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

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

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

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

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

can be used. One hiding scheme utilizing these states harness the permutation symmetry of these states. Define a permutation operator on two qubits as

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

This operator has +1 eigenvalue with all the Bell states except for [math]\displaystyle{ |\Psi^-\rangle }[/math], with which it has eigenvalue -1. The parity of a Bell state, therefore, hides one bit between two parties with with some degree of security 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 lacks perfect security however, because the parties can use LOCCs to measure the state with some certainty.

Security of a Generalized Scheme

Suppose the "hider" wishes to to hide a classical bit [math]\displaystyle{ b }[/math] in a two party quantum state. The hider can pick at random [math]\displaystyle{ n }[/math] Bell states to distribute to Alice and Bob, except that the hider ensures that the number of singlets is odd or even for hiding the classical bit [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 reveal the hidden bit.

Because of the locality requirement of measurements, it can be shown that the parties are restricted to a set of quantum measurements satisfying a positive partial transpose or separability. 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 laboratory, Bell states being constructible via optical down-conversion. 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 Werner 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, 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]

where [math]\displaystyle{ T }[/math] is the partial transpose operator over Bob's qubits. [math]\displaystyle{ \rho_0^{(n)} }[/math] is unentangled, while [math]\displaystyle{ \rho_1^{(n)} }[/math] is entangled. These states have permutation symmetry over two n-bit so hiding and revealing can be obtained using that symmetry. The quantum resources needed to prepare these states are on the order of one ebit which is much less than the bell state scheme ([math]\displaystyle{ n^2 }[/math] ebits).

Hiding Multiple Bits

It is easy to see that these schemes extend 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. However, it cannot be guaranteed that information cannot be obtained from LOCC measurements across the bit partitions unless [math]\displaystyle{ n }[/math] scales appropriately.

Multi-Party Data Hiding

Multi-party schemes are useful in allowing hiding and revealing to more complicated partitions. For example, a president of a bank may want a hiding scheme between four managers such that three of them can reveal a bank code by two of them cannot. In this case, with participants A, B, C, and D the hiding partitions are {{A},{B},{C},{D}}, {{A},{B},{C,D}}, and {{A,B},{C,D}} and the revealing partitions are {{A,B,C},{D}} and its permutations and { {A,B,C,D}}. In general, these schemes cannot be parameterized one-dimensionally as in a scheme which {{A,B},{C,D}} is authorized and {{A,B,C},{D}} is unauthorized.

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.