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Impossible Operations in the Quantum World-I and II Debasis Sarkar dsappmath@caluniv.ac.in Department of Applied Mathematics, University of Calcutta A List of Impossible Operations (No Go Principles) 1. Impossibility of Exact Cloning (No Cloning) 2. Impossibility of Exact Deletion (No Deleting) 3. Stronger No-Cloning 4. Non-Existence of Universal Exact Flipper (No Flipping) 5. No Partial Cloning 6. No Partial Erasure 7. Non-existence of Universal Hadamard Gate 8. Impossibility of Probabilistic Cloning 9. Impossibility of Broadcasting mixed states 10. No Splitting 11. No Hiding Etc….! Some Basic Notions about Quantum Systems Physical System- associated with a separable complex Hilbert space Observables are linear, self-adjoint operators acting on the Hilbert space States are represented by density operators acting on the Hilbert space Measurements are governed by two rules 1. Projection Postulate:- After the measurement of an observable A on a physical system represented by the state ρ, the system jumps into one of the eigen states of A. 2. Born Rule:- The probability of obtaining the system in an eigen state is given by Tr(ρP[ ]). The evolution is governed by an unitary operator or in other words by Schrodinger’s evolution equation. States of a Physical System Suppose H be the Hilbert space associated with the physical system. Then by a state ρ we mean a linear, Hermitian operator acting on the Hilbert space H such that It is non-negative definite and Tr(ρ)= 1. A state is pure iff ρ2 = ρ and otherwise mixed. Pure state has the form ρ=|, |H. Composite Systems Consider physical systems consist of two or more number of parties A, B, C, D, …… The associated Hilbert space is HAHB HC HD … States are then classified in two ways (I) Separable:- have the form, ρABCD =wi ρiA ρiB ρiC ρiD with 0 wi 1, and wi =1. (II) All other states are entangled. Pure Bipartite System States are represented by unique Schmidt form here A is the Schmidt rank, AB B and are the Schmidt coefficients, such that . Schmidt vector of this State is ( , , ) 1 2 d Pure Bipartite Entanglement Entanglement of pure state is uniquely measured by von Nuemann entropy of its subsystems, States are locally unitarily connected if and only if they have same Schmidt vector, hence their entanglement must be equal. Thermo-dynamical law of Entanglement : Amount of Entanglement of a state cannot be increased by any LOCC (local operations performed on subsystems together with classical communications between the subsystems). Some Use of Quantum Entanglement Quantum Teleportation,(Bennett et.al., PRL, 1993) Dense coding, (Bennett et.al., PRL, 1992) Quantum cryptography, (Ekert, PRL, 1991) Physical Operation Suppose a physical system is described by a state Krause describe the notion of a physical operation defined on as a completely positive map , acting on the system and described by where each Ak is a linear operator that satisfies the relation Separable Super operator If then the operation Ak is trace preserving. When the state is shared between a number of parties, say, A, B, C, D,. .... and each Ak has the form A B C D with all of Lk , Lk , Lk , Lk , are linear operators then the operation is said to be a separable super operator. Local operations with classical communications (LOCC) Consider a physical system shared between a number of parties situated at distant laboratories. Then the joint operation performed on this system is said to be a LOCC if it can be achieved by a set of some local operations over the subsystems at different labs together with the communications between them through some classical channel. A result : Every LOCC is a separable superoperator. But whether the converse is also true or not ? It is affirmed that there are separable superoperators which could not be expressed by finite LOCC. Bennett et. al. Phys. Rev. A 59, 1070 (1999). The number of Schmidt coefficients of a pure bipartite entangled states cannot be increased by any LOCC. Measure of Entanglement:- 1. For pure bipartite state entanglement is measured by the Von Neumann entropy of any of it’s subsystems. This is the unique measure for all pure bipartite states. 