arXiv:quant-ph/0605173v1 19 May 2006 n.I hswr eso htide rnilslk no- like principles LOCC under indeed entanglement signal- that of conservation show to and we signalling lead work this could In permanence information ing. of quantum violation If of that signalling. mean property would to that lead then also yes, could theorem information won- no-cloning quantum stronger may the in Since one of violation signalling whether information. implies der operation quantum deletion of the stronger together permanence taken The a theorems called implies no-deleting [16]. is the theorem This and non-orthogonal no-cloning of no-cloning stronger information. set a full the of supply as copy must proved a one was make states to it order theorem in [15]. no-cloning fol- that information the theorems quantum strengthening of no-deletion In conservation the the and from shown low no-cloning was it the Recently, of that non-increase [14]. the LOCC and under entanglement no-signalling been the has theorem from no-flipping obtained spatially The two established [13]. between parties was no-signalling separated of theorem principle the no-deleting 14]. from 13, the 12, 11, example, [10, systems For closed un- for entanglement operations of local non-increase der impossi- and like signalling principles of fundamental bility the from derived be quan- in [9]. operations’ theory Impossible Information unified ‘General tum been name have a theorems under the no-complementing in the no-conjugation, theorems ‘no-flipping’ the and other no-cloning, the few the have Moreover, and erasure’ we literature. [8], ‘no-partial ‘no-splitting’ particular, the the [6], In replication’ [7], ‘no-self 8]. the 5], 7, [4, 6, theorems’ theorems ‘no-go 5, 3]. other of heading many [4, 2, broad are this [1, there under come two as theorems which these known no-deleting to now addition the In are and which no-cloning states the quantum delete in and arbitrary information, clone an cannot classical we Unlike theory, information world. quantum impossibili- quantum become which in information operations ties digitized Many in feasible classi- information. are between quantum differences and various cal know to importance Introduction: o,w nwta hs mosbeterm a also can theorems impossible these that know we Now, togrn-lnn,n-inligadcnevto fqu of conservation and no-signalling no-cloning, Stronger rmtecnevto fqatmifrain hs observ no-sign the These to connected is information. information. information quantum quantum of of property sh conservation Furthermore, we the signalling. Here, to from lead signalling. could Al imply theorem would no-cloning information. theorem quantum no-deletion of the property permanence the provide ti nw httesrne ocoigtermadteno-d the and theorem no-cloning stronger the that known is It nrcn er ti ffundamental of is it years recent In 2 eglEgneigadSineUiest,Howrah-711103 University, Science and Engineering Bengal .Chakrabarty, I. 3 nttt fPyis hbnsa-505 Orissa,India Bhubaneswar-751005, Physics, of Institute 1 eiaeIsiueo ehooy okt,India Kolkata, Technology, of Institute Heritage ,2, 1, Dtd a ,2019) 3, May (Dated: ∗ .K Pati, K. A. ehncli aueadteei ocasclanalogue. quantum classical entirely no is is It there and this. nature to in mechanical akin is any is after information which even quantum something of exist Permanence to continue process. will physical and process place physical any taken has before like existed we already resemble neither which information energy that quantum the fact From viewpoint the information. objective quantum to destroy nor refers understand create quan- It could must of permanence we information. term Now informa- the tum by that mean quantity. us we actually physical tells what a which objective, is (ii) tion interacting and nothing after it, is system with a information about this obtained knowledge to but information: according regarding subjective, concepts fact, In (i) opposing only [17]. two not quantity are physical information a there treating as also are but we knowledge as that mind in keep information. quantum of permanence the establish sasto uennotooa tts hnteei no transformation is the there achieve then can states, | that non-orthogonal operation pure of unitary set a is hoe ttsta if that states theorem ann ootooa ar hntetransformation the then pair, orthogonal no taining spsil iff possible is where ef- in Thus copy. first of infor- cloning the fect supplementary of the independent from alone, mation generated be always can h ln tt.Here state. blank the rm osdrapyia rcs ecie yteequa- the by described tion process physical a Consider orem. the as known is [16]. This theorem’ input. no-cloning additional ‘stronger an as provided is iomna pc and space vironmental ψ i ealta h ocoigtermsae htif that states theorem no-cloning the that Recall should we all, of first details, the into