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Quantum Bit Commitment and the Reality of the Quantum State

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Quantum bit commitment is insecure in the standard non-relativistic quantum cryptographic framework, essentially because Alice can exploit quantum steering to defer making her commitment. Two assumptions in this framework are that: (a) Alice knows the ensembles of evidence E corresponding to either commitment; and (b) system E is quantum rather than classical. Here, we show how relaxing assumption (a) or (b) can render her malicious steering operation indeterminable or inexistent, respectively. Finally, we present a secure protocol that relaxes both assumptions in a quantum teleportation setting. Without appeal to an ontological framework, we argue that the protocol’s security entails the reality of the quantum state, provided retrocausality is excluded.

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The author thanks DST-SERB, Govt. of India, for financial support provided through the project EMR/2016/004019.

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Correspondence to R. Srikanth.

Appendix A: State of the Evidence

Appendix A: State of the Evidence

We conservatively assume that all states \(\mathinner {|{\phi ^{(a)}_j}\rangle }\) for a given a are identical, say \(\mathinner {|{0}\rangle }\). Under the stated assumptions, Alice’s evidence is in the state

$$\begin{aligned} \rho _B^a&= {\mathcal {C}}_W \left[ \left( \mathinner {|{0}\rangle }\mathinner {\langle {0}|}\right) ^{\otimes n} \otimes \left[ \frac{{\mathbb {I}}}{2}\right] ^{\otimes Q} \right] ,\nonumber \\&= \left( 2^{Q+n} - \sum _{j=1}^{n} {Q+n \atopwithdelims ()Q+j} \right) ^{-1} {\mathbb {I}}^*\nonumber \\&= \left( 2^{Q+n} - \sum _{j=0}^{n-1} {Q+n \atopwithdelims ()j} \right) ^{-1} {\mathbb {I}}^*\nonumber \\ \end{aligned}$$

where \({\mathbb {I}}^*\) is the density matrix in the Hilbert space \({\mathcal {H}}_2^{\otimes (Q+n)}\) of \(2^{Q+n}\) qubits, which is diagonal and equal-weighted in the computational basis, with precisely the components with Hamming weight greater than Q vanishing.

For a fixed integer t, and integer \(T \rightarrow \infty \), the truncated binomial series satisfies the bound [42]:

$$\begin{aligned} \lim _{T\rightarrow \infty } {T \atopwithdelims ()t}^{-1} \sum _{j=0}^t {T \atopwithdelims ()j}&= {{T \atopwithdelims ()t} + {T \atopwithdelims ()t-1} + {T \atopwithdelims ()t-2}+\dots \over {T \atopwithdelims ()t}} \nonumber \\&= {1 + {t \over T-t+1} + {t(t-1) \over (T-t+1)(T-t+2)} + \cdots } \nonumber \\&\le {1 + {t \over T-t+1} + \left( {t \over T-t+1} \right) ^2 + \cdots }\nonumber \\&= \frac{T-t+1}{T-2t+1}. \end{aligned}$$

Setting \(T \equiv n+Q\) and \(t \equiv n-1\) here, one finds

$$\begin{aligned} \sum _{j=0}^{n-1} {n+Q \atopwithdelims ()j} \le {n+Q \atopwithdelims ()n-1}\frac{Q+2}{Q-n+3}. \end{aligned}$$

Substituting this in Eq. (A1), we find that, for large \(Q \gg n\), the number of non-vanishing entries in \({\mathbb {I}}^*\) is bounded below by \(\upsilon (Q,n) \equiv 2^{Q+n}-{n+Q \atopwithdelims ()n-1}\frac{Q+2}{Q-n+3} \approx 2^{Q+n}-{Q+n \atopwithdelims ()n-1} \approx 2^{Q+n} - 2^{(Q+n)H(n/Q)} = 2^{Q+n}(1 - 2^{-(Q+n)[1-H(n/Q)]}\), where the Stirling approximation \({N \atopwithdelims ()Np} \approx NH(p)\), has been used.

The fidelity between states \(\rho \) and \(\sigma \) is given by Tr\((\sqrt{\sqrt{\sigma }\rho \sqrt{\sigma }}\). Setting \(\sigma \equiv 2^{-(Q+n)}{\mathbb {I}}\) and \(\rho \equiv \rho ^a_B\) in Eq. (A1) in the above approximation, we have fidelity \(F(Q,n) = 2^{-(Q+n)/2}\mathrm{Tr}(\sqrt{\rho ^a_B}) \gtrsim 2^{-(Q+n)/2}\sqrt{\upsilon (Q,n)}\), from which, one obtains Eq. (5).

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Srikanth, R. Quantum Bit Commitment and the Reality of the Quantum State. Found Phys 48, 92–109 (2018).

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