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Stronger No-Cloning, No-Signalling and Conservation of Quantum Information

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Stronger No-Cloning, No-Signalling and Conservation of Quantum Information Stronger no-cloning, no-signalling and conservation of quantum information I. Chakrabarty,1,2, ∗ A. K. Pati,3, † and S. Adhikari2, ‡ 1Heritage Institute of Technology, Kolkata, India 2Bengal Engineering and Science University, Howrah-711103, West Bengal, India 3Institute of Physics, Bhubaneswar-751005, Orissa,India (Dated: May 3, 2019) It is known that the stronger no-cloning theorem and the no-deleting theorem taken together provide the permanence property of quantum information. Also, it is known that the violation of the no-deletion theorem would imply signalling. Here, we show that the violation of the stronger no-cloning theorem could lead to signalling. Furthermore, we prove the stronger no-cloning theorem from the conservation of quantum information. These observations imply that the permanence property of quantum information is connected to the no-signalling and the conservation of quantum information. Introduction: In recent years it is of fundamental establish the permanence of quantum information. importance to know various differences between classi- Before going into the details, first of all, we should cal and quantum information. Many operations which keep in mind that we are treating information not only are feasible in digitized information become impossibili- as knowledge but also as a physical quantity [17]. In fact, ties in quantum world. Unlike classical information, in there are two opposing concepts regarding information: quantum information theory, we cannot clone and delete (i) subjective, according to this information is nothing an arbitrary quantum states which are now known as but knowledge obtained about a system after interacting the no-cloning and the no-deleting theorems [1, 2, 3]. with it, and (ii) objective, which tells us that informa- In addition to these two there are many other theorems tion is a physical quantity. Now we must understand which come under this broad heading of ‘no-go theorems’ what actually we mean by the term permanence of quan- [4, 5, 6, 7, 8]. In particular, we have the ‘no-flipping’ tum information. It refers to the fact that neither we [4, 5], the ‘no-self replication’ [6], the ‘no-partial erasure’ could create nor destroy quantum information. From the [7], the ‘no-splitting’ [8], and few other theorems in the objective viewpoint quantum information resemble like literature. Moreover, the no-cloning, the no-conjugation, energy which already existed before any physical process and the no-complementing theorems have been unified has taken place and will continue to exist even after any under a name ‘General Impossible operations’ in quan- physical process. Permanence of quantum information is tum Information theory [9]. something which is akin to this. It is entirely quantum Now, we know that these impossible theorems can also mechanical in nature and there is no classical analogue. Recall that the no-cloning theorem states that if ψi be derived from the fundamental principles like impossi- {| i} bility of signalling and non-increase of entanglement un- is a set of pure non-orthogonal states, then there is no der local operations for closed systems [10, 11, 12, 13, 14]. unitary operation that can achieve the transformation ψi ψi ψi [2]. However, the stronger no-cloning For example, the no-deleting theorem was established | i → | i| i theorem states that if ψi is a set of pure states con- from the principle of no-signalling between two spatially {| i} separated parties [13]. The no-flipping theorem has been taining no orthogonal pair, then the transformation arXiv:quant-ph/0605173v1 19 May 2006 obtained from the no-signalling and the non-increase of ψi ai ψi ψi (1) entanglement under LOCC [14]. Recently, it was shown | i| i → | i| i that the no-cloning and the no-deletion theorems fol- is possible iff ai ψi . That is the second copy ψi low from the conservation of quantum information [15]. can always be| generatedi → | i from the supplementary infor-| i In strengthening the no-cloning theorem it was proved mation alone, independent of the first copy. Thus in ef- that in order to make a copy of a set of non-orthogonal fect cloning of ψi is possible only when the second copy states one must supply full information. This is called is provided as an| i additional input. This is known as the as the stronger no-cloning theorem [16]. The stronger ‘stronger no-cloning theorem’ [16]. no-cloning and the no-deleting theorems taken together