going Before e sbifl eaiuaetesrne ocoigthe- no-cloning stronger the recapitulate briefly us Let | → i 3, † epoetesrne ocoigtheorem no-cloning stronger the prove we | n .Adhikari S. and ψ ligadtecnevto fquantum of conservation the and alling i ψ wta h ilto ftestronger the of violation the that ow i o ti nw httevoainof violation the that known is it so, i tosipyta h permanence the that imply ations i| stecp hc st ecoe and cloned be to is which copy the is | ψ ψ | i i ltn hoe ae together taken theorem eleting | a i| i ψ i 0 | → i etBna,India Bengal, West , i 2.Hwvr h togrno-cloning stronger the However, [2]. i | i| ψ spsil nywe h eodcopy second the when only possible is α i i| i i| a ψ {| C | i 2, i C nu information antum | → i | i ψ | → i α hti h eodcopy second the is That . ‡ i i i i} i steiiilsaeo h en- the of state initial the is ψ stesaewihalready which state the is sasto uesae con- states pure of set a is ψ i i| i i| ψ ψ i i i i| C i i , {| | 0 ψ | i ψ (1) (2) i i} i is 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 {|statesi | (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). As| a result,i the combined system given by (7) takes quantum information will lead to signalling. Alternately, the form we can say that the no-signalling leads to permanence of quantum information. In the second part, we will show χ ξ C | i| i| i → that the violation of the stronger no-cloning theorem will 1 [ ψi αi ( ψi ψi Ci2 )+ ψi αi ( ψi ψi Ci1 ) lead to the violation of the principle of conservation of en- 2 | i| i | i| i| i | i| i | i| i| i tanglement under LOCC (conservation of information). + ψi αi ( φi )+ ψi αi ( φi )]. (9) This clearly indicates that the principle of conservation | i| i | i | i| i | i Now if Alice finds her particles in the basis ψ1 , ψ1 of entanglement under LOCC implies the permanence of {| i | i} quantum information. and α1 , α1 , the reduced density matrix describing the Bob’s{| i system| i} is given by Stronger no-cloning from no-signalling: Suppose we have two singlet states χ , ξ shared by two distant 1 ρB = [ ψ1ψ1 ψ1ψ1 + ψ1ψ1 ψ1ψ1 parties Alice and Bob. Since| i the| singleti states are invari- 4 | ih | | ih | ant under local unitary operations, it remains same in all + φ1 φ1 + φ1 φ1 ]. (10) | ih | | ih | 3

Now if Alice finds her particles in the basis ψ2 , ψ2 The reduced density matrix on Alice side before the {| i | i} and α2 , α2 , the reduced density matrix of Bob’s sys- application of hypothetical machine is given by tem{| is giveni | byi} A 1 ρ = [ 0 0 + 1 0 ψi ψj αi αj + 1 2 | ih | | ih |h | ih | i ρB = [ ψ2ψ2 ψ2ψ2 + ψ2ψ2 ψ2ψ2 4 | ih | | ih | 0 1 ψj ψi αj αi + 1 1 ]. (16) | ih |h | ih | i | ih | + φ2 φ2 + φ2 φ2 ]. (11) | ih | | ih | Now after the application of the transformation defined Since the statistical mixture in (10) and (11) are differ- in (12) and (13) by Bob on his qubits the entangled state ent, this would allow Bob to distinguish in which basis takes the form, Alice has performed measurement, thus allowing for su- 1 per luminal signalling. Thus, we see that the violation of Ψ SNC = [ 0 A ψi B ψi B Ci B + | i √ | i | i | i | i stronger no-cloning theorem will lead to signalling. It had 2 1 A ψj B ψj B Cj B ]. (17) already been seen that the violation of the no-deleting | i | i | i | i theorem will lead to signalling [13], henceforth we can The reduced density matrix on Alice side by tracing out conclude that the violation of permanence property of Bob’s part is given by quantum information will make super luminal signalling possible. A 1 2 ρ = [ 0 0 + 1 0 ψi ψj Ci Cj Stronger no-cloning theorem from conservation SNC 2 | ih | | ih |h | i h | i of information: 2 Now we will show the principle of con- + 0 1 ψj ψi Cj Ci + 1 1 ]. (18) servation of entanglement ( conservation of information | ih |h | i h | i | ih | in a closed system) under LOCC will imply the stronger Now from equation (14) we can say that equations (16) no-cloning theorem in quantum information. This will and (18) are not identical. Since transformation defined prove that if we violate the permanence of quantum infor- in (12) and (13) are local operation and there is no clas- mation we will also violate the principle of conservation sical communication between two distant parties Alice of entanglement (information). and Bob, to maintain the principle of no signalling the Now negating the stronger no-cloning theorem is equiv- density matrix on Alice’s side must remain unchanged. alent of rejecting the unitary equivalence of two sets of A different density matrix on Alice’s side will imply that pure states αi and βi (say) having equal matrices the signalling has taken place. {| i} {| i} of inner product i.e αi αj = βi βj [16]. In other words Let us consider the entangled state given by equation we can say that thereh | existsi anh | unitaryi operator opera- (15), which is shared by two