aa r X i v : . [ qu a n t - ph ] J un Partial Swapping, Unitarity andNo-signalling
I.Chakrabarty ∗ Department of MathematicsHeritage Institute of Technology,Kolkata-107,West Bengal,India
Abstract
It is a well known fact that an quantum state | ψ ( θ, φ ) i is represented by a pointon the Bloch sphere, characterized by two parameters θ and φ . In a recent work wealready proved that it is impossible to partially swap these quantum parameters.Here in this work we will show that this impossibility theorem is consistent withprinciples like unitarity of quantum mechanics and no signalling principle. PACS numbers: 03.67.-a
In quantum information theory understanding the limits of fidelity of different operationshas become an important area of research. Noticing these kind of operations which arefeasible in classical world but have a much restricted domain in quantum informationtheory started with the famous ’no-cloning’ theorem [1]. The theorem states that onecannot make a perfect replica of a single quantum state. Later it was also shown byPati and Braunstein that we cannot delete either of the two quantum states when weare provided with two identical quantum states at our input port [2]. In spite of these ∗ Corresponding author: [email protected] θ and φ , then the partialswapping of quantum parameters θ and φ is given by, | A ( θ , φ ) i| ¯ A ( ¯ θ , ¯ φ ) i −→ | A ( θ , ¯ φ ) i| ¯ A ( ¯ θ , φ ) i (1) | A ( θ , φ ) i| ¯ A ( ¯ θ , ¯ φ ) i −→ | A ( ¯ θ , φ ) i| ¯ A ( θ , ¯ φ ) i (2)However in ref [6] we showed that this operation is impossible in the quantum domainfrom the linear structure of quantum theory.In this work we once again claim this impossibility from two different principles namely[i] unitarity of quantum mechanics [ii] no signalling principle. The organization of thework is as follows: In the first section we will prove this impossibility from the unitarityof quantum mechanics. In the second section we will do the same from the principle ofno signalling. Then the conclusion follows. Let us consider a set S consisting of two non orthogonal states S = {| A ( θ , φ ) i , | B ( θ , φ ) i} Let us assume that hypothetically it is possible to partially swap the parameters of thesetwo states | A ( θ , φ ) i , | B ( θ , φ ) i .First of all we will assume that at least in principle swapping of phase angles of two quan-tum states are possible, keeping the azimuthal angles fixed. Therefore the transformationdescribing such an action is given by, | A ( θ , φ ) i| ¯ A ( ¯ θ , ¯ φ ) i −→ | A ( θ , ¯ φ ) i| ¯ A ( ¯ θ , φ ) i| A ( θ , φ ) i| ¯ A ( ¯ θ , ¯ φ ) i −→ | A ( θ , ¯ φ ) i| ¯ A ( ¯ θ , φ ) i (3)2o preserve the unitarity of the above transformation, will preserve the inner product. h A ( θ , φ ) | A ( θ , φ ) ih ¯ A ( ¯ θ , ¯ φ ) | ¯ A ( ¯ θ , ¯ φ ) i = h A ( θ , ¯ φ ) | A ( θ , ¯ φ ) ih ¯ A ( ¯ θ , φ ) | ¯ A ( ¯ θ , φ ) i (4)The above equality will not hold for all values of ( θ, φ ). The equality will hold if i ) tan θ tan θ = tan ¯ θ tan ¯ θ or ii )( φ − φ ) = ( ¯ φ − ¯ φ ) ± kπ , where k is an inte-ger. These two conditions characterize the set of states on the Bloch sphere for which thepartial swapping of the phase angles are possible. However in general this is not true forall possible values of θ i , φ i where ( i = 1 , | A ( θ , φ ) i| ¯ A ( ¯ θ , ¯ φ ) i −→ | A ( ¯ θ , φ ) i| ¯ A ( θ , ¯ φ ) i| A ( θ , φ ) i| ¯ A ( ¯ θ , ¯ φ ) i −→ | A ( ¯ θ , φ ) i| ¯ A ( θ , ¯ φ ) i (5)Now once again, in order to preserve the unitarity of such a transformation we arrive at thesame conditions, (i) and (ii). This clearly indicates the fact that there are certain class ofstates on the bloch sphere for which partial swapping of phase angles and azimuthal anglesare possible. However in this context we cannot say that this is true for all such valuesof phase and azimuthal on the bloch sphere. Therefore it is evident that the unitarity ofquantum mechanics, doesn’t allow partial swapping of quantum parameters for all suchpairs of non orthogonal states on the bloch sphere. Suppose we have two identical singlet states | χ i shared by two distant parties Alice andBob. Since the singlet states are invariant under local unitary operations, it remains samein all basis. The states are given by | χ i| χ i = 12 ( | ψ ( θ , φ ) i| ¯ ψ ( ¯ θ , ¯ φ ) i − | ¯ ψ ( ¯ θ , ¯ φ ) i| ψ ( θ , φ ) i )( | ψ ( θ , φ ) i| ¯ ψ ( ¯ θ , ¯ φ ) i − | ¯ ψ ( ¯ θ , ¯ φ ) i| ψ ( θ , φ ) i )= 12 ( | ψ ( θ , φ ) i| ¯ ψ ( ¯ θ , ¯ φ ) i − | ¯ ψ ( ¯ θ , ¯ φ ) i| ψ ( θ , φ ) i )( | ψ ( θ , φ ) i| ¯ ψ ( ¯ θ , ¯ φ ) i − | ¯ ψ ( ¯ θ , ¯ φ ) i| ψ ( θ , φ ) i ) (6)3here {| ψ i , | ¯ ψ i} and {| ψ i , | ¯ ψ i} are two sets of mutually orthogonal spin states (qubitbasis). Alice possesses the first particle while Bob possesses the second particle. Alice canchoose to measure the spin in any one of the qubit basis namely {| ψ i , | ¯ ψ i} , {| ψ i , | ¯ ψ i} .The theorem of no signalling tells us that the measurement outcome of one of the twoparties are invariant under local unitary transformation done by other party on his or herqubit.The density matrix is invariant under local unitary operation by the other party.Hence the first party cannot distinguish two mixtures due to the unitary operation doneat remote place.At this point one may ask if Alice(Bob) partially swap the quantum parameters of her(his)particle and if Bob(Alice) measure his(her) particle in either of the two basis then is thereany possibility that Alice(Bob) know the basis in which Bob(Alice) measures his(her)qubit or in other words, is