2. For mixed bipartite entangled states, there is no unique way to define entanglement of a state. Two useful measure of entanglement: Entanglement of Formation and Distillable Entanglement. For multipartite case the situation is very complex. Cloning Basic tasks: Is to copy quantum information exactly in (arbitrary) quantum states. If there exists a machine which can perform such tasks, then we call it a quantum cloning machine. The question is: whether there exists quantum cloning machine which can copy exactly arbitrary quantum information or not? The No-Cloning Theorem Arbitrary quantum information cannot be copied. To prove it, suppose we have provided with two states , and assume that there is an exact cloning machine. Then we could write its action as: b M M b M M where |b>,|M> are the blank and machine states. Sketch of the proof Use the Unitarity of quantum operations, i.e., consider U be the unitary operation responsible for cloning two above states. Unitarity implies, = M M 2 If the states are different, then the above relation immediately implies, both must be orthogonal. One may also prove the theorem using linearity. Some points to Note Quantum mechanics prohibits exact cloning of arbitrary information encoded in quantum states. This does not imply inexact cloning is not possible. For qubit system, there exists optimal universal isotropic quantum cloning machine with fidelity 5/6. If it is restricted further to states in a great circle of Bloch sphere, then fidelity is increased further and is ½+¼(). There is a nice relation between impossibility of discriminating non-orthogonal states with certainty and no cloning theorem (exercise!). There are lot of results already in literature to study the possibility of doing inexact quantum cloning in different quantum systems(see, arXiv) One can also prove no-cloning theorem using some other physical constraints on the system. We will do some later. Ref:-Wootters & Zurek, Nature, 299(1982)802, Diecks, Phys. Lett. A, 92(1982)271, Yuen, Phys. Lett. A, 113(1986)405, Scarani et al, Rev. Mod. Phys., 77(2005)1225, N. Gisin & S. Massar, PRL, 79(1997), 2153, Bruss et al, PRA, 57(1998)2368. Quantum Deleting Here the task is: Given two copies of a quantum state (unknown) |, is it possible to delete the information of one of the part. In other words, given two distinct states , whether there is any quantum operation which could perform the following operation: | | | | 0 | | | | 0 No Deleting theorem Given two copies of an unknown quantum state, it is impossible to delete the information encoded in one of the copy. Suppose we assume there is a quantum operation which can perform the following task: 0 0 M 0 b M0 1 1 M 1 b M1 Ref.:-Pati & Braunstein, Nature, 404(2000)164. Sketch of the proof Use linearity of quantum operations; i.e., as the operation is linear, the above task is also possible for the states, |, where 2 2 . Then, we have, from M b M and using other two relations, { 2 00 ( 01 10 ) 2 11 } M ( 2 0bM 0 2 1bM 1 ) 2 ( 0b 1b ) M Clearly, the above expressions imply it is impossible to delete exactly arbitrary quantum information encoded in a state. Exercise:- What would be the status of ancilla states|M0, |M1 and |M? One may prove the result with other ways also; e.g., considering some constraints on the systems. Observe the differences between cloning and deleting operations. Have there any dual kind of relation between them? Stronger No-Cloning Theorem Here the question is: “How much or what kind of additional quantum information is needed to supplement one copy of a quantum state in order to be able to produce two copies of that state by a physical operation?” i.e., what information is needed initially to clone. Ref.:-R. Jozsa, quant-ph/0305114, quant-ph/0204153. Consider a finite set of non-orthogonal states {|i} and a set of states{i}(generally mixed). Then stronger no-cloning theorem states that: There is a physical operation i i i i i i i if and only if there is a physical operation i i i i.e., full information is needed to clone. Outline of the proof Use the lemma that: Given two sets of pure states {|i}, {|i}, if they have equal matrices of inner products, ijij for all i and j, then there is a unitary operation U on the direct sum of the state spaces with U|i=|i, for all i and vice-versa. Now consider first i be a pure state. Then prove the result using lemma. After that using i as mixture of pure states, prove the result in general. Some consequences For cloning assisted by