Quantum Data Hiding

(work in progress for academic purposed, please do not edit now without explicitly noting it as a comment)

Quantum data hiding refers to multiparty distributed encoding schemes of data, either classical or quantum, into quantum states (QSS), where unauthorized subsets of the party cannot reconstruct the hidden state using only LOCC operations. These differ from quantum secret sharing in that in QSS unauthorized subsets can reconstruct the state using LOCC operators. Like classical data hiding schemes, quantum data hiding relies on sharing of data between participants so as to limit limit information to unauthorized subsets of participants while allowing information to be revealed to authorized subsets. Security cannot in general be parameterized one-dimensionally, but rather by the revealing and hiding strength over authorized and unauthorized sets respectively. A quantum hiding data hiding scheme is called [math]\displaystyle{ \epsilon }[/math] secure if for any unauthorized set and analyzing operator [math]\displaystyle{ A }[/math],

[math]\displaystyle{ Tr(\rho_i A) - Tr(\rho_j A) \leq \epsilon }[/math]

for all [math]\displaystyle{ i,j }[/math] and [math]\displaystyle{ \delta }[/math] revealing or correct if for any authorized set and at least one analyzing operator [math]\displaystyle{ A }[/math],

[math]\displaystyle{ Tr(\rho_i A) - i \leq \delta }[/math]

for all [math]\displaystyle{ i }[/math].

Two Party Hiding

Hiding in Bell States

Quantum data hiding schemes can be implemented utilizing quantum entanglement. In the case of two party entanglement, the Bell states are an appropriate choice. Entangled states exhibit nonlocal statistics, as is apparent in the violation of Bell inequalities, which limit the amount of information available to those making local quantum measurements. Bell states are highly entangled quantum state of two qubits. There are four bell states:

[math]\displaystyle{ |\Phi^+\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B + |1\rangle_A \otimes |1\rangle_B) }[/math]

[math]\displaystyle{ |\Phi^-\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B - |1\rangle_A \otimes |1\rangle_B) }[/math]

[math]\displaystyle{ |\Psi^+\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |1\rangle_B + |1\rangle_A \otimes |0\rangle_B) }[/math]

[math]\displaystyle{ |\Psi^-\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B) }[/math]

A simple hiding scheme utilizing these states would involve the permutation symmetry of these states. Define a permutation operator

[math]\displaystyle{ V_{1 2} = \displaystyle\sum_{i,j}{|i j \rangle \langle j i |}. }[/math]

This operator has [math]\displaystyle{ +1 }[/math] eigenvalue with all the Bell states except for [math]\displaystyle{ |\Psi^-\rangle }[/math], which has eigenvalue [math]\displaystyle{ -1 }[/math]. The parity of a Bell state, therefore, can be hidden between two parties without quantum communication because the permutation symmetry cannot be directly measured without allowing state exchange. A hider only need to distribute a triplet among the parties if she wants to hide "0" and the singlet for "1". This scheme is not sufficient however, because the parties can use LOCCs to measure the state with some certainty.

Proof of Security

This can easily be remedied by increasing the number of states that Alice and Bob share. Suppose the "hider" wishes to to hide a classical bit [math]\displaystyle{ b }[/math]. The hider then distributes [math]\displaystyle{ n }[/math] random Bell states to Alice and Bob. The Distribution of Bell states are only random up to the number of singlets distributed: the hider makes sure to hide an odd or even number of singlets for [math]\displaystyle{ b = 1,0 }[/math] respectively. Alice and Bob need only open up a quantum channel between then and measure the parity of the state to get the secret.

Because of the locality requirement of measurements, it can be shown that the parties are restricted to a set of quantum measurement s satisfying a positive partial transpose. It can be shown that their measurements using only LOCCs is bounded by:

[math]\displaystyle{ - \delta \le p_{1 | 1} + p_{0 | 0} + 1 \le \delta }[/math]

where [math]\displaystyle{ p_{i | i } }[/math] is the probability of measuring [math]\displaystyle{ i }[/math] given [math]\displaystyle{ i }[/math] and [math]\displaystyle{ \delta = 1 / 2^{n - 1} }[/math]. So as [math]\displaystyle{ n \to \infty }[/math] the value of the bit can be hidden with certainty.

