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logical state as the main quantities of interest and places restrictions between them. For example, if q$_{i}$ is the heat dumped to the bath during a Landauer erasure when the initial logical state is i , then it can be shown that 

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an inequality which is first derived by Szilard [19]. The main advantage of this formulation is its independence of the coding of the information into the logical states of the device. The standard statement of LEP, however, is dependent on the coding: if the device is used for storing information in such a way that the logical state i is used with frequency p$_{i}$ , then the device has information storage capacity of I = − ∑ $_{i}$p$_{i}$ log$_{2}$ p$_{i}$ bits and the Landauer bound states that 

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where ¯ q = ∑ $_{i}$p$_{i}$ q$_{i}$ is the average heat dumped to the bath when the information in the device is erased. 

Note that Eq. (2) is valid for any distribution p , hence it provides an infinite number of restrictions on q$_{i}$ . It can be shown that Eq. (2) holds for all possible distributions p if and only if Eq. (1) is satisfied. Thus, Eqs. (1) and (2) are two equivalent formulations of LEP. The standard form of LEP, Eq. (2) is more profound as it directly connects two disparate disciplines, thermodynamics and information theory. Yet, Eq. (1) is also somewhat attractive as it is a concise inequality that captures all of the infinitely many inequalities in Eq. (2). Both inequalities can be generalized to arbitrary nondeterministic processes. It is observed that the generalization of Eq. (1) contains further restrictions on heat exchanges that cannot be derived from the entropic restrictions of the generalized LEP [14]. 

The purpose of this article is to adapt that coding-independent approach to quantum operations applied on devices holding quantum information. The major difference between the classical and quantum cases is in the description of the state: instead of a discrete index i , we now have a density matrix that describes the state of the device. Hence, in place of the numbers q$_{i}$ , we now have a hermitian operator Q whose expectation value with the density matrix gives the average heat dumped to the bath. This operator will be called as the heat transfer operator (HTO). 

The unitary transformation can be thought as moving information around in the com- bined system of the device and the bath. The resulting information exchange between the 

