Describing a Quantum Channel by State Tomography of a Single Probe State

A general law is presented for (composite) quantum systems which directly describes the time evolution of quantum states (with one or both components) through an arbitrary noisy quantum channel. It is shown that the time evolution of all quantum states through a quantum channel can be completely captured by the evolution of a single ’probe state’. Thus in order to grasp the information of the final output states subject to a quantum channel, especially an unknown one, it only requires quantum state tomography of a single probe state, which dramatically simplifies the practical operations in experiment. Quantum states are the basic carrier of quantum information [1]. The core of all the quantum information processing (QIP) including quantum communication [2] and quantum computation [3] is the controlled time evolution of quantum state in essence [4]. However, in realistic scenario, quantum states will be unavoidably and greatly disturbed by the undesired coupling to the uncontrolled degree of freedom usually termed as ’environment’ and described as a ’quantum channel’. As a consequence, besides the state itself the valuable properties of quantum states such as coherence [5,6], entanglement [7,8] of composite systems and so on will be greatly corrupted. The precise characterization of some properties of quantum states usually largely relies on the evaluation of quantum states, if these properties such as entanglement does not correspond to a direct observable for a general unknown quantum state [9-11]. Furthermore, quantum channel is not restricted to the previous interaction between the system and environment. It is a general notion of any a input/output device governed by quantum mechanics including the controlled interactions, for example, the dynamical action of a quantum gate in a quantum computer etc [12]. Therefore, it is of practical importance to precisely explore the time evolution of quantum states on which a reliable QIP task depends. In general cases, there is no direct way to evaluate the time evolution of quantum states. One has to begin with (a)[email protected]; [email protected] considering the dynamics of ’system of interests + environment’ governed by quantum principle [13-16]. It is implied that the concrete description of the quantum channel has been known by the quantum process tomography that includes a series of quantum state tomography [17] and is usually quite complex [3]. In experiment, the time evolution of quantum states could be described by determining the initial and final states in terms of quantum state tomography no matter whether the quantum channel is known. However, it is a drawback that the procedure needs to be repeated every time with different input states chosen. In the present Letter, we provide a direct and general scheme in terms of the evolution of a given probe state to describe the evolution of quantum states of an arbitrary (composite) quantum system which (the components of which) undergoes an arbitrary (especially unknown) quantum channel. The distinguished advantages of our scheme are as follows: 1) Especially for unknown quantum channels, it is only necessary to do state tomography of the single probe state, stead of repeating the same procedure for different input states or doing quantum process tomography (In other words, it is not necessary to know the concrete description of quantum channel). 2) The scheme can be directly applied to any quantum mechanical input/output process. Thus all information of the final states can be learned and the properties of interests such as coherence or entanglement etc. can be obtained by a sequent simple calculation [18,19]. Let us first consider an (N ⊗N) -dimensional bipartite