Local Indistinguishability and Possibility of Hiding cbits in Activable Bound Entangled States
aa r X i v : . [ qu a n t - ph ] N ov Local Indistinguishability and Possibility ofHiding cbits in Activable Bound EntangledStates
Indrani Chattopadhyay ∗ and Debasis Sarkar † Department of Applied Mathematics, University of Calcutta,92, A.P.C. Road, Kolkata- 700009, IndiaNovember 1, 2018
Abstract
In this letter we prove local indistinguishability of four orthogonalactivable bound entangled states shared among even number of par-ties. All reduced density matrices of such states are maximally mixed.We further proceed to establish a multipartite quantum data hidingscheme on those states and explore its power and limitations.PACS number(s): 03.67.Hk, 03.65.Ud, 03.65.Ta, 03.67.-a, 89.70+c.Keywords: Local Indistinguishability, Data hiding, LOCC, Entan-glement.
Keeping a data secret by sharing it among some parties is an importanttask in quantum information processing [1, 2]. Secrecy of a data is definedin two ways. Firstly against the attack of an eavesdropper [3] and secondlyagainst the cheating attempts of the parties sharing the data where the data iskept secret from the parties themselves. A well known task in classical secretsharing is to prepare a key, which is being distributed among some parties sothat to unlock the secret, i.e., to know the key, some parties (the number ofsuch parties can be pre-assigned) have to contribute their shared parts [1].Instead of classical key, if quantum states are used to encode classical data,then we find two different directions of research. In Quantum Secret Sharingthe hidden data can be explored by some of the parties concerned, by col-lective LOCC (i.e., Local operations with classical communications) on their ∗ [email protected] † [email protected] et.al [8], to show the local indistinguishability ofthe four Bell states. However, we have considered here a general class oforthogonal mixed multipartite bound entangled states shared between evennumber of parties.The local indistinguishable character and some other properties of ouractivable bound entangled states provide us the possibility of hiding classicalbits in quantum states. We consider here the task of quantum data hiding tohide classical data in quantum states with a much more secured scenario. Inquantum data hiding, classical information is kept secret in terms of quantumstates shared among some parties situated at distant locations. The involvedparties know which quantum state is used to encode which classical bit, but2o not know the actual state they are sharing. The security in such schemesmust guarantee the requirement that the parties can not retrieve the secretby LOCC only. This imply, in a quantum data hiding scheme the hidingstates must be necessarily locally indistinguishable. Such processes shouldnecessarily require some amount (which may be pre-assigned) of quantumcommunication [9], i.e., exchange of quantum information, to retrieve thehidden information. That pre-assigned amount of quantum communicationdefines the level of security of the hiding scheme. Previous works [10, 11]suggest that the hiding states may be chosen to be separable. In case ofpure states, maintaining the primary requirement of orthogonality and localindistinguishability property, it is impossible to find suitable pair of pureorthogonal entangled or separable states [5] to hide one cbit of information.In entanglement based hiding schemes where the hiding states are takento be entangled, it is expected that the scheme may be broken by a finiteamount of prior entanglement shared between the unfaithful parties who maycheat others and try to retrieve the data. The aim of such a hiding schemeis to build a considerably high level of security with a minimum number offaithful parties, required to maintain the secrecy. By faithful parties we meanthose who are not try to recover the hidden data by exchange of quantuminformation. In any such scheme, the hiding states are expected to have ahighly symmetrical structure to construct the security bound, independentof any permutation of unfaithful parties. For that reason, only the numberof unfaithful parties is important to establish the security of the protocol.Schemes are also proposed to encode quantum data in terms of qubits intohiding states and in bipartite case, it is found that hiding two classical bits isequivalent of hiding a qubit in a similar scenario [12]. Recently, Hayden et.al. [13] gave an