there any way by which Alice(Bob) using a perfect partialswapping machine can distinguish the statistical mixture in her(his) subsystem resultingfrom the measurement done by Bob(Alice). If Alice(Bob) can do this then signallingwill take place, which is impossible. Note that whatever measurement Bob(Alice) does,Alice(Bob) does not learn the results and her(his) description will remain as that of acompletely random mixture , i.e., ρ A ( B ) = I ⊗ I . In other words we can say that the localoperations performed on his(her) subspace has no effect on Alice’s(Bob’s) description ofher(his) states.Let us consider a situation where Alice is in possession of a hypothetical machine whichcan partially swap quantum parameters θ and φ .Let us first of all consider the case where with the help of the machine we can partiallyswap the phase angles keeping the azimuthal angles of the states fixed. The action ofsuch a machine is given by, | ψ i ( θ i , φ i ) i| ¯ ψ i ( ¯ θ i , ¯ φ i ) i −→ | ψ i ( θ i , ¯ φ i ) i| ¯ ψ i ( ¯ θ i , φ i ) i (7)where ( i = 1 , | χ i P S | χ i P S = 12 [( | ψ ( θ , φ ) ψ ( θ , φ ) i ) A ( | ¯ ψ ( ¯ θ , ¯ φ ) ¯ ψ ( ¯ θ , ¯ φ ) i ) B +( | ¯ ψ ( ¯ θ , ¯ φ ) ¯ ψ ( ¯ θ , ¯ φ ) i ) A ( | ψ ( θ , φ ) ψ ( θ , φ ) i ) B − ( | ψ ( θ , ¯ φ ) ¯ ψ ( ¯ θ , φ ) i ) A ( | ¯ ψ ( ¯ θ , ¯ φ ) ψ ( θ , φ ) i ) B − ( | ¯ ψ ( ¯ θ , φ ) ψ ( θ , ¯ φ ) i ) A ( | ψ ( θ , φ ) ¯ ψ ( ¯ θ , ¯ φ ) i ) B ]= 12 [( | ψ ( θ , φ ) ψ ( θ , φ ) i ) A ( | ¯ ψ ( ¯ θ , ¯ φ ) ¯ ψ ( ¯ θ , ¯ φ ) i ) B +( | ¯ ψ ( ¯ θ , ¯ φ ) ¯ ψ ( ¯ θ , ¯ φ ) i ) A ( | ψ ( θ , φ ) ψ ( θ , φ ) i ) B − ( | ψ ( θ , ¯ φ ) ¯ ψ ( ¯ θ , φ ) i ) A ( | ¯ ψ ( ¯ θ , ¯ φ ) ψ ( θ , φ ) i ) B − ( | ¯ ψ ( ¯ θ , φ ) ψ ( θ , ¯ φ ) i ) A ( | ψ ( θ , φ ) ¯ ψ ( ¯ θ , ¯ φ ) i ) B ] (8)where A, B denotes the particles in Alice’s and Bob’s possession respectively.Now, if Bob does his measurement on {| ψ i , | ¯ ψ i} qubit basis, then the reduced densitymatrix describing Alice’s subsystem is given by, ρ A = 14 [ | ψ ( θ , φ ) ψ ( θ , φ ) ih ψ ( θ , φ ) ψ ( θ , φ ) | + | ¯ ψ ( ¯ θ , ¯ φ ) ¯ ψ ( ¯ θ , ¯ φ ) ih ¯ ψ ( ¯ θ , ¯ φ ) ¯ ψ ( ¯ θ , ¯ φ ) | + | ψ ( θ , ¯ φ ) ¯ ψ ( ¯ θ , φ ) ih ψ ( θ , ¯ φ ) ¯ ψ ( ¯ θ , φ ) | + | ¯ ψ ( ¯ θ , φ ) ψ ( θ , ¯ φ ) ih ¯ ψ ( ¯ θ , φ ) ψ ( θ , ¯ φ ) | ] (9)Interestingly if Bob does his measurement in {| ψ i , | ¯ ψ i} qubit basis, then the densitymatrix representing Alice’s subsystem is given by, ρ A = 14 [ | ψ ( θ , φ ) ψ ( θ , φ ) ih ψ ( θ , φ ) ψ ( θ , φ ) | + | ¯ ψ ( ¯ θ , ¯ φ ) ¯ ψ ( ¯ θ , ¯ φ ) ih ¯ ψ ( ¯ θ , ¯ φ ) ¯ ψ ( ¯ θ , ¯ φ ) | + | ψ ( θ , ¯ φ ) ¯ ψ ( ¯ θ , φ ) ih ψ ( θ , ¯ φ ) ¯ ψ ( ¯ θ , φ ) | + | ¯ ψ ( ¯ θ , φ ) ψ ( θ , ¯ φ ) ih ¯ ψ ( ¯ θ , φ ) ψ ( θ , ¯ φ ) | ] (10)It is clearly evident that the equations (9) and (10) are not identical in any respect