It should be noted that given a certain amount of prior entanglement, the parties can determine information about the hidden bit. However, this information is on the order of the amount of information needed to establish quantum teleportation.

Construction in the Labratory

This scheme is realizable in the laborator, Bell states being constructible via optical downconvertion. Photons can then be sent through optical fibers to Alice and Bob. To reveal the secret, Alice can send her photons to Bob via a quantum channel and Bob can measure the number of singlets. These measurements can be preformed in modern labs, as complete bell measurements are unnecessary and technically unfeasible.

Hiding in more general states

It can be shown that hiding schemes are realizable without quantum entanglement. The basis of this is shown by () where it can be shown that LOCC measurements cannot distinguish between three non-entangled non-orthogonal states as well as with quantum communication. In this case, the non-entangled Werner states can be used, which are mixtures of Bell states:

[math]\displaystyle{ \rho_0^{(n)} = I+ 2^n H }[/math]

[math]\displaystyle{ \rho_1^{(n)} = I- 2^n H }[/math]

where

[math]\displaystyle{ H = (1 \otimes T)^{\otimes n} \left[ | \Psi^+ \rangle \langle \Psi^+|^{\otimes n} \right] }[/math]

Hiding Multiple Bits

It is easy to see that this scheme extends to hiding multiple bits. The preparer only need to apply the scheme to each bit to be hidden and designate each bit to different block of Bell or Werner states.

Multi-Party Data Hiding

The goal here is not necessarily to hide against all but the total number of participants, but to hide against certain subsets. In general a multi-party scheme can be tailored hide against certain subsets. For example, in a 4 party scheme with participants A, B, C, and D we can scheme that is hiding for partitions {{A},{B},{C},{D}} and {{A,B},{C,D}} but revealing for {{A,B,C},{D}} and Template:A,B,C,D. In general, these schemes cannot be parameterized so they are not threshold schemes.

CAT States

Bells states generalize to multiple d-bits (d dimensional qubits) as CATS states, which are maximally entangled. The bits can be hidden as the eigenstates of the X operator.

Werner States

Werner states are constructible from CAT states and are generalizable to n sites. If we pick Werner states which are eigenstates of permutation over subsets. For example, for a four party scheme we define permutation operations,

[math]\displaystyle{ V_\pi = \displaystyle\sum_{i j k l}{\pi |i j k l\rangle \langle i j k l |}. }[/math]

where [math]\displaystyle{ \pi }[/math] is a permutation over a subset. For insance,

[math]\displaystyle{ V_{123} = \displaystyle\sum_{i j k l}{|j k i l\rangle \langle i j k l |}. }[/math]

We then hide the states in eigenvectors of these operators:

[math]\displaystyle{ \rho_i = (r_2, r_{2 2}, r_3, r_4) }[/math]

and

[math]\displaystyle{ r_2 = tr \left(\rho_i V_{12} \right) }[/math] [math]\displaystyle{ r_{2 2} = tr \left(\rho_i V_{(12)(34)} \right) }[/math] [math]\displaystyle{ r_3 = tr \left(\rho_i V_{123} \right) }[/math] [math]\displaystyle{ r_4 = tr \left(\rho_i V_{1234} \right) }[/math]

As in the bipartite case, the parties are limited to measurements which have positive partial transpose over the allowed sets. The basis used to hide the states are non-orthogonal, however it can be shown that [math]\displaystyle{ Tr(V_{\pi} V_{\pi'}^{*} = \delta(\pi,\pi') + \mathcal{O}\left( \frac{1}{d} \right) }[/math]

So diagonal terms go to zero as the dimension is large.