classical information (i are required to be mutually commuting), supplementary data must contain full identity of the states as classical information. The proof of no-deleting theorem could be done using the lemma that used in stronger no cloning theorem. Both no-go theorems together establish permanence of quantum information. Spin Flipping Here the task is: Whether it is possible to flip spin directions of any arbitrary qubit or not? In other words, is it possible to construct a quantum device which could take an arbitrary(unknown) qubit and transform it into the orthogonal qubit or not? i.e., possibility of following operation: 0 1 * 0 * 1 No-Flipping theorem It is not possible to flip arbitrary (unknown) spin directions. To prove it, we must careful about the nature of the operation we are trying to do. Unlike, cloning here it is not possible to prove the result by considering only two non-orthogonal states. Why? Because flipping is possible for states lying on a great circle of the Bloch sphere. (exercise) Sketch of the proof One way is to show the operation we have written is in general an anti-unitary operation and proof is complete as anti-unitary operations are not Physical!!(Not a CP map!) Other way is to consider three states not lying on a great circle and prove the impossibility. Consider states like the following: | 0 | a | 0 b | 1 | c | 0 dei | 1 Then consider there is a flipping machine which could act on those three states. Now try to prove the operation is impossible. One may also prove by considering some constraints on the quantum systems. Ref.:-Buzek et al, PRA, 60(1999) R2626, Martini et al, Nature, 419(2002)815. Probabilistic Quantum Cloning Here the task is copying/cloning exactly but not with certainty. i.e., the condition is relaxed one. Interesting fact is that if one try to copy a set of states exactly but probabilistically, then the set should be a linearly independent one. Ref.:-Duan & Guo, PRL, 80 (1998)4999. The intuitive proof follows from a constraint on the system.(Pati, PLA, 270(2000)103; Hardy & Song, PLA, 259(1999)331. Partial Cloning Here we have to study the possibility of cloning partially, i.e., whether there is any quantum device which could perform the following operation or not: b M F ( ) M where 0 1 , are arbitrary and F is a function of the original. Ref:- Pati, PRA, 66(2002)062319. Outline of the proof Consider some cases of F first. Suppose, |F(, where K is a unitary or anti- unitary operator. Then using linearity and anti- linearity of the operation it is straight forward to prove. Then consider K as a combination (linear) unitary and anti-unitary operator and prove the result. One may also prove in general way. Non-Existence of Universal Hadamard Gate There are two ways of defining it: Check the possibility of the following operations: 1 ( ) 2 1 ( ) 2 Or, 1 ( i ) 2 1 (i ) 2 Result: There is no Hadamard gate of the above kind for arbitrary unknown qubits. The proof is straightforward, if we consider any two distinct qubits and then taking inner products of them and their orthogonals before and after the operations. (exercise) Ref:- Pati, PRA, 66(2002)062319. Partial Erasure Reversibility of a quantum operation(unitary) prohibits complete erasure of state (say, qubit). i.e., to transform any qubit to a standard state by unitary evolution is not possible. However, one may ask the following: Whether it is possible to erase partially quantum information, even by using irreversible ones? What do we mean by partial erasure? A partial erasure is a trace preserving completely positive map that maps all pure states of a n- dimensional Hilbert space to pure states in a m- dimensional subspace (m<n) via a constraint. In other words, partial erasure reduces the dimension of the parameter domain and does not leave the state entangled with other system. No-partial Erasure Theorem 1. Given any pair of non-orthogonal qudits, in general, there is no physical operation that can partially erase them. Linearity of quantum theory gives us even more than the above. Theorem 2. Any arbitrary qudit cannot be partially erased by an irreversible operation. Ref.:-A.K.Pati & B.C.Sanders, PLA, 359(2006)31. Outline of the proofs For theorem-1, Consider two states , in a n-dim Hilbert space and suppose there is a partial erasure with a constraint K. For simplicity, choose the