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Workshop on Physics and Computation (PhysComp)

Storage And Retrieval Of Quantum Information

Asher Peres

Published Aug 24, 2005• 1 citation
Publisher

Abstract

Information encoded in non-orthogonal quantum states cannot be duplicated, nor amplified, and in general at is only partly recoverable. The most efficient way of retrieving it is not a direct uquantum measurement” (as defined by von Neumann), but an indirect method similar to heterodyne detection in communications engineering. The mathematical representation of this process requires the introduction of a positive operator valued measure. The optimization of these measures is not yet fully undersiood. An interesting and potentially important application of quantum information is its use an cryptography.

DOI
10.1109/phycmp.1992.615515

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Information encoded in non-orthogonal quantum states cannot be duplicated, nor amplified, and in general at is only partly recoverable. The most efficient way of retrieving it is not a direct uquantum measurement” (as defined by von Neumann), but an indirect method similar to heterodyne detection in communications engineering. The mathematical representation of this process requires the introduction of a positive operator valued measure. The optimization of these measures is not yet fully undersiood. An interesting and potentially important application of quantum information is its use an cryptography.
In quantum information processing and quantum computing protocols information is encoded in the state of a quantum system. After processing the information it has to be read out which is equivalent to determining the final state of the system. When the possible final states are not orthogonal this is a highly nontrivial task that constitutes the general area of what is known as quantum state discrimination. It consists in finding measurement schemes that, according to some figure of merit, will determine the state of the system. Optimized measurement schemes often lead to generalized measurements (Positive Operator Valued Measures, POVMs). In this tutorial review we begin with a brief introduction to the theory of generalized measurements and illustrate the power of the concept on examples relevant to applications in quantum cryptography.
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.
We consider how the theory of optimal quantum measurements determines the maximum information available to the receiving party of a quantum key distribution (QKD) system employing linearly independent but non-orthogonal quantum states. Such a setting is characteristic of several practical QKD protocols. Due to non-orthogonality, the receiver is not able to discriminate unambiguously between the signals. To understand the fundamental limits that this imposes, the quantity of interest is the maximum mutual information between the transmitter (Alice) and the receiver, whether legitimate (Bob) or an eavesdropper (Eve). To find the optimal measurement we use a framework based on operator algebra and general results derived from singular value decomposition, achieving optimal solutions for von Neumann measurements and positive operator-valued measures (POVMs). The formal proof and quantitative analysis elaborated for two signals allow one to conclude that optimal von Neumann measurements are uniquely defined and provide a higher information gain compared to POVMs. Interestingly, collective measurements not only do not provide additional information gain with respect to individual ones, but also suffer from a gain reduction in the case of POVMs.
From the early days of quantum theory, it is felt that the classical ideas about information may need reorientation under a quantum theoretical framework. In quantum theory, where Heisenberg uncertainty principle is the guiding philosophy, a non-commuting observable cannot simultaneously have precisely defined values. In fact, if two observations do not commute, performing a measurement of one observable will necessarily influence the outcome of a subsequent measurement of the other observable. Hence, the act of acquiring information about a physical system inevitably disturbs the state of the system. This is because of the fact that the outcome of a measurement has some random elements due to uncertainty of the initial state of the system. Moreover, acquiring information causes a disturbance, and due to this disturbance the information cannot be copied with perfect fidelity. The information under this framework is termed as quantum information [1].