just two-state systems, with the two states called 0 and 1. By grouping bits togetherwe can represent arbitrary pieces of information, and by manipulating these bits we can perform arbitrary computations. The corresponding basic element used in quantum information is the quantum bit, or qubit . This is simply a quantum system with two orthonormal basis states, which we shall call |0⧽ and |1⧽. There are many possible physical implementations of a qubit, such as spin states of electrons or atomic nuclei, charge states of quantum dots, atomic energy levels, vibrational states of groups of atoms, polarization states of photons, or paths in an interferometer. At this stage the physical implementation is not important: the idea of a qubit is to abstract the discussion away from physical details. Taking the standard approach of quantum information theory, we shall begin by not worrying too much about the properties of these states, or even what their energies are; we shall simply assume that they are eigenstates of the system's Hamiltonian with known eigenvalues (that is, known energies). This approach allows us to concentrate on the fundamental properties of the system, without considering all the tedious details. We can in principle perform classical information processing on our quantum system by using the two states |0⧽ and |1⧽ as our logical states 0 and 1 and proceeding in the usual fashion, giving rise to the field of reversible computing , which will be explored briefly in Part II. This, however, misses the point.
A compete orthonormal basis of N-qutrit unitary operators drawn from the Pauli group consists of the identity and ${9}^{N}\ensuremath{-}1$ traceless operators. The traceless ones partition into 3N+1 maximally commuting subsets (MCS's) of ${3}^{N}\ensuremath{-}1$ operators each, whose joint eigenbases are mutually unbiased. We prove that Pauli factor groups of order 3N are isomorphic to all MCS's and show how this result applies in specific cases. For two qutrits, the 80 traceless operators partition into 10 MCS's. We prove that 4 of the corresponding basis sets must be separable, while 6 must be totally entangled (and Bell-like). For three qutrits, 728 operators partition into 28 MCS's with less rigid structure, allowing for the coexistence of separable, partially entangled, and totally entangled (GHZ-like) bases. However a minimum of 16GHZ-like bases must occur. Every basis state is described by an N-digit trinary number consisting of the eigenvalues of N observables constructed from the corresponding MCS.
This study presents a generalized n--bit superdense coding protocol that enables the transmission of n classical bits of information using an entangled n--qubit quantum system and the transmission of n−1 qubits. The protocol involves creating a maximally entangled n--qubit GHZ state, encoding the classical message with Pauli--Z and Pauli--X gates, and then transmitting and decoding the message via quantum communication, quantum operations, and measurements. The key novelty of this work is an explicit, scalable n--bit encoding routine, realized on the GHZ channel with a low--depth encoding layer using only single--qubit Pauli--X and Pauli--Z operations.
The protocol was tested on real quantum hardware using Qiskit 2.0 and the IBM--Fez quantum computer for message lengths of 4, 6, 8, and 10 bits, as a circuit--level proof of concept on a single processor; the physical transmission of qubits between the parties was not implemented. Results show that success rates decrease as message length, transpiled circuit depth and gate count increase; the latter two are set mainly by transpilation rather than by the encoding itself. A mutual information analysis shows that the transmitted information exceeds the classical bound of n−1 bits only for short messages. Strategies to improve performance include sending messages in shorter segments and advances in qubit coherence and gate fidelity. This work offers a practical and easily scalable quantum communication protocol with potential applications in quantum networks and communication systems.