constraint such that it reduces the parameter space at least 1 dim. The reduced dim. states then may not have in general equal inner product with the initial states. Now attach ancilla to see the effect of the quantum evolution and considering unitarity take inner product of both side. For theorem-2, take an orthonormal basis {|i}, i= 1,…,d. Then partial erasure machine yields, i A i A i Where|A> is the ancilla state. Now consider an arbitrary state |. Then, d n d n A e Cos i n n A e Cos i n n A n 2 n 2 n Clearly, the resultant state of the system is mixed that contradicts the definition of erasure to be pure. Some Consequences Check the result of theorem-2 for d=2 not using ancilla. For d=2 in theorem-1 an immediate consequence is “A universal NOT gate is impossible”, i.e., No-flipping theorem (Check it) Also from theorem-2 the no-splitting of quantum information , follows. There are also some other consequences. No-Hiding Perfect Hiding is a process i o , where i is arbitrary input state and o be the fixed output state. The input states forms a subspace of a larger Hilbert space and output state resides in a subspace where it has no dependence on input. No-Hiding theorem states, “Quantum information can run, but it can’t hide”. Sketch of the proof Use linearity and unitarity of quantum process. Consider only pure input states using linearity. Unitarity allows suitable choice of ancilla making the space larger. Then, d i pk k o Ak ( ) k 1 Examining physical nature of hiding, we have, d i pk k o ( qk 0) k 1 So, perfect hiding is impossible, as ancilla contains full information of hiding state, which by definition impossible. One may also study the imperfect hiding processes and found the implication of the result of no-hiding theorem. Further, it has severe implication in Black-hole information paradox. Ref.:-Braunstein & Pati, PRL, 98(2007)080502. Constraints over Physical Operations 1) No-Signalling 2) Non-increase of Entanglement under LOCC 3) Impossibility of inter conversion by deterministic LOCC of a pair of Incomparable states. No-Signalling Status : It is a very strong constraint over any physical system. This is not only a quantum mechanical constraint. Power : This is very powerful restriction and senses easily in most of the settings and for almost all kind of impossible operations. Even not using quantum mechanical formalism, e.g., linearity and unitary dynamics. Impossible Operations detected by it : UNIVERSAL EXACT CLONING, UNIVERSAL EXACT DELETING, UNIVERSAL EXACT SPIN-FLIPPING, e.t.c. Non-Increase of Entanglement under LOCC Status : It is quite similar to the thermo-dynamical constraint over physical systems. This is entirely a constraint of quantum information processing. The physical reason behind this restriction entirely depends on the existence of entangled states of a physical system. Entanglement is a measurable physical resource that can be applied to perform various kinds of information and computational tasks. As it may be viewed as an amount of non-locality of the system, therefore we have a principle that entanglement can not be increased by local operations with classical communications. Power : This is also a very powerful restriction over any quantum mechanically allowed Evolution of any physical system and it detects many impossible operations of quantum Information processing. As it is purely a quantum mechanical constraint so we can use it to detect any local operation to be physical or not and we have the facility of using the operation to act linearly on superposition level of quantum states. Impossible Operations detected by using the principle of non-increase of entanglement by LOCC : UNIVERSAL EXACT CLONING - In a pure state setting, by Horodecki et.al. In a much more relaxed mixed state scenario, by I. Chattopadhyay et. al. (in preparation). UNIVERSAL EXACT DELETING – By Horodecki et.al., considering the constraint that, `In a closed system the entanglement remain unchanged under LOCC’. Further, using `Non-increase of entanglement under LOCC’, in a very simple setting, with linearity assumption only on density matrix level and applying the deletion over only two arbitrary qubits, by I. Chattopadhyay et. al. UNIVERSAL EXACT SPIN-FLIPPING - Assuming the existence of exact flipping operation acted on only three states not lying in a great circle of the Bloch