distant parties Alice and tor U which describe the physical process (2) but violate Bob. The largest eigen value of the reduced density ma- the condition αi αj = βi βj . Now we look out for the trix (16) on Alice’s side is given by equivalent mathematicalh | i h expression| i for the statement like 1 a b ‘negating the stronger no-cloning theorem’. Consider the λA = + | || | (19) following transformations 2 2 and the largest eigen value of the density matrix (18) on ψi 0 αi C ψi ψi Ci (12) | i| i| i| i → | i| i| i Alice’s side after the application transformation defined ψj 0 αj C ψj ψj Cj (13) | i| i| i| i → | i| i| i in (12) and (13) is given by and let us negate the stronger no-cloning theorem, i.e., 1 a 2 c λA = + | | | |, (20) we assume that there exists a unitary operator, for which SNC 2 2 the inner products are not equal where a = ψi ψj , b = αi αj and c = Ci Cj . h | i h | i h | i αi αj = ψi ψj Ci Cj . (14) Now from equation (14) it is clear that equations (19) h | i 6 h | ih | i and (20) are not identical. This violates the principle Let us consider an entangled state shared by two distant of conservation of entanglement under LOCC. Thus we parties Alice and Bob again observe that by violating the permanence property 1 of quantum information we are violating the principle Ψ = [ 0 A ψi B αi B + 1 A ψj B αj B ], (15) of conservation of entanglement under LOCC or in other | i √2 | i | i | i | i | i | i words we can say that the principle of conservation of en- where Alice is in possession of the qubit ‘A’ and Bob is tanglement (conservation of information) under LOCC in possession of the qubit ‘B’. Now imagine that Bob is implies the permanence property of quantum informa- in possession of a hypothetical machine whose action on tion. his qubit is given by the transformation defined in (12) Conclusion: In summary, we have shown that if we and (13). He attaches two ancillas in the state 0 B C B could negate the stronger no-cloning theorem then we and carries out the transformation (12) and (13).| i | i could send signal faster than light. Thus, the stronger 4 no-cloning follows from the no-signalling. Earlier, it has † Electronic address: [email protected] been shown that negating the no-deletion theorem also ‡ Electronic address: [email protected] leads to signalling [13]. Since the stronger no-cloning and [1] W. K. Wootters and W. H. Zurek, Nature 299, the no-deleting theorems taken together provide perma- 802(1982). [2] H. P. Yuen, Phys. Lett. A 113, 405 (1986). nence to quantum information, this in turn implies that [3] A. K. Pati and S. L. Braunstein, Nature 404, 164(2000). the violation of the permanence property of quantum in- [4] V. Buzek, M. Hillery and R. F. Werner, Phys. Rev. A formation could also lead to signalling condition. In other 60, R2626 (1999). words the only possibility that we can import and export [5] N. Gisin and S. Popescu, Phys. Rev. Lett.83, 432 (1999). quantum information without being able to creat or de- [6] A. K. Pati and S. L. Braunstein, Quantum Mechanical stroy is consistent with no-signalling condition. Also, we Universal Constructor, quant-ph/0303124. have proved the stronger no-cloning theorem from the [7] A. K. Pati and B. C. Sanders, No partial erasure of quan- tum information, quant-ph/0503138. conservation of quantum information under LOCC oper- [8] D. Zhou, B. Zeng, and L. You, Quantum infor- ation. mation cannot be split into complementary parts, quant-ph/0503168. [9] A. K. Pati, Phys. Rev. A 66, 062319 (2002). [10] N. Gisin, Phys. Lett. A 242, 1 (1998). Acknowledgement: Indranil and Satyabrata acknowl- [11] L. Hardy and D. D. Song, Phys. Lett. A 259, 331(1999). edge Prof B. S. Choudhuri, Head of the Department [12] A. K. Pati, Phys. Lett. A 270, 103 (2000). of Mathematics, Bengal Engineering and Science Uni- [13] A. K. Pati and S. L. Braunstein, Phys. Lett. A 315, 208 versity. Indranil acknowledge Prof C. G. Chakraborti, (2003). S.N.Bose Professor of Theoretical Physics, Department [14] I. Chattopadhyay, S. K. Choudhary, G. Kar, S. Kunkri, of Applied Mathematics, University of Calcutta for be- and D. Sarkar, Phys. Lett. A 351, 384 (2006). ing the source of inspiration in carrying out research. [15] M. Horodecki, R. Horodecki, A. Sen(De) and U. Sen, No- deleting and no-cloning principles as consequences of con- This work is done in Institute of Physics, Bhubaneshwar, servation of quantum information, quant-ph/0306044. Orissa. Indranil and Satyabrata gratefully acknowledge [16] R. Jozsa, A Stronger no-cloning theorem, their hospitality and also acknowledge Dr. P. Agarwal for quant-ph/0204153. having various useful discussions. Authors also acknowl- [17] R. Landauer, Physics Today 44(5), 22 (1991). edge Dr. G. P. Kar for giving various useful suggestions and encouragement for the improvement of the work.

∗ Electronic address: [email protected]