Relativistic restrictions on the distinguishability of orthogonal quantum states

We analyze the restrictions on distinguishability of quantum states imposed by special relativity. An explicit expression relating the error probability for distinguishing between two orthogonal single-photon states to the time interval T between the beginning of the measurement procedure and the moment when the measurement result is obtained by the observer. Many problems of non-relativistic quantum information theory involve the task of distinguishing between two quantum states. Basic principles of non-relativistic classical physics implicitly assume that any measurement of a physical system can in principle be carried out with arbitrarily high accuracy and without disturbing the system state. Moreover, because of the absence of any restrictions on the maximum possible speed any measurement (even spatially non-local) can in principle be carried out arbitrarily fast (formally, in zero time). Therefore any two states of a physical system can be can be distinguished reliably, instantaneously, and without disturbing them. In the non-relativistic quantum mechanics any measurement of a quantum system generally disturbs its state. There is a fundamental difference between the distinguishing of a pair of orthogonal and a pair of non-orthogonal quantum states. For orthogonal states the state of a quantum system can be reliably (with zero error probability) identified without disturbing it [1,2]. The very possibility of obtaining the measurement result instantaneously (in zero time) implicitly contains the absence of any restrictions on the maximum possible speed. Non-orthogonal states are fundamentally non-distinguishable reliably, i.e., it is never possible to distinguish between the two non-orthogonal states of a quantum system with zero error probability. The exact lower boundary for the probability of this error has been established long ago [3–5]. That is why all the non-relativistic quantum cryptographic protocols are based on non-orthogonal states. There exist no fundamental restrictions on instantaneous distinguishing (although with non-zero error probability) between a pair of non-orthogonal states. The relativistic quantum theory (which actually arises as the quantum field theory, since no meaningful interpretation of relativistic quantum mechanics can be proposed) should also contain additional (compared with the non-relativistic quantum mechanics) restrictions on the time required for distinguishing of quantum states. The fundamental restrictions imposed by special relativity on the measurability of dynamical observables of quantum systems were first considered in 1931 in the paper by Landau and Peierls [6]. Qualitative analysis of Ref. [6] based on the uncertainty relations together with the limitation on the maximum possible speed led to the conclusion that, in contrast …