A3/No measurement without disturbance

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A3

Communication
Introduction

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In the previous chapter we identified the fact that any interference with a quantum signal will lead to a disturbance in the signal as the source for the security in quantum information. We will now explain this principle from a more general point of view, as it is one of the fundamental principles of quantum physics. The principle “no measurement without disturbance” constitutes a strong contrast between classical and quantum physics.

What is a measurement device?

Until now, we have only considered measurements, in which the measured system was discarded after the measurement. This resembles many experimental situations, in which a signal is e.g. absorbed by a detector, as in the examples with polarized light.

In general one can consider a measurement device, which produces a measurement result (usually a classical result) and returns the measured system. If we want to discuss the influence of measurement, we need to consider such devices so that we may compare the state prior measurement and after measurement. The disturbance of the measurement will then be defined as the amount how the state changes.

Let us formalise this: We consider a system that is combined from a quantum part and a classical register, which denotes the result of the measurement. Prior to measurement, the quantum part is in the state \rho_{in} where the classical register is in a neutral position 0. After the measurement, the classical register holds the measurement value x and the quantum state will be in the state \rho_{out, x}. We introduce this notation to represent, that the output state may be different for each classical result.

We call a channel between systems ideal, if the channel transmits any input state without disturbance. If we call the channel T_{id} then this would mean that for the ideal channel T_{id} \rho_{in} = \rho_{in} holds for all input states. Such a channel is also called an identical channel or an identity.

The "opposite" of an identical channel is a depolarizing channel. It is defined by being completely forgetful and simply preparing an output state independent of the input state. If we call the channel T_{dep}, then there exists a state \sigma such that T_{dep} \rho_{in} = \sigma holds for all input states. This behaviour implies that there is no possibility to reconstruct the input state of a depolarizing channel by its output state.

Theorem

Now we have gathered all notions to describe the connection between measurement and disturbance in a formal manner. We again consider a measurement as a mapping from an input state to an output state and a classical register for the measurement value. Then the simplest version tf the theorem says that:

Whenever the channel from the input state to the output state is ideal, the channel from the input state to the classical register is completely depolarising.

In other words, any measurement device that leaves the input state unchanged will only produce results that are independent of the input state. Yet in this simplest form the theorem would be too weak for practical purposes, as it does not provide a quantitative relation between the information gained by the measurement and the induced disturbance. We will come back to giving the precise mathematical trade-off in one of the following lectures, but we note here, that the theorem in its final form will state that:

Whenever the channel from input to output state is almost ideal, the channel from input to classical register is almost depolarising.

Special cases

We should stress here, that the important restriction of the theorem is, that it shall hold for all input states. If one eases on this condition, then the theorem does not hold anymore. If the measurement shall only cover a certain subset of states, then measurement of disturbance is possible, if all states that shall be measured are distinguishable (i.e. the states are orthogonal). In the polarisation example, only those states that lie antipodal on the Bloch sphere can be distinguished with certainty, so if one has the guarantee that only e.g. horizontal and vertical polarisation are to be distinguished there are measurements that can perform this without disturbance. But the same measurement would induce disturbance, if applied to other states. This corresponds to restricting the qubit to the degrees of freedom of a classical bit, namely two states that can be distinguished with certainty. It is a general feature in quantum physics, that systems become quasi classical of one restricts to orthogonal states.

Connection to quantum cryptography

This principle provides an intuitive reason, why quantum cryptography is secure: Any attempt for an attacker to extract information from a quantum signal unknown to him, this interaction will always change the state of the signal. The quantitative version of the theorem (that we only sketched here), will provide a quantitative bound on how much information an attacker can extract from a given disturbance. In classical physics no such lower bound on the extractable information exists. There it is in principle possible to extract an arbitrary amount of information without causing any disturbance.