andhenceforth we will conclude that Alice can distinguish the basis in which Bob has per-formed the measurement, This is impossible in principle as this will violate the causality.5ence we arrive at a contradiction with the assumption that the partial swapping of phaseangle is possible.Next we see that whether the partial swapping of azimuthal angles is consistent with theprinciple of no signalling or not.If we assume that the partial swapping of phase angles ispossible keeping the azimuthal angles fixed, then its action is given by, | ψ i ( θ i , φ i ) i| ¯ ψ i ( ¯ θ i , ¯ φ i ) i −→ | ψ i ( ¯ θ i , φ i ) i| ¯ ψ i ( θ i , φ i ) i (11)Let us assume that this hypothetical machine is in possession of Alice, and she appliesthe transformation (11) on her particles as a result of which the entangled state (6) takesthe form, | χ i P S | χ i P S = 12 [( | ψ ( θ , φ ) ψ ( θ , φ ) i ) A ( | ¯ ψ ( ¯ θ , ¯ φ ) ¯ ψ ( ¯ θ , ¯ φ ) i ) B +( | ¯ ψ ( ¯ θ , ¯ φ ) ¯ ψ ( ¯ θ , ¯ φ ) i ) A ( | ψ ( θ , φ ) ψ ( θ , φ ) i ) B − ( | ψ ( ¯ θ , φ ) ¯ ψ ( θ , φ ) i ) A ( | ¯ ψ ( ¯ θ , ¯ φ ) ψ ( θ , φ ) i ) B − ( | ¯ ψ ( θ , φ ) ψ ( ¯ θ , ¯ φ ) i ) A ( | ψ ( θ , φ ) ¯ ψ ( ¯ θ , ¯ φ ) i ) B ]= 12 [( | ψ ( θ , φ ) ψ ( θ , φ ) i ) A ( | ¯ ψ ( ¯ θ , ¯ φ ) ¯ ψ ( ¯ θ , ¯ φ ) i ) B +( | ¯ ψ ( ¯ θ , ¯ φ ) ¯ ψ ( ¯ θ , ¯ φ ) i ) A ( | ψ ( θ , φ ) ψ ( θ , φ ) i ) B − ( | ψ ( ¯ θ , ¯ φ ) ¯ ψ ( θ , φ ) i ) A ( | ¯ ψ ( ¯ θ , ¯ φ ) ψ ( θ , φ ) i ) B − ( | ¯ ψ ( θ , φ ) ψ ( ¯ θ , ¯ φ ) i ) A ( | ψ ( θ , φ ) ¯ ψ ( ¯ θ , ¯ φ ) i ) B ] (12)If Bob does his measurement in any one of the two basis {| ψ i , | ¯ ψ i} and {| ψ i , | ¯ ψ i} ,then the respective density matrix representing Alice’s subsystem is given as, ρ A = 14 [ | ψ ( θ , φ ) ψ ( θ , φ ) ih ψ ( θ , φ ) ψ ( θ , φ ) | + | ¯ ψ ( ¯ θ , ¯ φ ) ¯ ψ ( ¯ θ , ¯ φ ) ih ¯ ψ ( ¯ θ , ¯ φ ) ¯ ψ ( ¯ θ , ¯ φ ) | + | ψ ( ¯ θ , φ ) ¯ ψ ( θ , ¯ φ ) ih ψ ( ¯ θ , φ ) ¯ ψ ( θ , ¯ φ ) | + | ¯ ψ ( θ , ¯ φ ) ψ ( ¯ θ , φ ) ih ¯ ψ ( θ , ¯ φ ) ψ ( ¯ θ , φ ) | ]= 14 [ | ψ ( θ , φ ) ψ ( θ , φ ) ih ψ ( θ , φ ) ψ ( θ , φ ) | + | ¯ ψ ( ¯ θ , ¯ φ ) ¯ ψ ( ¯ θ , ¯ φ ) ih ¯ ψ ( ¯ θ , ¯ φ ) ¯ ψ ( ¯ θ , ¯ φ ) | + | ψ ( ¯ θ , φ ) ¯ ψ ( θ , ¯ φ ) ih ψ ( ¯ θ , φ ) ¯ ψ ( θ , ¯ φ ) | + | ¯ ψ ( θ , ¯ φ ) ψ ( ¯ θ , φ ) ih ¯ ψ ( θ , ¯ φ ) ψ ( ¯ θ , φ ) | ] (13)6lice can easily distinguish two statistical mixtures and as a consequence of which shecan easily understand in which basis Bob has performed his measurement. This is notpossible in principle as this will violate causality. Hence we conclude that the partialswapping of azimuthal angles is not possible. I.C acknowledges Almighty and Prof. C.G.Chakrabarti for being the source of inspirationin carrying out research. [1] W.K.Wootters and W.H.Zurek,Nature ,802(1982).[2] A.K.Pati and S.L.Braunstein, Nature404