sphere and applying this operation locally on a joint system using a less restricted linearity assumption the entanglement content of the system can be increased which establishes the impossibility of universal exact spin-flipping. ***In all the settings we assume linearity of the operations only on density matrix level Quantum Deleting and Signalling Alice-Bob share 2 copy of Singlet state and Bob has 1-2 copy exact deletion machine acting on the arbitrary qubit M b M cos 0 sin 1 as, M b M M M Bob apply the deleting machine on joint state 12 34 M ; 1 2 If Alice measure her particles on either {0,1} or in qubit basis and Bob traced out Machine state, then the remaining 2 qubit mixed state depends on choice of basis() Which can be distinguished exactly from =0. 24 tr135 out 12345 out 1 4 I2 b 4 b tr5 245 tr5 245 Thus assumption of perfect deletion machine can lead to signalling. Ref.:- Pati & Braunstein, PLA, 315(2003)208. No Signalling and Probabilistic Quantum Cloning Alice, Bob share a large no. of entangled state AB N N AB 1 N n 1 n A Bn B 1 N n n 1 A BN n B BnB are arbitrary (not necessarily orthogonal) If Bob have a 1-2 copy PQCM then he have a 1 to copy PQCM, for a large and for a less (but > 0) probability as ( 1) prob 0 Bn b Bn for some values n=1,2,···,N and N+1 Alice has 1 cbit information, corresponding to it she make a measurement on each of her particle, but do not communicate to Bob. BOB apply PQCM on his particle. Cbit Measurement Basis of Input set for Bob’s PQCM value Alice 0 A1 : {1, 2,····, N} B1={B1, B2,····, BN} 1 A2 :{1´, 2´,····, N´} B2={BN+1, BN+2,····, B2N} For general input Bk Bob’s output N 1 c Bl k l d k k where k Bl 0 l 1,2,, N 1 i 1 Bob succeeds to clone exactly copies of particle B, if input is one of Bn ; n=1,2,….,N+1 P0(A1) = n=1N Prob(n| A1) = 1 , P1(A1) = P(N+1| A1) = 0, P0(A2)= N P(n| A2) 1- P(N+1| A2) , P1(A2)= P(N+1| A2) 0 After many repetition P0(A2)→1, Bob can correctly know the bit without any actual communication imply superluminal signaling. Ref.:- L. Hardy & D. D. Song, PLA 259(1999) 331. See also Pati, PLA, 270(2000)103. Impossibility of Exact Cloning Consider a mixed state ρABC with A in one location and B, C are in another location so that the system may be considered as a bipartite system. 00 11 00 11 ABC | | P (1 ) | | P 2 BC 2 where = a0+b1 and = c0+dei 1 are two non- orthogonal qubit state on which we assume exact cloning is possible. The logarithmic negativity (an upper bound of distillable entanglement) of the state is 1 LN ( ) N ( ) 1 bc ad 1 2 2 If exact cloning is possible then we can extract one ebit of entanglement, however in AB:C cut LN() 1. Partial Cloning We define the partial cloning machine on two qubit states , as follows: | | b | | F ( ) | | b | | F ( ) where F(), F() are some functions of , resp. Suppose two distant parties A, B share a state, 1 | AB | 0 A | 0 B |1 A |1 B | b B 2 where the first particle is with A and other two with B and bB is the blank state. Applying partial cloning machine the final state has the form: | AB f 1 2 | 0 A | 0 F ( ) B ei |1 A |1 F ( ) B Tracing out the last two qubits of initial and final states, the reduced state between A and B are AB P 00 P 11 | 2 | 0011| | 2 |1100 | i 1 2 And P 00 P 11 | F | F | 0011| 1 f AB 2 | F | F |1100 | So, whenever 0, i.e., two states remain non- orthogonal, the entanglement content of the final reduced state is always greater than the initial one. Comment: violation of principle of non-increase of entanglement. Quantum Deletion Consider a pure entangled state shared between three parties A, B, C. 1 | ABC | A |1 B |1 C | A | 0 B | 0 C 2 Assume A has a exact deleting machine for two non- orthogonal states , , acting as, | | | | 0 | | | | 0 Final state is then, 1 | ABC d | 0 A |1 B |1 C | 0 A | 0 B | 0 C 2 Tracing out first two qubits of A, the reduced state between B, and C are, BC TrA | ABC | and 1 2 P 00 P 11 | 2 | 0011| | 2 |1100 | d BC TrA | d ABC d | 1 2 P 00 P 11 | | 0011| | |1100 | The concurrence (used to measure entanglement of formation) of initial and final states are, C() =2 and C() = therefore, C ( ) C ( ) which implies, E ( ) E ( ) f f Spin Flipping Consider three arbitrary qubit states not lying in one great circle of the Bloch sphere, | 0 | a | 0 b | 1 | c | 0 dei | 1 where a0, c0, 0, b,d are real nos., such that a2 +b2 =1, c2 +d2 =1 and 0, 1 are