Impossible Operations in the Quantum World-I and II

1. Impossible Operations in the Quantum World-I and II Debasis Sarkar dsappmath@caluniv.ac.in Department of Applied Mathematics, University of Calcutta

2. 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

3. 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….!

4. 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

5. 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.

6. 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.

7. Composite Systems • Consider physical systems consist of two or more number of parties A, B, C, D, …… • The associated Hilbert space is HAHB HC HD … • States are then classified in two ways • (I) Separable:- have the form, ρABCD =wiρiAρiBρiCρiD with 0 wi 1, and wi =1. • (II) All other states are entangled.

8. Pure Bipartite System States are represented by unique Schmidt form here is the Schmidt rank, and are the Schmidt coefficients, such that . Schmidt vector of this State is A B

9. 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).

10. Some Use of Quantum Entanglement • Quantum Teleportation,(Bennett et.al., PRL, 1993) • Dense coding, (Bennett et.al., PRL, 1992) • Quantum cryptography, (Ekert, PRL, 1991)

11. 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 is a linear operator that satisfies the relation

12. Separable Super operator If then the operation is trace preserving. When the state is shared between a number of parties, say, A, B, C, D,. .... and each has the form with all of are linear operators then the operation is said to be a separable super operator.

13. 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.

14. A result : Every LOCC is a separable superoperator. But whether the converse isalso 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).

15. 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.

16. 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?

17. 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: where |b>,|M> are the blank and machine states.

18. 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, = If the states are different, then the above relation immediately implies, both must be orthogonal. One may also prove the theorem using linearity.

19. 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().

20. 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.

21. 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:

22. 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: Ref.:-Pati & Braunstein, Nature, 404(2000)164.

23. 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 and using other two relations,

24. 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?

25. 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.

26. 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 if and only if there is a physical operation i.e., full information is needed to clone.

27. Outline of the proof • Use the lemma that: Given two sets of pure states {|i}, {|i}, if they have equal matrices of inner products, ijij 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.

28. 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.

29. 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:

30. 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)

31. 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:

32. 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.

33. 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.

34. 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: where ,  are arbitrary and F is a function of the original. Ref:- Pati, PRA, 66(2002)062319.

35. 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.

36. Non-Existence of Universal Hadamard Gate • There are two ways of defining it: • Check the possibility of the following operations: • Or,

37. 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.

38. 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?

39. 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.

40. 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.

41. 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.

42. For theorem-2, take an orthonormal basis {|i}, i= 1,…,d. Then partial erasure machine yields, Where|A> is the ancilla state. • Now consider an arbitrary state |. Then, • Clearly, the resultant state of the system is mixed that contradicts the definition of erasure to be pure.

43. 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.

44. No-Hiding • Perfect Hiding is a process , where is arbitrary input state and 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”.

45. 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, • Examining physical nature of hiding, we have,

46. 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.

47. 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.

48. 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.

49. 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.

50. 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.