asymptotically secured data hiding scheme for a large amountof quantum data in multipartite setting. However, we consider here onlyhiding classical information in multipartite quantum states. Multiparty datahiding is quite an interesting as well as challenging job because of the strongsecurity requirement. Earlier, Eggeling et.al. [11] proposed a method forhiding a classical bit in multipartite separable quantum states, explicitly for N = 4. In this work, a protocol is proposed for hiding two classical bits ratherthan one cbit on activable bound entangled states in multi-qubit systems. Asa generalization of Smolin state [14], we found in any 2 N qubit systems for N ≥
2, there are always four orthogonal activable bound entangled states[15]. The states are locally indistinguishable. But, there are some limitationsin providing a hiding scheme. We investigate the possibility of hiding twobits of classical information in those four states of 2 N qubit system sharedbetween 2 N number of distant parties.Firstly, let us describe the class of activable bound entangled states of3ulti-qubit system. The four qubit states are shared among four distantparties by sharing equiprobable mixture of pairs of Bell states taken in properorder. ρ ± = { P [Φ + ] ⊗ P [Φ ± ] + P [Φ − ] ⊗ P [Φ ∓ ] + P [Ψ + ] ⊗ P [Ψ ± ]+ P [Ψ − ] ⊗ P [Ψ ∓ ] } σ ± = { P [Φ + ] ⊗ P [Ψ ± ] + P [Φ − ] ⊗ P [Ψ ∓ ] + P [Ψ + ] ⊗ P [Φ ± ]+ P [Ψ − ] ⊗ P [Φ ∓ ] } (1)where | Φ ± i ≡ | i±| i√ and | Ψ ± i ≡ | i±| i√ are the Bell states, written intheir usual basis and P [ · ] represents projectors on those states. The state ρ +4 ,known as Smolin state [14], is used to perform various quantum informationtheoretic tasks like secret key distillation, remote information concentration,etc., [16, 17]. Afterwards it is generalized to a class of activable boundentangled states in multiqubit systems [15]. In any even number of qubitsystem starting from four, there are exactly four states belonging to thisclass. A nice Bell-correlation is seen in this class between the states of twosuccessive systems, that provides the generalization scheme. If we denote the2 N qubit states as ρ ± N , σ ± N then the next four states of 2 N + 2 qubit systemare given by, ρ ± N +2 = { ρ +2 N ⊗ P [Φ ± ] + ρ − N ⊗ P [Φ ∓ ] + σ +2 N ⊗ P [Ψ ± ]+ σ − N ⊗ P [Ψ ∓ ] } σ ± N +2 = { ρ +2 N ⊗ P [Ψ ± ] + ρ − N ⊗ P [Ψ ∓ ] + σ +2 N ⊗ P [Φ ± ]+ σ − N ⊗ P [Φ ∓ ] } (2)This correlation enables one to generate the whole class of states from thefour qubit states by a recursive process. Our aim is to explore some specialfeatures of this class of states together with some practical usefulness. Permutation Symmetry:
The whole class of states are symmetric overall the parties concerned, i.e., the states remain invariant under the inter-change of any two parties. The four states ρ ± N +2 , σ ± N +2 can be expressedas ρ ± N +2 = 12 N { N X i =1 P [ | α i N +2 i ± | α i N +2 i ] } σ ± N +2 = 12 N { N X i =1 P [ | β i N +2 i ± | β i N +2 i ] } where {| α k N +2 i , | α k N +2 i , | β j N +2 i , | β j N +2 i ; k, j = 1 , , · · · , N } is the usualbasis of 2 N + 2 qubit system, divided in four equal parts of N +2 = 2 N number of states, so that | α k N +2 i , | β j N +2 i can be expressed as | α k N +2 i = | p k i ⊗ | p k i ⊗ · · · ⊗ | p k N +2 i ∀ k = 1 , , · · · , N (3)4here p ki ∈ { , } ∀ i = 1 , , · · · , N + 2 with p k = 0 and | β j N +2 i = | q j i ⊗ | q j i ⊗ · · · ⊗ | q j N +2 i ∀ j = 1 , , · · · , N (4)where q ji ∈ { , } ∀ i = 1 , , · · · , N + 2 with q j = 0, such that N +2 X i =1 p ki = 0( mod , N +2 X i =1 q ji = 1( mod α k N +2 ( β j N +2 ) is even(odd)). The states | α k N +2 i and | β j N +2 i are orthogonal to the states | α k N +2 i and | β j N +2 i respectively forall possible values of k and j . In the above form, if we permute any two partiesthen all | α k N +2 i for k = 1 , , · · · , N , are interchanged within themselves andtheir orthogonals | α k N +2 i . Similarly for all | β j N +2 i ’s for j = 1 , , · · · , N .This simple property implies the permutation symmetry of all the four statesin 2 N + 2 qubit system. In particular, the explicit form of the 4-qubit statesare, ρ ± = ( P [0000 ± P [0011 ± P [0101 ± P [0110 ± σ ± = ( P [0001 ± P [0010 ± P [0100 ± P [0111 ± Orthogonality:
From Eq.(2) it is clear that the four states of 2 N + 2qubit system are orthogonal to each other if the 2 N qubit states are so. Alsofrom Eq.(1) we observe that the four states ρ ± , σ ± of four qubit system aremutually orthogonal. Thus in a recursive way it provides orthogonality ofthe four activable bound entangled states of any even qubit systems startingfrom four. Local Indistinguishability:
The four states of 2 N qubit system, for N ≥ N ≥
2, the four states ρ ± N , σ ± N are locally distinguishable. Now,consider the state, ρ +2 N +2 = 14 { ρ +2 N ⊗ P [Φ + ] + ρ − N ⊗ P [Φ − ] + σ +2 N ⊗ P [Ψ + ] + σ − N ⊗ P [Ψ − ] } where the first 2 N parties are A , A , . . . , A N − , B and last two parties are A N , B , i.e., the state is separable by construction in A A . . . A N − B : A N B cut. Again, the state is symmetric with respect to the interchange5f any two parties, i.e., ρ +2 N +2 has the same form if the first 2 N parties are A , A , . . . , A N and last two parties are B , B . If, the four states ρ ± N , σ ± N are locally distinguishable, then by LOCC only, A , A , . . . , A N are abledistinguish between the states ρ ± N , σ ± N . The remaining state between B and B , is then any one of the Bell states correlated according as above,so that A , A , . . . , A N are able to share a Bell state among B and B ,which is impossible as initially there is no entanglement in between B and B . So, all the four states ρ ± N , σ ± N are locally indistinguishable for any N ≥
2. Our protocol also suggest that the states are even probabilisticallyindistinguishable for any N ≥
2, as it is impossible to share any entanglementby LOCC between B and B . Let us assume that the four states are locallyindistinguishable with some probability p >
0, then having shared the state ρ +2 N +2 among the 2 N +2 parties, any set of 2 N parties may able to distinguishtheir joint local system with that probability 1 > p > ρ +2 N +2 . Thus the four states of 2 N qubit system are evenprobabilistically locally indistinguishable. Maximal Ignorance:
Ignorance of any one party(i.e., by tracing outone qubit system) from any of the four states ρ ± N +2 , σ ± N +2 will results in thestate N +1 I N +1 . To establish this result let us first show that it is true forthe 4-qubit states. The first of the four qubit state is, ρ +4 = { P [Φ + ] ⊗ P [Φ + ] + P [Φ − ] ⊗ P [Φ − ] + P [Ψ + ] ⊗ P [Ψ + ]+ P [Ψ − ] ⊗ P [Ψ − ] } = { P [ | i + | i√ ] ⊗ P [Φ + ] + P [ | i−| i√ ] ⊗ P [Φ − ]+ P [ | i + | i√ ] ⊗ P [Ψ + ] + P [ | i−| i√ ] ⊗ P [Ψ − ] } = { ( P [ | i ] + P [ | i ]) ⊗ ( P [Φ + ] + P [Φ − ]) + ( P [ | i ]+ P [ | i ]) ⊗ ( P [Ψ + ] + P [Ψ − ]) + ( | ih | + | ih | ) ⊗ ( P [Φ + ] − P [Φ − ]) + ( | ih | + | ih | ) ⊗ ( P [Ψ + ] − P [Ψ − ]) } (6)Thus tracing out first qubit system of the above state we will obtain, ρ ′ = { ( P [ | i ] + P [ | i ]) ⊗ ( P [Φ + ] + P [Φ − ])+ ( P [ | i ] + P [ | i ]) ⊗ ( P [Ψ + ] + P [Ψ − ]) } = ( P [ | i ] + P [ | i ]) ⊗ ( P [Φ + ] + P [Φ − ] + P [Ψ + ] + P [Ψ − ])= I ⊗ I = I (7)Similarly all the other three four qubit states have this property. Thenext step is to prescribe a mathematical induction process to prove this6roperty for the whole class of states, taken into consideration. The processwill ensure that if the statement of the property is true for the 2 N qubitstates then so also the 2 N + 2 qubit states and thus proceeding from the 4qubit states to the 6 qubit states, then from 6 qubit to 8 qubit and so on.For this purpose, we assume that for some integer N , the four states ρ ± N , σ ± N have this property. Thus tracing out the first qubit system of ρ ± N , σ ± N ,will results in N − I N − . Then applying the relation (2) we will show that,tracing out the first qubit system of the state ρ ± N +2 will give N +1 I N +1 .Taking trace over the first qubit system of ρ ± N +2 will produce, ρ ′ N +1 = · N − I N − ⊗ ( P [Φ + ] + P [Φ − ] + P [Ψ + ] + P [Ψ − ])= N +1 I N − ⊗ I = N +1 I N +1 (8)In a similar manner it can be shown that all the four states ρ ± N +2 , σ ± N +2 havethis property, if it holds for the 2 N qubit states. Now, we already found theresult that the four qubit states have this property and assuming the validityof this property for the four states of 2 N qubit system, we find the propertyis also true for the 2 N + 2 qubit states. Thus through a recursive method weobtain, the property is true for the whole class of states. As the states aresymmetric over permutation of all parties, thus tracing out any one partyresults the same. This will also imply that the individual density matricesof each party is a maximally mixed state, i.e., I .This class of states appears to be very suitable to construct a data hidingprotocol. Instead of finding two orthogonal mixed states to hide one cbit ofinformation, here we want to use all the four orthogonal, highly symmetricmixed entangled states, to hide classical bits. Our protocol is to hide twocbit of information b = 0 , , , N + 2 number of parties separatedby distance, by sharing the four states ρ ± N +2 , σ ± N +2 , for N ≥ N +1parties as the hidden data can not retrieved perfectly, until and unless allthe parties remain separated or all of the 2 N + 2 parties are dishonest. Security against LOCC:
The class of four states of 2 N + 2 qubit sys-tem, for N ≥
1, used for sharing the data are locally indistinguishable notonly deterministically but also probabilistically (shown earlier). So consider-ing all the parties to be dishonest, they can not even probabilistically recoverthe hidden data perfectly by local operations on their subsystems and com-municating each other through some classical channel. However imperfectknowledge of hidden data may be obtained by LOCC.
Security against Global operation:
The data remains secure underthe action of any 2 N +1 number of dishonest distant parties, who are allowed7o make global operations, by joining in some labs and make collective opera-tion on their joint system. This follows from the maximal ignorance propertyof the activable bound entangled states, as ignorance of the system of thehonest party (there should be at least one such or otherwise the states areobviously globally distinguishable as being orthogonal to each other) givesthe reduced density matrix of the others to be the maximally mixed state.Here the quantum communication is allowable among a maximum numberof parties, i.e., 2 N + 1.It is interesting to note that in the above protocol we need only one honestparty, not allowed to communicate with the others through some quantumchannel. The hider may not be a part of the system. It is also not necessarythat the hider herself encrypt the bit in the quantum state and thus knowsthe hidden data. Limitations regarding Inconclusive distinguishability:
Althoughour protocol appears to be quite nice to maintain the secrecy of the hiddendata in a very stronger manner, but it has some limitations. So far we haveonly considered perfect distinguishability of the states. Precisely, it impliesthat we have to discriminate the state supplied, from the whole set (here theset of four states of 2 N + 2 qubit system). However, it may be possible todetermine whether the given state belongs to some particular subset of thewhole set of states. i.e., although it is impossible to distinguish perfectly thefour states ρ ± N +2 , σ ± N +2 , for N ≥ ρ +2 N +2 , ρ − N +2 , σ +2 N +2 and σ − N +2 orfrom other two. For example, if ρ +2 N +2 and ρ − N +2 are in a group and σ +2 N +2 , σ − N +2 are in another group, then by measuring on σ z basis in each party andchecking only the parity (even or odd number of zeros or ones), it is possibleto discriminate any state from the four ρ ± N +2 , σ ± N +2 , the group it belongs.The basic fact of this set discrimination, taken two together, is that the fourstates ρ ± N +2 , σ ± N +2 are locally Pauli connected.In conclusion we have obtained a class of highly symmetric activablebound entangled states in any even number of parties that are locally indis-tinguishable, if single copy of the states are given. We have formulated ascheme to hide two bits of classical information by sharing it among 2 N + 2number of parties, for any N ≥
1. The advantage of our protocol is that thenumber of parties can be extended in pairs up to any desired level keepingthe individual systems only with dimension two. The hidden information cannot be exactly revealed by any classical attack of the corresponding partiesand also against every quantum attack, as long as one party remains honest.However, the hiding scheme has some limitations from the viewpoint of setdistinguishability. The states are nice for practical preparation by sharing8ell mixtures among distant parties. This class of locally Pauli connected butlocal indistinguishable states with the power of activable boundness opensa new direction in the study of the relation between nonlocality and localdistinguishability.
Acknowledgement.
The authors thank the referees for their valuablesuggestions and comments. They are also grateful to G. Kar for useful dis-cussions concerning the issue of this paper. I.C. acknowledges CSIR, Indiafor providing fellowship during this work.
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