orthogonal to each other. Assume the most general flipping operation for those three states as: 0 M 1 M0, Mei M , M ei M . Consider a bipartite pure entangled state shared between two spatially separated parties A, B: |i AB =(13)[|0A |0B+|1A|B+|2A|B ] |MB where A holds 3-dim. system with basis {|0A , |1A,|2A}. Applying exact flipper on the side of B the state takes the form |f AB =(13)[|0A |1M0B+ ei |1A|MB+ ei|2A|MB ] Now if we calculate the reduced density matrices of A’s side, then it is easy to check that the initial and final density matrices are different unless when b=0, or, d=0, or sin()=0. Conclusion: No-signalling implies no-flipping Note:- the above setting does not show any change in entanglement. Next consider a bipartite state shared between A, B with A has a 2-dim. system and B has four qubit system(say, 1,2,3,4), |i AB =(18)[(|000+ |111)A12 |1034 – (|010+|100+|101)A12|34 – (|011+|110+|001)A12|34] |MB Assume B has a exact flipping machine that acts on last qubit(4-th). Then the final joint system takes the form: |f AB =(18)[(|000+ |111)A12 |11M0 34B – ei(|010+|100+|101)A12|M34B – ei(|011+|110+|001)A12|M34B] Now if we calculate the reduced density matrices of A’s side we find that the largest eigen values of initial and final density matrices are related by: f i, which implies E(|fA:B ) E(|iA:B) i.e., entanglement has increased by local operations in A:B cut. Incomparability ~ As a New Constraint over Pure Bipartite System Incomparability of two pure bipartite entangled states imply that either of the two states can not be converted to the other by LOCC with certainty. We propose this as a constraint over the joint system through LOCC. Thus if a local operation can inter-convert an incomparable pair of states, then the operation is certainly an unphysical operation. Nielsen’s criteria Let AB and AB be any two bipartite pure states of Schmidt rank d with Schmidt vectors, and respectively, where 1 i , j 0 i, j , i 1 j and (PRL, 83, 436 (1999)) Then the state can be deterministically AB transformed to the state by LOCC (denoted by ) if and only if majorized by (denoted by ) i.e., if If this criteria is not satisfied for a pair of states then we denote it as AB AB Though it may happen that . The pair of states are said to be comparable if either AB AB or AB AB . From thermo-dynamical law of entanglement one may conclude, For any pair of states, implies rank of AB is greater or equals to the rank of AB . Incomparability of two bipartite pure states :- If for some pair of bipartite pure states we have both |Ψ |Φ and |Φ |Ψ, (i.e., either α1β1 and i=k1 βi i=k1 αi for some k2,3,…,n or β1α1 and i=s1 αi i=s1 βi for some s2,3,…,n Then (|Ψ, |Φ) is said to be an incomparable pair of states and this phenomenon is denoted as |Ψ|Φ. It is to be noted that from the incomparability criteria |Ψ|Φ will not indicate either E(|Ψ) E(|Φ) or E(|Φ) E(|Ψ). If (|Ψ, |Φ) are mn states, where min{m,n}=3, then the criteria for |Ψ |Φ can be expressed as, either α1 β1 and α3 β3 or β1 α1 and β3 α3 Consider an operation ( A) defined on the set of input states { ; i = 1,2,…k} in the following manner A(| i ) | 'i A Now consider the pure bipartite state | AB i 1i | i A | i B k shared between two distant parties Alice and Bob. Applying locally on Bob’s system the joint state transforms to another pure bipartite state | AB i 1 i | i A | 'i B k Now the Schmidt coefficients corresponding to the two pure bipartite states are 'i ; i 1, 2,....k i ; i 1, 2,....k If the two states | AB ,| AB are incomparable, then the operation ( A ) is an impossible one. Universal exact Flipping via Incomparability Special feature: Flipping operation is defined only on a minimum number of three qubit states as in the earlier cases. | 0 | a | 0 b | 1 | c | 0 dei | 1 The flipping operation is described as: | 0 |1 | ei | ei b | 0 a |1 | ei | ei de i | 0 c |1 We omit the machine part as we assume the quantum formalism. Now consider a pure bipartite state shared between A, B: 1 | AB i | 0 A | 00 B |1 A | B | 2 A | B 3 After applying the flipping operation on the second qubit of B, the state takes the form: | f AB 1 3 | 0 A | 01 B ei |1 A | B ei | 2 A | B Now if we calculate the eigen values of the initial and final density matrices of A, then we find that they satisfies the Incomparability criteria unless they are in a great circle. So incomparability