Let us briefly recapitulate the stronger no-cloning the- implies a permanence of quantum information. Since orem. Consider a physical process described by the equa- the deletion operation implies signalling one may won- tion der whether violation of the stronger no-cloning theorem in quantum information could also lead to signalling. If ψi 0 αi C ψi ψi Ci , (2) yes, then that would mean that violation of permanence | i| i| i| i → | i| i| i property of quantum information could lead to signal- where ψi is the copy which is to be cloned and 0 is ing. In this work we show that indeed principles like no- the blank| i state. Here C is the initial state of the| i en- | i signalling and conservation of entanglement under LOCC vironmental space and αi is the state which already | i 2 existed in some other part of the world. Now Ci is the basis. The states are given by, | i output state of the two register which initially contained 1 α , C . Since we have included the environmental space χ = ( ψ1 ψ1 ψ1 ψ1 ) i | i √2 | i| i − | i| i in| thisi | physicali process, we can consider the transforma- 1 tion (2) as a unitary transformation. Hence from unitar- = ( ψ2 ψ2 ψ2 ψ2 ) (5) √2 | i| i − | i| i ity, the sets αi and ψi , Ci have equal matrices of {| i} {| i | i} 1 inner product. Then, αi and ψi , Ci ] are unitar- ξ α α α α {| i} {| i | i } = ( 1 1 1 1 ) ily equivalent and in advance we can say that ψi is gen- | i √2 | i| i − | i| i | i erated from αi (discarding the ancillary system Ci ). 1 | i | i = ( α2 α2 α2 α2 ), (6) This is known as the ‘Stronger no-cloning’ theorem [16]. √2 | i| i − | i| i Now if we look into the process of deletion of non- where ψ1 , ψ1 , ψ2 , ψ2 are two sets of mutually orthogonal states we find that the permanence of quan- orthogonal{| i spin| i} states{| ori polarizations| i} in case of photons tum information plays an important role. Suppose we are (qubit basis) for the first entangled pair given by (5) provided with two copies ψi ψi of a quantum state and shared between Alice and Bob. Similarly α1 , α1 , we wish to delete one copy| i| byi some physical operation {| i | i} α2 , α2 are two sets of mutually orthogonal spin (a completely positive, trace preserving map) like states{| i | (qubiti} basis) for the second entangled pair (6). Let us write the combined state of the system in an arbitrary ψi ψi ψi 0 . (3) | i| i → | i| i qubit basis. 1 Such a physical operation will corresponds to a linear χ ξ = ( ψi ψi ψi ψi )( αi αi αi αi ) (7) operation if we include ancilla and would be given by | i| i 2 | i| i − | i| i | i| i − | i| i where i = 1, 2. Note that whatever measurement Alice ψi ψi A ψi 0 Ai . (4) does, Bob does not learn the results and his description | i| i| i → | i| i| i will remain as that of a completely random mixture , i.e., I I The no-deleting principle tells us that the second copy ρB = . In other words we can say that the local 2 ⊗ 2 ψi can never be deleted in the sense that ψi can always operations performed on her subspace has no effect on | i | i be obtained from Ai . Bob’s description of his states. | i Thus, we see in both deletion and cloning we obtain a Now we look out for the equivalent mathematical ex- similarity: the first copy provided no assistance in con- pression for the statement like ‘negating the stronger no- struction of the second copy and in deletion the first copy cloning theorem’. Let us define has not provided any assistance in deleting the second ψi αi C ψi ψi Ci1 copy. Thus, we can say that the quantum information | i| i| i → | i| i| i ψi αi C ψi ψi Ci2 has some property called permanence which tells us that | i| i| i → | i| i| i the creation of copies will only be possible if we import ψi αi C φi | i| i| i → | i information from some other part which already existed ψi αi C φi , (8) and deletion can only be possible by exporting informa- | i| i| i → | i where i = 1, 2 and φ , φ are some general states of tion to some other part where it will continue to exist. i i the combined system.| i The| abovei equations corresponds The aim of this work is basically two fold. In the to a situation where we are negating stronger no-cloning first part we will show that negating the stronger no- theorem by allowing the cloning of the input state ψ cloning theorem will lead to signalling. Since we already i to be dependent on the register α . Now, suppose Bob| i know that negating the no-deletion theorem will lead to i does some local operation on his| i system by attaching signalling [13]. Therefore, together with our proof we ancilla C and applying the transformations given by can say that the violation of the permanence property of (8).
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