Randomizing quantum states: Constructions and applications

TL;DR

This work shows that near-perfect quantum state randomization requires far fewer secret key bits than perfect encryption, with an $l$-qubit state encryptible using $l+o(l)$ bits of key. It constructs ε-randomizing maps from a compact set of random unitaries, enabling approximate private quantum channels and informing how randomization affects classical and quantum correlations. The authors apply these maps to quantum data hiding, achieving LOCC-invisible hides of roughly $l$ qubits in $2l$ qubits, and to locking classical correlations, highlighting the volatility of accessible information. They also discuss a path to remote state preparation and outline future directions toward efficient implementations and optimality results. The results collectively reveal deep links between randomization, correlation structure, and secure quantum information processing.

Abstract

The construction of a perfectly secure private quantum channel in dimension d is known to require 2 log d shared random key bits between the sender and receiver. We show that if only near-perfect security is required, the size of the key can be reduced by a factor of two. More specifically, we show that there exists a set of roughly d log d unitary operators whose average effect on every input pure state is almost perfectly randomizing, as compared to the d^2 operators required to randomize perfectly. Aside from the private quantum channel, variations of this construction can be applied to many other tasks in quantum information processing. We show, for instance, that it can be used to construct LOCC data hiding schemes for bits and qubits that are much more efficient than any others known, allowing roughly log d qubits to be hidden in 2 log d qubits. The method can also be used to exhibit the existence of quantum states with locked classical correlations, an arbitrarily large amplification of the correlation being accomplished by sending a negligibly small classical key. Our construction also provides the basic building block for a method of remotely preparing arbitrary d-dimensional pure quantum states using approximately log d bits of communication and log d ebits of entanglement.

Figures (2)

• Figure 1: A private quantum channel built on the randomization map $R$. Alice and Bob share knowledge of the secret key $j$, using it to encrypt and decrypt the state. Because an eavesdropper does not have access to $j$, her view is $R(\varphi) \approx {\mathbb I} / d$. If $\varphi$ is a $d$-dimensional quantum state, then the key length need only be $\log d + o(\log d)$.
• Figure 2: Quantum data hiding. (a) depicts the encoding procedure. A random $U_j$ is applied to the state $|\varphi\rangle$ drawn from subspace $S$. The output, $R(\varphi)$, is almost indistinguishable from the maximally mixed state using LOCC alone. (b) The different subspaces $\{U_j S\}$ have very small overlaps, however, so a collective operation on ${\cal H}_A \otimes {\cal H}_B$ can be used to distinguish them without causing much distortion to the encoded states $U_j |\varphi\rangle$.

Theorems & Definitions (14)

• Definition 2.1
• Theorem 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 3.1
• Lemma 3.2
• Theorem 3.3
• Definition 4.1: Adapted from Ref. DHT02
• Theorem 4.2
• Theorem 5.1