implies no-flipping. Also we find the reverse case. We illustrating it by examples. Consider three qubits representing three axes of the Bloch sphere as |Z =|0 , |x = ½ (|0 +|1) and |Y = ½ (|0 + i|1). We define our flipping operation on those three state as follows, |Z |Z |x eiμ |x . ……(1) |Y eiν |Y The flipped state |i is orthogonal to the initial state |i, i= z,x,y and eiμ , eiν are arbitrary phase factors. Let |χi = 1/ 3 {|0A |ZZ B+|1A |xyB+|2A |yxB} be a 34 ( 322 ) state between Alice and Bob situated in two distant labs. Now if Bob applies the flipping operation defined in (1) on the 2nd qubit of his local system then the joint state between them will transform as follows, |χi |χf = 1/ 3 {|0A |ZZ B + eiμ |1A |xyB + eiν |2A|yxB} Schmidt vector corresponding to |χi and |χf are λf = {1/ 3 ( 1/ 3 + 1/2 ), 1/3 , 1/ 3 ( 1/ 3 - 1/2 )} and λi = {2/3 , 1/6 , 1/6 }. Obviously, 1/ 3 ( 1/ 3 - 1/2 ) < 1/6 < 1/3 <1/ 3 ( 1/ 3 + 1/2 ) < 2/3 , which imply that the initial and final joint state between Alice and Bob are incomparable. Hence it is not possible to transform |χi to |χf by LOCC, though |χf is being prepared from |χi by applying the flipping operation locally. Thus the operation defined in (1) is not a valid (physical) operation. So we have |χi |χf Universal Flipping is not possible. Incomparability from No-Flipping Principle Lets consider two pure state of 34 system shared between two spatially separated parties Alice and Bob. |1 = ·51 |0A |0B + ·30 |1A |1B + ·19 |2A |2B |2 = ·49 |0A |0B + ·36 |1A |1B + ·15 |2A |2B Bob’s local system has the following form |0B= |ΨB1 |ΨB2 , |1B = |ΨB1 |ΨB2 , |2B = |ΨB1 |ΨB2 where |Ψ be a arbitrary qubit state |ΨΨ|= ½ [ I + n · σ ]. Local systems of Bob’s first qubit (B1) corresponding to the two joint state |1 and |2 between Alice and Bob are, 1= Tr A B2 ( |1 1| ) = ½ [ I + 0.2 n · σ ] 2 = Tr A B2 ( |2 2| ) = ½ [ I - 0.2 n · σ ] Hence local transformation of |1 to |2 will flip the spin direction of 1 associated with |Ψ. Conclusion : If somehow we extend the LOCC transformation criteria so that either |1 |2 or |2 |1 then consequently on a subsystem, the direction of spin-polarization of an arbitrary state 1 will be reversed. That is equivalent of preparing an exact universal flipping machine. So ``No-Flipping principle’’ |1 |2. This two result together shows an equivalence between the constraint on LOCC state transformation criteria and the No-Flipping principle. The relation is drawn from the unphysical nature of flipping operation as it is not an unitary operation but an anti-unitary operation. Cloning and Deleting via Incomparability This work connects the Cloning and Deleting with the concept of Incomparability as a constraint imposed over the system by LOCC. We define the Cloning operation to act on just two arbitrary input qubits | 0,| in the following manner | 0 | b | 0 | 0 | | b | | Now consider a pure bipartite state shared between Alice and Bob situated at distant labs in the form 1 | AB | 1 A (| 00 | 00) B N | 2 A (| 00 | 00 ) B | 3 A (| 00 | 00) B | b B Let Bob applies the Cloning operation on his last two qubits. Then the joint state transforms to another pure bipartite state 1 | AB | 1 A (| 00 | 000 ) B N | 2 A (| 000 | 00 ) B | 3 A (| 00 | 000) B Tracing out Bob’s subsystem we observe the initial and final form of Alice’s local system and compute the Schmidt vector corresponding to the two joint states | AB ,| AB are and , and find that | AB | AB Thus the cloning operation defined over | 0,| is unphysical in nature. If instead of Cloning we consider the Deleting operation defined on the two arbitrary input states in the following way that Consider now that Alice and Bob will initially share the joint state .Then by assuming the existence of the exact Deleting operation and applying it on Bob’s system the final joint state will be . The incomparability of the pair of states will imply that the exact deleting operation defined on the two arbitrary non-orthogonal states does not exists. Question:- Is it possible to extend the proof via incomparability for general anti-unitary operations and inner- product preserving operations? Answer:- Yes. (ref.- Quant. Inf. & Comp., 7 (2007), 392-400) Thanks to the organizers for inviting me in IPQI-2010, January 4-30, 2010 at Institute of Physics, Bhubaneswar.

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